# Viscosity

Last updated
Viscosity
A simulation of liquids with different viscosities. The liquid on the right has higher viscosity than the liquid on the left.
Common symbols
,
Derivations from
other quantities
μ = G · t

The viscosity of a fluid is a measure of its resistance to gradual deformation by shear stress or tensile stress. [1] For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. [2]

In physics, a fluid is a substance that continually deforms (flows) under an applied shear stress, or external force. Fluids are a phase of matter and include liquids, gases and plasmas. They are substances with zero shear modulus, or, in simpler terms, substances which cannot resist any shear force applied to them.

In fluid dynamics, drag is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. This can exist between two fluid layers or a fluid and a solid surface. Unlike other resistive forces, such as dry friction, which are nearly independent of velocity, drag forces depend on velocity. Drag force is proportional to the velocity for a laminar flow and the squared velocity for a turbulent flow. Even though the ultimate cause of a drag is viscous friction, the turbulent drag is independent of viscosity.

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

## Contents

Viscosity can be conceptualized as quantifying the frictional force that arises between two adjacent layers of fluid that are in relative motion. For instance, when a fluid is forced through a tube, the fluid flows more quickly near the tube's axis and more slowly 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.

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:

Pressure is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure is the pressure relative to the ambient pressure.

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; [3] [4] 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.

In physics, cryogenics is the production and behaviour of materials at very low temperatures. A person who studies elements that have been subjected to extremely cold temperatures is called a cryogenicist.

Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without loss of kinetic energy. When stirred, a superfluid forms cellular vortices that continue to rotate indefinitely. Superfluidity occurs in two isotopes of helium when they are liquified by cooling to cryogenic temperatures. It is also a property of various other exotic states of matter theorized to exist in astrophysics, high-energy physics, and theories of quantum gravity. The phenomenon is related to Bose–Einstein condensation, but neither is a specific type of the other: not all Bose-Einstein condensates can be regarded as superfluids, and not all superfluids are Bose–Einstein condensates. The theory of superfluidity was developed by Lev Landau.

The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time. The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.

## Etymology

The word "viscosity" is derived from the Latin "viscum", meaning mistletoe and also a viscous glue made from mistletoe berries. [5]

Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, and ultimately from the Phoenician alphabet.

Mistletoe is the English common name for most obligate hemiparasitic plants in the order Santalales. They are attached to their host tree or shrub by a structure called the haustorium, through which they extract water and nutrients from the host plant. Their parasitic lifestyle have led to some dramatic changes in their metabolism.

## Definition

### Simple definition

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.

The interdisciplinary field of materials science, also commonly termed materials science and engineering is the design and discovery of new materials, particularly solids. The intellectual origins of materials science stem from the Enlightenment, when researchers began to use analytical thinking from chemistry, physics, and engineering to understand ancient, phenomenological observations in metallurgy and mineralogy. Materials science still incorporates elements of physics, chemistry, and engineering. As such, the field was long considered by academic institutions as a sub-field of these related fields. Beginning in the 1940s, materials science began to be more widely recognized as a specific and distinct field of science and engineering, and major technical universities around the world created dedicated schools of the study, within either the Science or Engineering schools, hence the naming.

Engineering is the application of knowledge in the form of science, mathematics, and empirical evidence, to the innovation, design, construction, operation and maintenance of structures, machines, materials, devices, systems, processes, and organizations. The discipline of engineering encompasses a broad range of more specialized fields of engineering, each with a more specific emphasis on particular areas of applied mathematics, applied science, and types of application. See glossary of engineering.

In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material. For example, when a solid vertical bar is supporting an overhead weight, each particle in the bar pushes on the particles immediately below it. When a liquid is in a closed container under pressure, each particle gets pushed against by all the surrounding particles. The container walls and the pressure-inducing surface push against them in (Newtonian) reaction. These macroscopic forces are actually the net result of a very large number of intermolecular forces and collisions between the particles in those molecules. Stress is frequently represented by a lowercase Greek letter sigma (σ).

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 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 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.

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. [6] 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 viscosity of the fluid, with units of ${\displaystyle {\text{Pa}}\cdot {\text{s}}}$ (pascal-second). 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 viscosity is common among mechanical and chemical engineers, as well as physicists. [7] [8] [9] However, the Greek letter eta (${\displaystyle \eta }$) is also used by chemists, physicists, and the IUPAC. [10] The viscosity ${\displaystyle \mu }$ is sometimes also referred to as 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. [11]

### General definition

In very general terms, the viscous stresses in a fluid are defined as those resulting from the relative velocity of different fluid particles. As such, the viscous stresses must depend on spatial gradients of the flow velocity. If the velocity gradients are small, then to a first approximation the viscous stresses depend only on the first derivatives of the velocity. [12] (For Newtonian fluids, this is also a linear dependence.) In Cartesian coordinates, the general relationship can then be written as

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

where ${\displaystyle \mu _{ijkl}}$ is a viscosity tensor that maps the strain rate tensor ${\displaystyle \partial v_{k}/\partial r_{l}}$ onto the viscous stress tensor ${\displaystyle \tau _{ij}}$. [13] Since the indices in this expression can vary from 1 to 3, there are 81 "viscosity coefficients" ${\displaystyle \mu _{ijkl}}$ 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 ${\displaystyle \mu }$ and the bulk viscosity ${\displaystyle \kappa }$:

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

where ${\displaystyle \mathbf {\delta } }$ is the unit tensor, and the dagger ${\displaystyle \dagger }$ denotes the transpose. [14] [15] This equation can be thought of as a generalized form of Newton's law of viscosity.

The bulk 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. [14] 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 and independent means.

### Dynamic and kinematic viscosity

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 units ${\displaystyle \mathrm {(length)^{2}/time} }$:

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

Consistent with this nomenclature, the viscosity ${\displaystyle \mu }$ is frequently called the dynamic viscosity or absolute viscosity, and has units force × time/area.

## Momentum transport

Transport theory provides an alternate 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. [16] To see this, note that in Newton's law of viscosity, ${\displaystyle \tau =\mu (\partial u/\partial y)}$, the shear stress ${\displaystyle \tau }$ has units equivalent to a momentum flux, i.e. momentum per unit time per unit area. Thus, ${\displaystyle \tau }$ can be interpreted as specifying the flow of momentum in the ${\displaystyle y}$ 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:

{\displaystyle {\begin{aligned}\mathbf {J} &=-D{\frac {\partial \rho }{\partial x}}\qquad \;\;\;\,{\text{(Fick's law of diffusion)}}\\\mathbf {q} &=-k_{t}{\frac {\partial T}{\partial x}}\qquad \;\;\,{\text{(Fourier's law of heat conduction)}}\\\tau &=\mu {\frac {\partial u}{\partial y}}\qquad \qquad {\text{(Newton's law of viscosity)}}\\\end{aligned}}}

where ${\displaystyle \rho }$ is the density, ${\displaystyle \mathbf {J} }$ and ${\displaystyle \mathbf {q} }$ are the mass and heat fluxes, and ${\displaystyle D}$ and ${\displaystyle k_{t}}$ are the mass diffusivity and thermal conductivity. [17]

The fact that mass, momentum, and energy (heat) transport are among the most relevant processes in continuum mechanics is not a coincidence: these are among the few physical quantities that are conserved at the microscopic level in interparticle collisions. Thus, rather than being dictated by the fast and complex microscopic interaction timescale, their dynamics occurs on macroscopic timescales, as described by the various equations of transport theory and hydrodynamics.

## Newtonian and non-Newtonian fluids

Newton's law of viscosity is not a fundamental law of nature, but rather a constitutive equation (like Hooke's law, Fick's law, and Ohm's law) which serves to define the viscosity ${\displaystyle \mu }$. Its form is motivated by experiments which show that for a wide range of fluids, ${\displaystyle \mu }$ 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.

The Trouton ratio or Trouton's ratio is the ratio of extensional viscosity to shear viscosity. [18] For a Newtonian fluid, the Trouton ratio is 3. [19]

Shear-thinning liquids are very commonly, but misleadingly, described as thixotropic. [20]

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.

## In solids

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. [21] Such materials are therefore best described as possessing both elasticity (reaction to deformation) and viscosity (reaction to rate of deformation); that is, being viscoelastic.

Indeed, some authors have claimed that amorphous solids, such as glass and many polymers, are actually liquids with a very high viscosity (greater than 1012 Pa·s). [22] 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, [23] 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. [24]

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. [25]

## Measurement

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.

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. [26]

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.

## Units

The SI unit of dynamic viscosity is Pa·s or kg·m−1·s−1. The cgs unit is called the poise [27] (P), named after Jean Léonard Marie Poiseuille. It is commonly expressed, particularly in ASTM standards, as centipoise (cP) since the latter is equal to the SI multiple millipascal seconds (mPa·s).

The SI unit of kinematic viscosity is m2/s, whereas the cgs unit for kinematic viscosity is the stokes (St), named after Sir George Gabriel Stokes. [28] It is sometimes expressed in terms of centistokes (cSt). In U.S. usage, stoke is sometimes used as the singular form.

The reciprocal of viscosity is fluidity, usually symbolized by ${\displaystyle \phi =1/\mu }$ or ${\displaystyle F=1/\mu }$, 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.

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.

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). [29] 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. [30]

## Molecular origins

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 1985. [31] 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. [32]

Momentum transport in gases is generally mediated by discrete molecular collisions, and in liquids by attractive forces which bind molecules close together. [16] Because of this, the dynamic viscosities of liquids are typically much larger than those of gases.

### Gases

Viscosity in gases arises principally from the molecular diffusion that transports momentum between layers of flow. An elementary calculation for a dilute gas at temperature ${\displaystyle T}$ 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. [16] [33]

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. [34]

#### Chapman–Enskog theory

A technique developed by Sydney Chapman and David Enskog in the early 1900s allows a more refined calculation of ${\displaystyle \mu }$. [32] It is based on the Boltzmann equation, which provides a systematic statistical description of a dilute gas in terms of intermolecular interactions. [35] 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. [16] 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, [36] 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 ${\displaystyle T_{0}}$. 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 and Cowling [32] 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. [37] [38]

### Liquids

In contrast with gases, there is no simple yet accurate picture for the molecular origins of viscosity in liquids.

At the simplest level of description, the relative motion of adjacent layers in a liquid is opposed primarily by attractive molecular forces acting across the layer boundary. In this picture, one (correctly) expects viscosity to decrease with increasing temperature. This is because increasing temperature increases the random thermal motion of the molecules, which makes it easier for them to overcome their attractive interactions. [39]

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. [40] 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

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

(2)

where ${\displaystyle A}$ and ${\displaystyle B}$ are constants fit from data. [40] [41] One 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. [40] More fundamentally, the physical assumptions underlying equation ( 1 ) have been extensively criticized. [42] 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. [43] [44]

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. [45] 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. [46]

### Mixtures, blends, and suspensions

#### Gaseous mixtures

The same molecular-kinetic picture of a single component gas can also be applied to a gaseous mixture. For instance, in the Chapman-Enskog approach the viscosity ${\displaystyle \mu _{\text{mix}}}$ 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. [47] 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.

#### Blends of liquids

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

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

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. [48]

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. [48]

#### Suspensions

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. [49] 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). [50] 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. [51]

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), [51] or in suspensions of elongated or flexible particles. [49]

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. [52]

### Amorphous materials

In the high and low temperature limits, viscous flow in amorphous materials (e.g. in glasses and melts) [54] [55] [56] has the Arrhenius form:

${\displaystyle \mu =Ae^{\frac {Q}{RT}},}$

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 QH at low temperatures (in the glassy state) to a low value QL at high temperatures (in the liquid state).

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. [56]

### Eddy viscosity

In the study of turbulence in fluids, a common practical strategy is to ignore the small-scale vortices (or eddies) in the motion and to calculate a large-scale motion with an effective viscosity, called the "eddy viscosity", which characterizes the transport and dissipation of energy in 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]

## Selected substances

Observed values of viscosity vary over several orders of magnitude, even for common substances. For instance, a 70% sucrose (sugar) solution has a viscosity over 400 times that of water, and 26000 times that of air. [62] More dramatically, pitch has been estimated to have a viscosity 230 billion times that of water. [61]

### Water

The viscosity of water is about 0.89 mPa·s at room temperature (25 °C). As a function of temperature, the viscosity can be estimated using the semi-empirical relation:

${\displaystyle \mu =A\times 10^{B/(T-C)},}$

where A = 2.414×10−5 Pa·s, B = 247.8 K, and C = 140 K.[ citation needed ]

Experimentally determined values of the viscosity at various temperatures are given below.

Viscosity of water
at various temperatures [62]
Temperature (°C)Viscosity (mPa·s)
101.3059
201.0016
300.79722
500.54652
700.40355
900.31417

### Air

Under standard atmospheric conditions (25 °C and pressure of 1 bar), the 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.

### Other common substances

SubstanceViscosity (mPa·s)Temperature (°C)
Benzene 0.60425
Water [62] 1.001620
Mercury 1.52625
Whole milk [63] 2.1220
Olive oil [63] 56.226
Honey [64] ${\displaystyle \approx }$ 2000-1000020
Ketchup [lower-alpha 1] [65] ${\displaystyle \approx }$ 5000-2000025
Peanut butter [lower-alpha 1] [66] ${\displaystyle \approx }$ 104-106
Pitch [61] 2.3×101110-30 (variable)
1. These materials are highly non-Newtonian.

## Related Research Articles

Rheology is the study of the flow of matter, primarily in a liquid state, but also as "soft solids" or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an applied force. It is a branch of physics which deals with the deformation and flow of materials, both solids and liquids.

A Newtonian fluid is a fluid in which the viscous stresses arising from its flow, at every point, are linearly proportional 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.

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 the results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of earth sciences.

A cross fluid is a type of generalized Newtonian fluid whose viscosity depends upon shear rate according to the following equation:

Carreau fluid is a type of generalized Newtonian fluid where viscosity, , depends upon the shear rate, , by the following equation:

The temperature dependence of liquid viscosity is the phenomenon by which liquid viscosity tends to decrease as its temperature increases. This can be observed, for example, by watching how cooking oil appears to move more fluidly upon a frying pan after being heated by a stove.

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, astrophysics, and biology.

The Bagnold number (Ba) is the ratio of grain collision stresses to viscous fluid stresses in a granular flow with interstitial Newtonian fluid, first identified by Ralph Alger Bagnold.

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 becomes important only for such effects where fluid compressibility is essential. Volume viscosity is mainly related to the vibrational energy of the molecules. It is zero for monatomic gases at low density, but can be large for fluids with larger molecules. The volume viscosity is important in describing sound attenuation, and the absorption of sound energy into the fluid depends on the sound frequency i.e. the rate of fluid compression and expansion. Volume viscosity is also important in describing the fluid dynamics of liquids containing gas bubbles. For an incompressible liquid the volume viscosity is superfluous, and does not appear in the equation of motion.

In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. It can be successfully applied to air flow in lung alveoli, or the flow through a drinking straw or through a hypodermic needle. It was experimentally derived independently by Jean Léonard Marie Poiseuille in 1838 and Gotthilf Heinrich Ludwig Hagen, and published by Poiseuille in 1840–41 and 1846.

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.

In fluid dynamics, hydrodynamic stability is the field which analyses the stability and the onset of instability of fluid flows. The study of hydrodynamic stability aims to find out if a given flow is stable or unstable, and if so, how these instabilities will cause the development of turbulence. The foundations of hydrodynamic stability, both theoretical and experimental, were laid most notably by Helmholtz, Kelvin, Rayleigh and Reynolds during the nineteenth century. These foundations have given many useful tools to study hydrodynamic stability. These include Reynolds number, the Euler equations, and the Navier–Stokes equations. When studying flow stability it is useful to understand more simplistic systems, e.g. incompressible and inviscid fluids which can then be developed further onto more complex flows. Since the 1980s, more computational methods are being used to model and analyse the more complex flows.

The Reynolds number is an important dimensionless quantity in fluid mechanics used to help 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 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. The Reynolds number has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing. It is used to predict the transition from laminar to turbulent flow, and is used in the scaling of similar but different-sized flow situations, such as between an aircraft model in a wind tunnel and the full size version. The predictions of the onset of turbulence and the ability to calculate scaling effects can be used to help predict fluid behaviour on a larger scale, such as in local or global air or water movement and thereby the associated meteorological and climatological effects.

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.

## References

1. "viscosity". Merriam-Webster Dictionary.
2. Symon, Keith (1971). Mechanics (3rd ed.). Addison-Wesley. ISBN   978-0-201-07392-8.
3. Balescu, Radu (1975), Equilibrium and Nonequilibrium Statistical Mechanics, John Wiley & Sons, pp. 428–429, ISBN   978-0-471-04600-4
4. Landau, L.D.; Lifshitz, E.M. (1987), Fluid Mechanics (2nd ed.), Pergamon Press, ISBN   978-0-08-033933-7
5. "viscous". Etymonline.com. Retrieved 2010-09-14.
6. Jan Mewis; Norman J. Wagner (2012). Colloidal Suspension Rheology. Cambridge University Press. p. 19. ISBN   978-0-521-51599-3.
7. Streeter, Victor Lyle; Wylie, E. Benjamin; Bedford, Keith W. (1998). Fluid Mechanics. McGraw-Hill. ISBN   978-0-07-062537-2.
8. Holman, J. P. (2002). Heat Transfer. McGraw-Hill. ISBN   978-0-07-122621-9.
9. Incropera, Frank P.; DeWitt, David P. (2007). Fundamentals of Heat and Mass Transfer. Wiley. ISBN   978-0-471-45728-2.
10. Nič, Miloslav; Jirát, Jiří; Košata, Bedřich; Jenkins, Aubrey, eds. (1997). "dynamic viscosity, η". IUPAC Compendium of Chemical Terminology. Oxford: Blackwell Scientific Publications. doi:10.1351/goldbook. ISBN   978-0-9678550-9-7.
11. Bird, R. Byron; Stewart, Warren E.; Lightfoot, Edwin N. (2007), Transport Phenomena (2nd ed.), John Wiley & Sons, Inc., p. 19, ISBN   978-0-470-11539-8
12. Landau, L.D.; Lifshitz, E.M. (1987), Fluid Mechanics (2nd ed.), Pergamon Press, pp. 44–45, ISBN   978-0-08-033933-7
13. Bird, Steward, & Lightfoot, p. 18 (Note that this source uses a alternate sign convention, which has been reversed here.)
14. Bird, Steward, & Lightfoot, p. 19
15. Landau & Lifshitz p. 45
16. Bird, R. Byron; Stewart, Warren E.; Lightfoot, Edwin N. (2007), Transport Phenomena (2nd ed.), John Wiley & Sons, Inc., ISBN   978-0-470-11539-8
17. Daniel V. Schroeder (1999). An Introduction to Thermal Physics. Addison Wesley. ISBN   978-0-201-38027-9.
18. "Archived copy". Archived from the original on 2008-12-02. Retrieved 2009-10-12.CS1 maint: Archived copy as title (link)
19. Jan Mewis; Norman J. Wagner (2012). Colloidal Suspension Rheology. Cambridge University Press. pp. 228–230. ISBN   978-0-521-51599-3.
20. Kumagai, Naoichi; Sasajima, Sadao; Ito, Hidebumi (15 February 1978). "Long-term Creep of Rocks: Results with Large Specimens Obtained in about 20 Years and Those with Small Specimens in about 3 Years". Journal of the Society of Materials Science (Japan). 27 (293): 157–161. Retrieved 2008-06-16.
21. Elert, Glenn. "Viscosity". The Physics Hypertextbook.
22. Gibbs, Philip. "Is Glass a Liquid or a Solid?" . Retrieved 2007-07-31.
23. Plumb, Robert C. (1989). "Antique windowpanes and the flow of supercooled liquids". Journal of Chemical Education. 66 (12): 994. Bibcode:1989JChEd..66..994P. doi:10.1021/ed066p994.
24. Scherer, George W.; Pardenek, Sandra A.; Swiatek, Rose M. (1988). "Viscoelasticity in silica gel". Journal of Non-Crystalline Solids. 107 (1): 14. Bibcode:1988JNCS..107...14S. doi:10.1016/0022-3093(88)90086-5.
25. "Viscosity" (PDF). BYK-Gardner.
26. "Poise". IUPAC definition of the Poise. 2009. doi:10.1351/goldbook.P04705. ISBN   978-0-9678550-9-7 . Retrieved 2010-09-14.
27. Gyllenbok, Jan (2018). "Encyclopaedia of Historical Metrology, Weights, and Measures". Encyclopaedia of Historical Metrology, Weights, and Measures, Volume 1. Birkhäuser. p. 213. ISBN   9783319575988.
28. ASTM D 2161 (2005) "Standard Practice for Conversion of Kinematic Viscosity to Saybolt Universal Viscosity or to Saybolt Furol Viscosity", p. 1
29. "Quantities and Units of Viscosity". Uniteasy.com. Retrieved 2010-09-14.
30. Evans, Denis J.; Morriss, Gary P. (October 15, 1988). "Transient-time-correlation functions and the rheology of fluids". Physical Review A. 38 (8): 4142–4148. Bibcode:1988PhRvA..38.4142E. doi:10.1103/PhysRevA.38.4142. PMID   9900865.
31. Chapman, Sydney; Cowling, T.G. (1970), The Mathematical Theory of Non-Uniform Gases (3rd ed.), Cambridge University Press
32. Bellac, Michael; Mortessagne, Fabrice; Batrouni, G. George (2004), Equilibrium and Non-Equilibrium Statistical Thermodynamics, Cambridge University Press, ISBN   978-0-521-82143-8
33. Chapman & Cowling, p. 103
34. Cercignani, Carlo (1975), Theory and Application of the Boltzmann Equation, Elsevier, ISBN   978-0-444-19450-3
35. The discussion which follows draws from Chapman & Cowling, pp. 232-237.
36. Bird, Steward, & Lightfoot, p. 25-27
37. Chapman & Cowling, pp. 235 - 237
38. Reid, Robert C.; Sherwood, Thomas K. (1958), The Properties of Gases and Liquids, McGraw-Hill Book Company, Inc., p. 202
39. Bird, Steward, & Lightfoot, pp. 29-31
40. Reid & Sherwood, pp. 203-204
41. Hildebrand, Joel Henry (1977), Viscosity and Diffusivity: A Predictive Treatment, John Wiley & Sons, Inc., ISBN   978-0-471-03072-0
42. Hildebrand p. 37
43. Egelstaff, P.A. (1992), An Introduction to the Liquid State (2nd ed.), Oxford University Press, p. 264, ISBN   978-0-19-851012-3
44. Irving, J.H.; Kirkwood, John G. (1949), "The Statistical Mechanical Theory of Transport Processes. IV. The Equations of Hydrodynamics", J. Chem. Phys., 18 (6): 817–829, doi:10.1063/1.1747782
45. Reid & Sherwood, pp. 206-209
46. Chapman & Cowling (1970)
47. Zhmud, Boris (2014), "Viscosity Blending Equations" (PDF), Lube-Tech, 93
48. Bird, Steward, & Lightfoot pp. 31-33
49. Bird, Steward, & Lightfoot p. 32
50. Mueller, S.; Llewellin, E. W.; Mader, H. M. (2009). "The rheology of suspensions of solid particles". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 466 (2116): 1201–1228. doi:10.1098/rspa.2009.0445. ISSN   1364-5021.
51. Bird, Steward, & Lightfoot p. 33
52. Fluegel, Alexander. "Viscosity calculation of glasses". Glassproperties.com. Retrieved 2010-09-14.
53. Doremus, R. H. (2002). "Viscosity of silica". J. Appl. Phys. 92 (12): 7619–7629. Bibcode:2002JAP....92.7619D. doi:10.1063/1.1515132.
54. Ojovan, M. I.; Lee, W. E. (2004). "Viscosity of network liquids within Doremus approach". J. Appl. Phys. 95 (7): 3803–3810. Bibcode:2004JAP....95.3803O. doi:10.1063/1.1647260.
55. Ojovan, M. I.; Travis, K. P.; Hand, R. J. (2000). "Thermodynamic parameters of bonds in glassy materials from viscosity-temperature relationships". J. Phys.: Condens. Matter. 19 (41): 415107. Bibcode:2007JPCM...19O5107O. doi:10.1088/0953-8984/19/41/415107. PMID   28192319.
56. Bird, Steward, & Lightfoot, p. 163
57. Marcel Lesieur (6 December 2012). Turbulence in Fluids: Stochastic and Numerical Modelling. Springer Science & Business Media. pp. 2–. ISBN   978-94-009-0533-7.
58. Sivashinsky, V.; Yakhot, G. (1985). "Negative viscosity effect in large-scale flows". The Physics of Fluids. 28 (4): 1040. doi:10.1063/1.865025.
59. Xie, Hong-Yi; Levchenko, Alex (23 January 2019), "Negative viscosity and eddy flow of the imbalanced electron-hole liquid in graphene", Phys. Rev. B, 99 (4): 045434, arXiv:, doi:10.1103/PhysRevB.99.045434
60. Edgeworth, R.; Dalton, B. J.; Parnell, T. (1984). "The pitch drop experiment". European Journal of Physics. 5 (4): 198–200. Bibcode:1984EJPh....5..198E. doi:10.1088/0143-0807/5/4/003 . Retrieved 2009-03-31.
61. John R. Rumble, ed. (2018). CRC Handbook of Chemistry and Physics (99th ed.). Boca Raton, FL: CRC Press. ISBN   978-1138561632.
62. Fellows, P.J. (2009), Food Processing Technology: Principles and Practice (3rd ed.), Woodhead Publishing, ISBN   978-1845692162
63. Yanniotis, S.; Skaltsi, S.; Karaburnioti, S. (February 2006). "Effect of moisture content on the viscosity of honey at different temperatures". Journal of Food Engineering. 72 (4): 372–377. doi:10.1016/j.jfoodeng.2004.12.017.
64. Koocheki, Arash; Ghandi, Amir; Razavi, Seyed M. A.; Mortazavi, Seyed Ali; Vasiljevic, Todor (2009), "The rheological properties of ketchup as a function of different hydrocolloids and temperature", International Journal of Food Science & Technology, 44 (3): 596–602, doi:10.1111/j.1365-2621.2008.01868.x
65. Citerne, Guillaume P.; Carreau, Pierre J.; Moan, Michel (2001), "Rheological properties of peanut butter", Rheologica Acta, 40 (1): 86–96, doi:10.1007/s003970000120