Diffraction

Diffraction refers to various phenomena associated with wave propagation, such as the bending, spreading and interference of waves passing by an object or aperture that disrupts the wave. It occurs with any type of wave, including sound waves, water waves, electromagnetic waves such as visible light, x-rays and radio waves. Diffraction also occurs with matter – according to the principles of quantum mechanics, any physical object has wave-like properties. While diffraction always occurs, its effects are generally most noticeable for waves where the wavelength is on the order of the feature size of the diffracting objects or apertures. The complex patterns in the intensity of a diffracted wave are a result of interference between different parts of a wave that traveled to the observer by different paths.

Examples of diffraction in everyday life

The effects of diffraction can be readily seen in everyday life. The most colorful examples of diffraction are those involving light; for example, the closely spaced tracks on a CD or DVD act as a diffraction grating to form the familiar rainbow pattern we see when looking at a disk. This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the hologram on a credit card is an example. Diffraction in the atmosphere by small particles in it can cause a bright ring to be visible around a bright light source like the sun or the moon. A shadow of a solid object, using light from a compact source, shows small fringes near its edges. All these effects are a consequence of the fact that light is a wave.

Diffraction can occur with any kind of wave. Ocean waves diffract around jetties and other obstacles. Sound waves can diffract around objects, this is the reason we can still hear someone calling us even if we are hiding behind a tree. Diffraction can also be a concern in some technical applications; it sets a fundamental limit to the resolution of a camera, telescope, or microscope.

History

Thomas Young's sketch of two-slit diffraction, which he presented to the Royal Society in 1803

The effects of diffraction of light were first carefully observed and characterized by Francesco Maria Grimaldi, who also coined the term diffraction, from the Latin diffringere, 'to break into pieces', referring to light breaking up into different directions. The results of Grimaldi's observations were published posthumously in 1665.^[1][2] Isaac Newton studied these effects and attributed them to inflexion of light rays. James Gregory (1638–1675) observed the diffraction patterns caused by a bird feather, which was effectively the first diffraction grating. In 1803 Thomas Young did his famous experiment observing diffraction from two closely spaced slits. Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves. Augustin-Jean Fresnel did more definitive studies and calculations of diffraction, published in 1815 and 1818, and thereby gave great support to the wave theory of light that had been advanced by Christian Huygens and reinvigorated by Young, against Newton's particle theory.

The mechanism of diffraction

Photograph of single-slit diffraction in a circular ripple tank

The very heart of the explanation of all diffraction phenomena is interference. When two waves combine, their displacements add, causing either a lesser or greater total displacement depending on the phase difference between the two waves. The effect of diffraction from an opaque object can be seen as interference between different parts of the wave beyond the diffraction object. The pattern formed by this interference is dependent on the wavelength of the wave, which for example gives rise to the rainbow pattern on a CD. Most diffraction phenomena can be understood in terms of a few simple concepts that are illustrated below.

The most conceptually simple example of diffraction is single-slit diffraction in which the slit is narrow, that is, significantly smaller than a wavelength of the wave. After the wave passes through the slit a pattern of semicircular ripples is formed, as if there were a simple wave source at the position of the slit. This semicircular wave is a diffraction pattern.

If we now consider two such narrow apertures, the two radial waves emanating from these apertures can interfere with each other. Consider for example, a water wave incident on a screen with two small openings. The total displacement of the water on the far side of the screen at any point is the sum of the displacements of the individual radial waves at that point. Now there are points in space where the wave emanating from one aperture is always in phase with the other, i.e. they both go up at that point, this is called constructive interference and results in a greater total amplitude. There are also points where one radial wave is out of phase with the other by one half of a wavelength, this would mean that when one is going up, the other is going down, the resulting total amplitude is decreased, this is called destructive interference. The result is that there are regions where there is no wave and other regions where the wave is amplified.

Another conceptually simple example is diffraction of a plane wave on a large (compared to the wavelength) plane mirror. The only direction at which all electrons oscillating in the mirror are seen oscillating in phase with each other is the specular (mirror) direction – thus a typical mirror reflects at the angle which is equal to the angle of incidence of the wave. This result is called the law of reflection. Smaller and smaller mirrors diffract light over a progressively larger and larger range of angles.

Slits significantly wider than a wavelength will also show diffraction which is most noticeable near their edges. The center part of the wave shows limited effects at short distances, but exhibits a stable diffraction pattern at longer distances. This pattern is most easily understood and calculated as the interference pattern of a large number of simple sources spaced closely and evenly across the width of the slit.

This concept is known as the Huygens–Fresnel principle: The propagation of a wave can be visualized by considering every point on a wavefront as a point source for a secondary radial wave. The subsequent propagation and interference of all these radial waves form the new wavefront. This principle mathematically results from interference of waves along all allowed paths between the source and the detection point (that is, all paths except those that are blocked by the diffracting objects).

Qualitative observations of diffraction

Several qualitative observations can be made of diffraction in general:

• The angular spacing of the features in the diffraction pattern is inversely proportional to the dimensions of the object causing the diffraction, in other words: the smaller the diffracting object the 'wider' the resulting diffraction pattern and vice versa. (More precisely, this is true of the sines of the angles.)
• The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object.
• When the diffracting object has a periodic structure, for example in a diffraction grating, the features generally become sharper. The third figure, for example, shows a comparison of a double-slit pattern with a pattern formed by five slits, both sets of slits having the same spacing, between the center of one slit and the next.

Quantitative description of diffraction

For more details on this topic, see Diffraction formalism.

To determine the pattern produced by diffraction we must determine the phase and amplitude of each of the Huygens wavelets at each point in space. That is, at each point in space, we must determine the distance to each of the simple sources on the incoming wavefront. If the distance to each of the simple sources differs by an integer number of wavelengths, all the wavelets will be in phase, resulting in constructive interference. If the distance to each source is an integer plus one half of a wavelength, there will be complete destructive interference. Usually it is sufficient to determine these minimums and maximums to explain the effects we see in nature. The simplest descriptions of diffraction are those in which the situation can be reduced to a 2 dimensional problem. For water waves, this is already the case, water waves propagate only on the surface of the water. For light, we can often neglect one direction if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes we will have to take into account the full three dimensional nature of the problem.

Multiple-slit arrangements can be described as multiple simple wave sources, if the slits are narrow enough. For light, a slit is an opening that is infinitely extended in one dimension, which has the effect of reducing a wave problem in 3-space to a simpler problem in 2-space. The simplest case is that of two narrow slits, spaced a distance d apart. To determine the maxima and minima in the amplitude we must determine the difference in path length to the first slit and to the second one. In the Fraunhofer approximation, with the observer far away from the slits, the difference in path length to the two slits can be seen from the image to be

ΔS = asinθ

Maxima in the intensity occur if this path length difference is an integer number of wavelengths.

asinθ = nλ		where n is an integer that labels the order of each maximum, λ is the wavelength, a is the distance between the slits and θ is the angle at which constructive interference occurs

And the corresponding minima are at path differences of an integer number plus one half of the wavelength:

For an array of slits, positions of the minima and maxima are not changed, the fringes visible on a screen however do become sharper as can be seen in the image. The same is true for a surface that is only reflective along a series of parallel lines; such a surface is called a reflection grating.

2-slit and 5-slit diffraction of red laser light

We see from the formula that the diffraction angle is wavelength dependent. This means that different colors of light will diffract in different directions, which allows us to separate light into its different color components. Gratings are used in spectroscopy to determine the properties of atoms and molecules, as well as stars and interstellar dust clouds by studying the spectrum of the light they emit or absorb. Another application of diffraction gratings is to produce a monochromatic light source. This can be done by placing a slit at the angle corresponding to the constructive interference condition for the desired wavelength.

Diagram of two slit diffraction problem, showing the angle to the first minimum, where a path length difference of a half wavelength causes destructive interference.

Single-slit diffraction

Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.

Graph and image of single-slit diffraction

Slits wider than a wavelength will show diffraction at their edges. The pattern is most easily understood and calculated as the interference pattern of a large number of simple sources spaced closely and evenly across the width of the slit. We can determine the minima of the resulting intensity pattern by using the following reasoning. If for a given angle a simple source located at the left edge of the slit interferes destructively with a source located at the middle of the slit, then a simple source just to the right of the left edge will interfere destructively with a simple source located just to the right of the middle. We can continue this reasoning along the entire width of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The result is a formula that looks very similar to the one for diffraction from a grating with the important difference that it now predicts the minima of the intensity pattern.

dsin(θ_min) = nλ n is now an integer greater than 0.

The same argument does not hold for the maxima. To determine the location of the maxima and the exact intensity profile, a more rigorous treatment is required; a diffraction formalism in terms of integration over all unobstructed paths is required. The intensity profile is then given by

Multiple extended slits

For an array of slits that are wider than the wavelength of the incident wave, we must take into account interference of wave from different slits as well as interference between waves from different locations in the same slit. Minima in the intensity occur if either the single slit condition or the grating condition for complete destructive interference is met. A rigorous mathematical treatment shows that the resulting intensity pattern is the product of the grating intensity function with the single slit intensity pattern.

When doing experiments with gratings that have a slit width being an integer fraction of the grating spacing, this can lead to 'missing' orders. If for example the width of a single slit is half the separation between slits, the first minimum of the single slit diffraction pattern will line up with the first maximum of the grating diffraction pattern. This expected diffraction peak will then not be visible. The same is true in this case for any odd numbered grating-diffraction peak.

Particle diffraction

See also: neutron diffraction and electron diffraction

Quantum theory tells us that every particle exhibits wave properties. In particular, massive particles can interfere and therefore diffract. Diffraction of electrons and neutrons stood as one of the powerful arguments in favor of quantum mechanics. The wavelength associated with a particle is the de Broglie wavelength

where h is Planck's constant and p is the momentum of the particle (mass × velocity for slow-moving particles) . For most macroscopic objects, this wavelength is so short that it is not meaningful to assign a wavelength to them. A Sodium atom traveling at about 3000 m/s would have a De Broglie wavelength of about 5 pico meters.

Because the wavelength for even the smallest of macroscopic objects is extremely small, diffraction of matter waves is only visible for small particles, like electrons, neutrons, atoms and small molecules. The short wavelength of these matter waves makes them ideally suited to study the atomic crystal structure of solids and large molecules like proteins.

Relatively recently, larger molecules like buckyballs,^[3] have been shown to diffract. Currently research is underway into the diffraction of viruses, which, being huge relative to electrons and other more commonly diffracted particles, have tiny wavelengths so must be made to travel very slowly through an extremely narrow slit in order to diffract. It is believed that there is a limit to the largest object that can be diffracted.

Bragg diffraction

For more details on this topic, see Bragg diffraction.

Diffraction from a three dimensional periodic structure such as atoms in a crystal is called Bragg diffraction. It is similar to what occurs when waves are scattered from a diffraction grating. Bragg diffraction is a consequence of interference between waves reflecting from different crystal planes. The condition of constructive interference is given by Bragg's law:

mλ = 2dsinθ

where

λ is the wavelength,

d is the distance between crystal planes,

θ is the angle of the diffracted wave.

and m is an integer known as the order of the diffracted beam.

Bragg diffraction may be carried out using either light of very short wavelength like x-rays or matter waves like neutrons whose wavelength is on the order of the atomic spacing. The pattern produced gives information of the separations of crystallographic planes d, allowing one to deduce the crystal structure.

Coherence

Main article: Coherence (physics)

The description of diffraction relies on the interference of waves emanating from the same source taking different paths to the same point on a screen. In this description, the difference in phase between waves that took different paths is only dependent on the effective path length. This does not take into account the fact that waves that arrive at the screen at the same time were emitted by the source at different times. The initial phase with which the source emits waves can change over time in an unpredictable way. This means that waves emitted by the source at times that are too far apart can no longer form a constant interference pattern since the relation between their phases is no longer time independent.

The length over which the phase in a beam of light is correlated, is called the coherence length. In order for interference to occur, the path length difference must be smaller than the coherence length. This is sometimes referred to as spectral coherence as it is related to the presence of different frequency components in the wave. In the case light emitted by an atomic transition, the coherence length is related to the lifetime of the excited state from which the atom made its transition.

If waves are emitted from an extended source this can lead to incoherence in the transversal direction. When looking at a cross section of a beam of light, the length over which the phase is correlated is called the transverse coherence length. In the case of Young's double slit experiment this would mean that if the transverse coherence length is smaller than the spacing between the two slits the resulting pattern on a screen would look like two single slit diffraction patterns.

In the case of particles like electrons, neutrons and atoms, the coherence length is related to the spacial extent of the wave function that describes the particle.

Diffraction limit of telescopes

The Airy disc around each of the stars from the 2.56m telescope aperture can be seen in this lucky image of the binary star zeta Boötis.

For diffraction through a circular aperture, there is a series of concentric rings surrounding a central Airy disc. The mathematical result is similar to a radially symmetric version of the equation given above in the case of single-slit diffraction.

A wave does not have to pass through an aperture to diffract; for example, a beam of light of a finite size also undergoes diffraction and spreads in diameter. This effect limits the minimum diameter d of spot of light formed at the focus of a lens, known as the diffraction limit:

where λ is the wavelength of the light, f is the focal length of the lens, and a is the diameter of the beam of light, or (if the beam is filling the lens) the diameter of the lens. The diameter given is enough to contain about 70% of the light energy; it is the radius to the first null of the Airy disk, in approximate agreement with the Rayleigh criterion. Twice that diameter, the diameter to the first null of the Airy disk, within which 83.8% of the light energy is contained, is also sometimes given as the diffraction spot diameter.

By use of Huygens' principle, it is possible to compute the diffraction pattern of a wave from any arbitrarily shaped aperture. If the pattern is observed at a sufficient distance from the aperture, it will appear as the two-dimensional Fourier transform of the function representing the aperture.

Viscosity

Viscosity is a measure of the resistance of a fluid to deform under shear stress. It is commonly perceived as "thickness", or resistance to flow. Viscosity describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction. Thus, water is "thin", having a lower viscosity, while vegetable oil is "thick" having a higher viscosity. All real fluids (except superfluids) have some resistance to shear stress, but a fluid which has no resistance to shear stress is known as an ideal fluid or inviscid fluid (Symon 1971).

When looking at a value for viscosity the number that one most often sees is the coefficient of viscosity, simply put this is the ratio between the pressure exerted on the surface of a fluid, in the lateral or horizontal direction, to the change in velocity of the fluid as you move down in the fluid (this is what is referred to as a velocity gradient). For example, at "room temperature", water has a nominal viscosity of 1.0 x 10^-3 Pa∙s and motor oil has a nominal apparent viscosity of 250 x 10^-3 Pa∙s.

The word "viscosity" derives from the Latin word "viscum" for mistletoe. A viscous glue was made from mistletoe berries and used for lime-twigs to catch birds.

Newton's theory

Laminar shear of fluid between two plates. Friction between the fluid and the moving boundaries causes the fluid to shear. The force required for this action is a measure of the fluid's viscosity. This type of flow is known as a Couette flow.

Laminar shear, the non-linear gradient, is a result of the geometry the fluid is flowing through (e.g. a pipe).

In general, in any flow, layers move at different velocities and the fluid's viscosity arises from the shear stress between the layers that ultimately opposes any applied force.

Isaac Newton postulated that, for straight, parallel and uniform flow, the shear stress, τ, between layers is proportional to the velocity gradient, ∂u/∂y, in the direction perpendicular to the layers, in other words, the relative motion of the layers.

.

Here, the constant η is known as the coefficient of viscosity, the viscosity, or the dynamic viscosity. Many fluids, such as water and most gases, satisfy Newton's criterion and are known as Newtonian fluids. Non-Newtonian fluids exhibit a more complicated relationship between shear stress and velocity gradient than simple linearity.

The relationship between the shear stress and the velocity gradient can also be obtained by considering two plates closely spaced apart at a distance y, and separated by a homogeneous substance. Assuming that the plates are very large, with a large area A, such that edge effects may be ignored, and that the lower plate is fixed, let a force F be applied to the upper plate. If this force causes the substance between the plates to undergo shear flow (as opposed to just shearing elastically until the shear stress in the substance balances the applied force), the substance is called a fluid. The applied force is proportional to the area and velocity of the plate and inversely proportional to the distance between the plates. Combining these three relations results in the equation F = η(Au/y), where η is the proportionality factor called the absolute viscosity (with units Pa·s = kg/(m·s) or slugs/(ft·s)). The absolute viscosity is also known as the dynamic viscosity, and is often shortened to simply viscosity. The equation can be expressed in terms of shear stress; τ = F/A = η(u/y). The rate of shear deformation is u / y and can be also written as a shear velocity, du/dy. Hence, through this method, the relation between the shear stress and the velocity gradient can be obtained.

In many situations, we are concerned with the ratio of the viscous force to the inertial force, the latter characterised by the fluid density ρ. This ratio is characterised by the kinematic viscosity, defined as follows:

.

James Clerk Maxwell called viscosity fugitive elasticity because of the analogy that elastic deformation opposes shear stress in solids, while in viscous fluids, shear stress is opposed by rate of deformation.

Measuring viscosity

Viscosity is measured with various types of viscometer. Close temperature control of the fluid is essential to accurate measurements, particularly in materials like lubricants, whose viscosity (-40 <>

For some fluids, it is a constant over a wide range of shear rates. The fluids without a constant viscosity are called Non-Newtonian fluids.

In paint industries, viscosity is commonly measured with a Zahn cup, in which the efflux time is determined and given to customers. The efflux time can also be converted to kinematic viscosities (cSt) through the conversion equations.

Also used in paint, a Stormer viscometer uses load-based rotation in order to determine viscosity. It uses units, Krebs units (KU), unique to this viscometer.

Units

Viscosity (dynamic/absolute viscosity): η or μ

The IUPAC symbol for viscosity is the Greek symbol eta (η), and dynamic viscosity is also commonly referred to using the Greek symbol mu (μ). The SI physical unit of dynamic viscosity is the pascal-second (Pa·s), which is identical to 1 kg·m⁻¹·s⁻¹. If a fluid with a viscosity of one Pa·s is placed between two plates, and one plate is pushed sideways with a shear stress of one pascal, it moves a distance equal to the thickness of the layer between the plates in one second. The name poiseuille (Pl) was proposed for this unit (after Jean Louis Marie Poiseuille who formulated Poiseuille's law of viscous flow), but not accepted internationally. Care must be taken in not confusing the poiseuille with the poise named after the same person!

The cgs physical unit for dynamic viscosity is the poise^[1] (P; IPA: [pwaz])) named after Jean Louis Marie Poiseuille. It is more commonly expressed, particularly in ASTM standards, as centipoise (cP). The centipoise is commonly used because water has a viscosity of 1.0020 cP (at 20 °C; the closeness to one is a convenient coincidence).

1 P = 1 g·cm⁻¹·s⁻¹

The relation between Poise and Pascal-second is:

10 P = 1 kg·m⁻¹·s⁻¹ = 1 Pa·s

1 cP = 0.001 Pa·s = 1 mPa·s

Kinematic viscosity: ν

Kinematic viscosity (Greek symbol: ν) has SI units (m²·s⁻¹). The cgs physical unit for kinematic viscosity is the stokes (abbreviated S or St), named after George Gabriel Stokes. It is sometimes expressed in terms of centistokes (cS or cSt). In U.S. usage, stoke is sometimes used as the singular form.

1 stokes = 100 centistokes = 1 cm²·s⁻¹ = 0.0001 m²·s⁻¹.

1 centistokes = 1 mm²/s

Dynamic versus kinematic viscosity

Conversion between kinematic and dynamic viscosity, is given by νρ = η. Note that the parameters must be given in SI units not in P, cP or St.

For example, if ν = 1 St (=0.0001 m²·s^-1) and ρ = 1000 kg m^-3 then η = νρ = 0.1 kg·m⁻¹·s⁻¹ = 0.1 Pa·s [1].

For a plot of kinematic viscosity of air as a function of absolute temperature, see James Ierardi's Fire Protection Engineering Site

Molecular origins

Pitch has a viscosity approximately 100 billion times that of water.

The viscosity of a system is determined by how molecules constituting the system interact. There are no simple but correct expressions for the viscosity of a fluid. The simplest exact expressions are the Green-Kubo relations for the linear shear viscosity or the Transient Time Correlation Function expressions derived by Evans and Morriss in 1985. Although these expressions are each exact in order to calculate the viscosity of a dense fluid, using these relations requires the use of molecular dynamics computer simulation.

Gases

Viscosity in gases arises principally from the molecular diffusion that transports momentum between layers of flow. The kinetic theory of gases allows accurate prediction of the behaviour of gaseous viscosity, in particular that, within the regime where the theory is applicable:

• Viscosity is independent of pressure; and
• Viscosity increases as temperature increases.

Liquids

In liquids, the additional forces between molecules become important. This leads to an additional contribution to the shear stress though the exact mechanics of this are still controversial.^{[citation needed]} Thus, in liquids:

• Viscosity is independent of pressure (except at very high pressure); and
• Viscosity tends to fall as temperature increases (for example, water viscosity goes from 1.79 cP to 0.28 cP in the temperature range from 0 °C to 100 °C); see temperature dependence of liquid viscosity for more details.

The dynamic viscosities of liquids are typically several orders of magnitude higher than dynamic viscosities of gases.

Viscosity of materials

The viscosity of air and water are by far the two most important materials for aviation aerodynamics and shipping fluid dynamics. Temperature plays the main role in determining viscosity.

Viscosity of air

The viscosity of air depends mostly on the temperature. At 15.0 °C, the viscosity of air is 1.78 × 10⁻⁵ kg/(m·s). You can get the viscosity of air as a function of altitude from the eXtreme High Altitude Calculator

Viscosity of water

The viscosity of water is 8.90 × 10^-4 Pa·s or 8.90 × 10^-3 dyn·s/cm²at about 25 °C.
as a function of temperature: μ=A × 10^{B/(T-C)
Where A=2.414 × 10-5 N*s/m2 ; B = 247.8 Kelvin ; C = 140 Kelvin}

Viscosity of various materials

Example of the viscosity of milk and water. Liquids with higher viscosities will not make such a splash when poured at the same velocity.

Honey being drizzled.

Peanut butter is a semi-solid and so can hold peaks.

The Sutherland's formula can be used to derive the dynamic viscosity as a function of the temperature:

where:

• η = viscosity in (Pa·s) at input temperature T
• η₀ = reference viscosity in (Pa·s) at reference temperature T₀
• T = input temperature in kelvin
• T₀ = reference temperature in kelvin
• C = Sutherland's constant

Valid for temperatures between 0 < T <>

Sutherland's constant and reference temperature for some gases

Gas	C [K]	T0 [K]	η0 [10-6 Pa s]
air	120	291.15	18.27
nitrogen	111	300.55	17.81
oxygen	127	292.25	20.18
carbon dioxide	240	293.15	14.8
carbon monoxide	118	288.15	17.2
hydrogen	72	293.85	8.76
ammonia	370	293.15	9.82
sulphur dioxide	416	293.65	12.54

Some dynamic viscosities of Newtonian fluids are listed below:

Gases (at 0 °C):

	viscosity [Pa·s]
hydrogen	8.4 × 10-6
air	17.4 × 10-6
xenon	21.2 × 10-6

Liquids (at 25 °C):

	viscosity [Pa·s]	viscosity [cP]
liquid nitrogen @ 77K	?	0.158
acetone	a 0.306 × 10−3	a 0.306
methanol	a 0.544 × 10−3	a 0.544
benzene	a 0.604 × 10−3	a 0.604
ethanol	a 1.074 × 10−3	a 1.074
mercury	a 1.526 × 10−3	a 1.526
nitrobenzene	a 1.863 × 10−3	a 1.863
propanol	a 1.945 × 10−3	a 1.945
sulfuric acid	a 24.2 × 10−3	a 24.2
olive oil	81 × 10−3	81
glycerol	a 934 × 10−3	a 934
castor oil	985 × 10−3	985
HFO-380	2022 × 10−3	2022
pitch	2.3 × 108	230 × 109

^a Data from CRC Handbook of Chemistry and Physics, 73^rd edition, 1992-1993.

Fluids with variable compositions, such as honey, can have a wide range of viscosities.

A more complete table can be found here, including the following:

	viscosity [cP]
honey	2,000–10,000
molasses	5,000–10,000
molten glass	10,000–1,000,000
chocolate syrup	10,000–25,000
chocolate*	45,000–130,000 [2]
ketchup*	50,000–100,000
peanut butter	~250,000
shortening*	~250,000

* These materials are highly non-Newtonian.

Can solids have a viscosity?

If on the basis that all solids flow to a small extent in response to shear stress then yes, substances known as Amorphous solids, such as glass, may be considered to have viscosity. This has led some to the view that solids are simply liquids with a very high viscosity, typically greater than 10¹² Pa•s. This position is often adopted by supporters of the widely held misconception that glass flow can be observed in old buildings. This distortion is more likely the result of glass making process rather than the viscosity of glass.

However, others argue that solids are, in general, elastic for small stresses while fluids are not. Even if solids flow at higher stresses, they are characterized by their low-stress behavior. Viscosity may be an appropriate characteristic for solids in a plastic regime. The situation becomes somewhat confused as the term viscosity is sometimes used for solid materials, for example Maxwell materials, to describe the relationship between stress and the rate of change of strain, rather than rate of shear.

These distinctions may be largely resolved by considering the constitutive equations of the material in question, which take into account both its viscous and elastic behaviors. Materials for which both their viscosity and their elasticity are important in a particular range of deformation and deformation rate are called viscoelastic. In geology, earth materials that exhibit viscous deformation at least three times greater than their elastic deformation are sometimes called rheids.

One example of solids flowing which has been observed since 1930 is the Pitch drop experiment.

Bulk viscosity

The trace of the stress tensor is often identified with the negative-one-third of the thermodynamic pressure,

,

which only depends upon the equilibrium state potentials like temperature and density (equation of state). In general, the trace of the stress tensor is the sum of thermodynamic pressure contribution plus another contribution which is proportional to the divergence of the velocity field. This constant of proportionality is called the bulk viscosity.

Eddy viscosity

In the study of turbulence in fluids, a common practical strategy for calculation is to ignore the small-scale vortices (or eddies) in the motion and to calculate a large-scale motion with an eddy viscosity that characterizes the transport and dissipation of energy in the smaller-scale flow (see large eddy simulation). Values of eddy viscosity used in modeling ocean circulation may be from 5x10⁴ to 10⁶ Pa·s depending upon the resolution of the numerical grid.

Fluidity

The reciprocal of viscosity is fluidity, usually symbolized by φ = 1 / η or F = 1 / η, depending on the convention used, measured in reciprocal poise (cm·s·g^-1), sometimes called the rhe. Fluidity is seldom used in engineering practice.

The concept of fluidity can be used to determine the viscosity of an ideal solution. For two components a and b, the fluidity when a and b are mixed is

which is only slightly simpler than the equivalent equation in terms of viscosity:

where χ_a and χ_b is the mole fraction of component a and b respectively, and η_a and η_b are the components pure viscosities.

The linear viscous stress tensor

(See Hooke's law and strain tensor for an analogous development for linearly elastic materials.)

Viscous forces in a fluid are a function of the rate at which the fluid velocity is changing over distance. The velocity at any point is specified by the velocity field . The velocity at a small distance from point may be written as a Taylor series:

where is shorthand for the dyadic product of the del operator and the velocity:

This is just the Jacobian of the velocity field. Viscous forces are the result of relative motion between elements of the fluid, and so are expressible as a function of the velocity field. In other words, the forces at are a function of and all derivatives of at that point. In the case of linear viscosity, the viscous force will be a function of the Jacobian tensor alone. For almost all practical situations, the linear approximation is sufficient.

If we represent x, y, and z by indices 1, 2, and 3 respectively, the i,j component of the Jacobian may be written as where is shorthand for . Note that when the first and higher derivative terms are zero, the velocity of all fluid elements is parallel, and there are no viscous forces.

Any matrix may be written as the sum of an antisymmetric matrix and a symmetric matrix, and this decomposition is independent of coordinate system, and so has physical significance. The velocity field may be approximated as:

where Einstein notation is now being used in which repeated indices in a product are implicitly summed. The second term on the left is the asymmetric part of the first derivative term, and it represents a rigid rotation of the fluid about with angular velocity ω where:

For such a rigid rotation, there is no change in the relative positions of the fluid elements, and so there is no viscous force associated with this term. The remaining symmetric term is responsible for the viscous forces in the fluid. Assuming the fluid is isotropic (i.e. its properties are the same in all directions), then the most general way that the symmetric term (the rate-of-strain tensor) can be broken down in a coordinate-independent (and therefore physically real) way is as the sum of a constant tensor (the rate-of-expansion tensor) and a traceless symmetric tensor (the rate-of-shear tensor):

where δ_ij is the unit tensor. The most general linear relationship between the stress tensor and the rate-of-strain tensor is then a linear combination of these two tensors (Landau & Lifshitz 1997):

where ζ is the coefficient of bulk viscosity (or "second viscosity") and η is the coefficient of (shear) viscosity.

The forces in the fluid are due to the velocities of the individual molecules. The velocity of a molecule may be thought of as the sum of the fluid velocity and the thermal velocity. The viscous stress tensor described above gives the force due to the fluid velocity only. The force on an area element in the fluid due to the thermal velocities of the molecules is just the hydrostatic pressure. This pressure term (pδ_ij) must be added to the viscous stress tensor to obtain the total stress tensor for the fluid.

The infinitesimal force dF_i on an infinitesimal area dA_i is then given by the usual relationship: