Continuum mechanics

Conservation of mass Conservation of momentum Navier–Stokes equations

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Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or extensional stress. Continuum mechanics is a branch of Mechanics that deals with the analysis of the Kinematics and mechanical behavior of materials modeled as a continuum e The law of conservation of mass/matter, also known as law of mass/matter conservation (or the Lomonosov - Lavoisier law says that the Mass of In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such In Fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a Solid object through a Fluid (a FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material Stress is a measure of the average amount of Force exerted per unit Area. 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. Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e Thus, water is "thin", having a lower viscosity, while vegetable oil is "thick" having a higher viscosity. Water is a common Chemical substance that is essential for the survival of all known forms of Life. All real fluids (except superfluids) have some resistance to stress, but a fluid which has no resistance to shear stress is known as an ideal fluid or inviscid fluid. Superfluidity is a phase of matter or description of Heat capacity in which unusual effects are observed when Liquids, typically of Helium-4 Stress is a measure of the average amount of Force exerted per unit Area. For example a high viscosity magma will create a tall volcano, because it cannot spread fast enough, low viscosity lava will create a shield volcano, which is large and wide. ^[1] The study of viscosity is known as rheology. Rheology is the study of the flow of matter mainly liquids but also soft solids or solids under conditions in which they flow rather than deform elastically

Contents 1 Etymology 2 Viscosity coefficients 3 Newton's theory 4 Viscosity measurement 4.1 Units of measure 4.1.1 Viscosity (dynamic/absolute viscosity) 4.1.2 Kinematic viscosity 4.1.3 Saybolt Universal Viscosity 4.1.4 Relation to Mean Free Path of Diffusing Particles 4.1.5 Dynamic versus kinematic viscosity 4.1.6 Example: viscosity of water 5 Molecular origins 5.1 Gases 5.1.1 Effect of temperature on the viscosity of a gas 5.1.2 Viscosity of a dilute gas 5.2 Liquids 5.2.1 Viscosity of blends of liquids 6 Viscosity of selected substances 6.1 Viscosity of air 6.2 Viscosity of water 6.3 Viscosity of various materials 7 Viscosity of solids 8 Viscosity of amorphous materials 9 Volume (bulk) viscosity 10 Eddy viscosity 11 Fluidity 12 The linear viscous stress tensor 13 See also 14 References 15 Additional reading 16 External links

Etymology

The word "viscosity" derives from the Latin word "viscum" for mistletoe. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. Mistletoe is the common name for a group of hemi-parasitic Plants in the order Santalales that grow attached to and within the A viscous glue was made from mistletoe berries and used for lime-twigs to catch birds. ^[2]

Viscosity coefficients

When looking at a value for viscosity, the number that one most often sees is the coefficient of viscosity. There are several different viscosity coefficients depending on the nature of applied stress and nature of the fluid. They are introduced in the main books on hydrodynamics^[3][4] and rheology. Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion Rheology is the study of the flow of matter mainly liquids but also soft solids or solids under conditions in which they flow rather than deform elastically ^[5]

• Dynamic viscosity determines the dynamics of an incompressible Newtonian fluid;
• Kinematic viscosity is the dynamic viscosity divided by the density for a Newtonian fluid;
• Volume viscosity determines the dynamics of a compressible Newtonian fluid;
• Bulk viscosity is the same as volume viscosity
• Shear viscosity is the viscosity coefficient when the applied stress is a shear stress (valid for non-Newtonian fluids);
• Extensional viscosity is the viscosity coefficient when the applied stress is an extensional stress (valid for non-Newtonian fluids). In Thermodynamics and Fluid mechanics, compressibility is a measure of the relative volume change of a Fluid or Solid as a response A Newtonian fluid (named for Isaac Newton) is a Fluid whose stress versus rate of strain curve is linear and passes through the origin Volume viscosity (also called bulk viscosity or second viscosity) appears in the Navier-Stokes equation if it is written for Compressible fluid A Newtonian fluid (named for Isaac Newton) is a Fluid whose stress versus rate of strain curve is linear and passes through the origin Volume viscosity (also called bulk viscosity or second viscosity) appears in the Navier-Stokes equation if it is written for Compressible fluid A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material Extensional viscosity is a Viscosity coefficient when applied stress is Extensional stress. Stress is a measure of the average amount of Force exerted per unit Area.

Shear viscosity and dynamic viscosity are much better known than the others. That is why they are often referred to as simply viscosity. Simply put, this quantity 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). In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar For example, at room temperature, water has a nominal viscosity of 1. 0 × 10^-3 Pa∙s and motor oil has a nominal apparent viscosity of 250 × 10^-3 Pa∙s. ^[6]

Extensional viscosity is widely used for characterizing polymers.

Volume viscosity is essential for Acoustics in fluids, see Stokes' law (sound attenuation) ^[7]

Newton's theory

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. In Physics, velocity is defined as the rate of change of Position.

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. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements In Physics, velocity is defined as the rate of change of Position. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent

.

Here, the constant η is known as the coefficient of viscosity, the viscosity, the dynamic viscosity, or the Newtonian viscosity. Many fluids, such as water and most gases, satisfy Newton's criterion and are known as Newtonian fluids. FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code Water is a common Chemical substance that is essential for the survival of all known forms of Life. This page is about the physical properties of gas as a state of matter A Newtonian fluid (named for Isaac Newton) is a Fluid whose stress versus rate of strain curve is linear and passes through the origin Non-Newtonian fluids exhibit a more complicated relationship between shear stress and velocity gradient than simple linearity. A non-Newtonian fluid is a Fluid whose flow properties are not described by a single constant value of Viscosity.

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. Heterogeneous is an adjective used to describe an object or system consisting of multiple items having a large number of structural variations 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. In Materials science, deformation is a change in the shape or size of an object due to an applied force. A material is said to be elastic if it deforms under stress (e 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.

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. James Clerk Maxwell (13 June 1831 &ndash 5 November 1879 was a Scottish mathematician and theoretical physicist. A solid' object is in the States of matter characterized by resistance to Deformation and changes of Volume. FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code

Viscosity measurement

Dynamic viscosity is measured with various types of viscometer. A viscometer (also called viscosimeter) is an instrument used to measure the Viscosity of a Fluid. Close temperature control of the fluid is essential to accurate measurements, particularly in materials like lubricants, whose viscosity can double with a change of only 5 °C. For some fluids, it is a constant over a wide range of shear rates. These are Newtonian fluids. A Newtonian fluid (named for Isaac Newton) is a Fluid whose stress versus rate of strain curve is linear and passes through the origin

The fluids without a constant viscosity are called Non-Newtonian fluids. A non-Newtonian fluid is a Fluid whose flow properties are not described by a single constant value of Viscosity. Their viscosity 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 paint industries, viscosity is commonly measured with a Zahn cup, in which the efflux time is determined and given to customers. Zahn cup is a Viscosity measurement device widely used in the paint industry 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. The viscosity is reported in Krebs units (KU), which are unique to Stormer viscometers.

Vibrating viscometers can also be used to measure viscosity. These models use vibration rather than rotation to measure viscosity.

Extensional viscosity can be measured with various rheometers that apply extensional stress

Volume viscosity can be measured with acoustic rheometer. Today a rheometer is a laboratory device used to measure the way in which a liquid suspension or slurry flows in response to applied forces Stress is a measure of the average amount of Force exerted per unit Area. Volume viscosity (also called bulk viscosity or second viscosity) appears in the Navier-Stokes equation if it is written for Compressible fluid Acoustic rheometer employes Piezo-electric Crystal that can easily launch a successive Wave of extensions and contractions into the Fluid

Units of measure

Viscosity (dynamic/absolute viscosity)

Dynamic viscosity and absolute viscosity are synonymous. 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 International Union of Pure and Applied Chemistry ( IUPAC) (aɪjuːpæk or ay-yoo-pec) is an international Non-governmental organization The SI physical unit of dynamic viscosity is the pascal-second (Pa·s), which is identical to kg·m⁻¹·s⁻¹. The second ( SI symbol s) sometimes abbreviated sec, is the name of a unit of Time, and is the International System of Units 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. FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material The second ( SI symbol s) sometimes abbreviated sec, is the name of a unit of Time, and is the International System of Units

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. Jean Louis Marie Poiseuille \pwä-'zəi\ ( April 22, 1799 - December 26, 1869) was a French Physician and Physiologist Jean Louis Marie Poiseuille \pwä-'zəi\ ( April 22, 1799 - December 26, 1869) was a French Physician and Physiologist The Hagen-Poiseuille equation is a Physical law that describes slow Viscous Incompressible flow through a constant circular cross-section Care must be taken in not confusing the poiseuille with the poise named after the same person. The poise (symbol P pwɑːz is the unit of dynamic Viscosity in the Centimetre gram second system of units.

The cgs physical unit for dynamic viscosity is the poise^[8] (P), named after Jean Louis Marie Poiseuille. The centimetre-gram-second system ( CGS) is a system of physical units. Jean Louis Marie Poiseuille \pwä-'zəi\ ( April 22, 1799 - December 26, 1869) was a French Physician and Physiologist It is more commonly expressed, particularly in ASTM standards, as centipoise (cP). ASTM International ( ASTM) originally known as the American Society for Testing and Materials is an international Standards organization that develops and publishes 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-seconds is:

10 P = 1 kg·m⁻¹·s⁻¹ = 1 Pa·s

1 cP = 0. 001 Pa·s = 1 mPa·s

Kinematic viscosity

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. The vis insita or innate force of matter is a power of resisting by which every body as much as in it lies endeavors to preserve in its present state whether it be of rest or of moving FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different This ratio is characterised by the kinematic viscosity (ν), defined as follows:

.

where μ is the (dynamic or absolute) viscosity (in centipoise cP), and ρ is the density (in grams/cm^3), and ν is the kinematic viscosity (in centistokes cSt ).

Kinematic viscosity (Greek symbol: ν) has SI units Pa. s/(kg/m³) = m²·s⁻¹. The cgs physical unit for kinematic viscosity is the stokes (abbreviated S or St), named after George Gabriel Stokes. Sir George Gabriel Stokes 1st Baronet FRS ( 13 August 1819 &ndash 1 February 1903) was a mathematician and physicist 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^-1 = 10^-6m²·s⁻¹

Saybolt Universal Viscosity

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). ^[9] Kinematic viscosity in centistoke can be converted from SUS according to the arithmetic and the reference tabel provided in ASTM D 2161. ASTM International ( ASTM) originally known as the American Society for Testing and Materials is an international Standards organization that develops and publishes It can also be converted in computerized method, or vice versa. ^[10]

Relation to Mean Free Path of Diffusing Particles

In relation to diffusion, the kinematic viscosity provides a better understanding of the behavior of mass transport of a dilute species. Viscosity is related to shear stress and the rate of shear in a fluid, which illustrates its dependence on the mean free path, λ, of the diffusing particles.

From fluid mechanics, shear stress, τ, is the rate of change of velocity with distance perpendicular to the direction of movement. Fluid mechanics is the study of how Fluids move and the Forces on them A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material

.

Interpreting shear stress as the time rate of change of momentum,p, per unit area (rate of momentum flux) of an arbitrary control surface gives

. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product

Further manipulation will show

where

is the rate of change of mass

ρ is the density of the fluid

is the average molecular speed

μ is the dynamic viscosity.

Dynamic versus kinematic viscosity

Conversion between kinematic and dynamic viscosity is given by νρ = μ.

For example,

if ν = 0. 0001 m²·s^-1 and ρ = 1000 kg m^-3 then μ = νρ = 0. 1 kg·m⁻¹·s⁻¹ = 0. 1 Pa·s

if ν = 1 St (= 1 cm²·s⁻¹) and ρ = 1 g cm^-3 then μ = νρ = 1 g·cm⁻¹·s⁻¹ = 1 P

A plot of the kinematic viscosity of air as a function of absolute temperature is available on the Internet. ^[11]

Example: viscosity of water

Because of its density of ρ = 1 g/cm³ (varies slightly with temperature), and its dynamic viscosity is near 1 mPa·s (see #Viscosity of water section), the viscosity values of water are, to rough precision, all powers of ten:

Dynamic viscosity:

μ = 1 mPa·s = 10^-3 Pa·s = 1 cP = 10^-2 poise

Kinematic viscosity:

ν = 1 cSt = 10^-2 stokes = 1 mm²/s

Molecular origins

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. Green–Kubo relations give exact mathematical expression for transport coefficients in terms of integrals of time correlation functions 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. Molecular dynamics ( MD) is a form of Computer simulation in which atoms and molecules are allowed to interact for a period of time by approximations of

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 behavior of gaseous viscosity.

Within the regime where the theory is applicable:

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

James Clerk Maxwell published a famous paper in 1866 using the kinetic theory of gases to study gaseous viscosity. James Clerk Maxwell (13 June 1831 &ndash 5 November 1879 was a Scottish mathematician and theoretical physicist. (Reference: J. C. Maxwell, "On the viscosity or internal friction of air and other gases", Philosophical Transactions of the Royal Society of London, vol. 156 (1866), pp. 249-268. )

Effect of temperature on the viscosity of a gas

Sutherland's formula can be used to derive the dynamic viscosity of an ideal gas 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 for the gaseous material in question

Valid for temperatures between 0 < T < 555 K with an error due to pressure less than 10% below 3. These four properties that constitute an ideal gas can be easily remembered by the acronym RIPE which stands for - R andom Motion (molecules are in constant random motion 45 MPa

Sutherland's constant and reference temperature for some gases

Gas	C [K]	T0 [K]	η0 [10-6 Pa s]
air	120	291. Temperature and layers The temperature of the Earth's atmosphere varies with altitude the mathematical relationship between temperature and altitude varies among five 15	18. 27
nitrogen	111	300. Nitrogen (ˈnaɪtɹəʤɪn is a Chemical element that has the symbol N and Atomic number 7 and Atomic weight 14 55	17. 81
oxygen	127	292. Oxygen (from the Greek roots ὀξύς (oxys (acid literally "sharp" from the taste of acids and -γενής (-genēs (producer literally begetteris the 25	20. 18
carbon dioxide	240	293. Carbon dioxide ( Chemical formula:) is a Chemical compound composed of two Oxygen Atoms covalently bonded to a single 15	14. 8
carbon monoxide	118	288. Carbon monoxide, with the chemical formula CO is a colorless odorless tasteless yet highly toxic Gas. 15	17. 2
hydrogen	72	293. Hydrogen (ˈhaɪdrədʒən is the Chemical element with Atomic number 1 85	8. 76
ammonia	370	293. Ammonia is a compound with the formula N[[hydrogen H3]] It is normally encountered as a Gas with a characteristic pungent Odor 15	9. 82
sulfur dioxide	416	293. 65	12. 54
helium	79. Helium ( He) is a colorless odorless tasteless non-toxic Inert Monatomic Chemical 4 [12]	273	19 [13]

(also see: ^[14])

Viscosity of a dilute gas

The Chapman-Enskog equation^[15] may be used to estimate viscosity for a dilute gas. This equation is based on semi-theorethical assumption by Chapman and Enskoq. The equation requires three empirically determined parameters: the collision diameter (σ), the maximum energy of attraction divided by the Boltzmann constant (є/к) and the collision integral (ω(T*)). Bridge from macroscopic to microscopic physics Boltzmann's constant k is a bridge between Macroscopic and microscopic physics

• T*=κT/ε Reduced temperature (dimensionless)
• η₀ = viscosity for dilute gas (uP)
• M = molecular mass (g/mol)
• T = temperature (K)
• σ = the collision diameter (Å)
• ε / κ = the maximum energy of attraction divided by the Boltzmann constant (K)
• ω_η = the collision integral

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. 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 Temperature dependence of liquid viscosity is the phenomenon by which liquid Viscosity tends to fall (or alternatively its fluidity tends to increase

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

Viscosity of blends of liquids

The viscosity of the blend of two or more liquids can be estimated using the Refutas equation^[16][17]. The calculation is carried out in three steps.

The first step is to calculate the Viscosity Blending Number (VBN) (also called the Viscosity Blending Index) of each component of the blend:

(1)

where v is the kinematic viscosity in centistokes (cSt). It is important that the kinematic viscosity of each component of the blend be obtained at the same temperature.

The next step is to calculate the VBN of the blend, using this equation:

(2)

where x_X is the mass fraction of each component of the blend. In Chemistry the mass fraction is the fraction of one substance (x_A with mass m_A to the total mixture mass m_{tot} would be defined

Once the viscosity blending number of a blend has been calculated using equation (2), the final step is to determine the kinematic viscosity of the blend by solving equation (1) for v:

(3)

where VBN_Blend is the viscosity blending number of the blend.

Viscosity of selected substances

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) or 1. 78 × 10⁻⁴ P. One can get the viscosity of air as a function of temperature from the Gas Viscosity Calculator

Viscosity of water

The viscosity of water is 8. 90 × 10⁻⁴ Pa·s or 8. 90 × 10⁻³ dyn·s/cm² or 0. 890 cP at about 25 °C.
As a function of temperature T (K): μ(Pa·s) = A × 10^B/(T−C)
where A=2. 414 × 10⁻⁵ Pa·s ; B = 247. 8 K ; and C = 140 K.

Viscosity of water at different temperatures is listed below.

Temperature [ºC]	viscosity [Pa·s]
10	1. 308 × 10−3
20	1. 003 × 10−3
30	7. 978 × 10−4
40	6. 531 × 10−4
50	5. 471 × 10−4
60	4. 668 × 10−4
70	4. 044 × 10−4
80	3. 550 × 10−4
90	3. 150 × 10−4
100	2. 822 × 10−4

Viscosity of various materials

Some dynamic viscosities of Newtonian fluids are listed below:

Gases (at 0 °C):

	viscosity [Pa·s]
hydrogen	8. This page is about the physical properties of gas as a state of matter The Celsius Temperature scale was previously known as the centigrade scale. Hydrogen (ˈhaɪdrədʒən is the Chemical element with Atomic number 1 4 × 10−6
air	17. Temperature and layers The temperature of the Earth's atmosphere varies with altitude the mathematical relationship between temperature and altitude varies among five 4 × 10−6
xenon	2. Xenon (ˈzɛnɒn or) is a Chemical element represented by the symbol Xe. 12 × 10−5

Liquids (at 25 °C):

	viscosity [Pa·s]	viscosity [cP]
liquid nitrogen @ 77K	1. Liquid is one of the principal States of matter. A liquid is a Fluid that has the particles loose and can freely form a distinct surface at the boundaries of The Celsius Temperature scale was previously known as the centigrade scale. Liquid nitrogen (liquid density at the Triple point is 0707 g/mL is the liquid produced industrially in large quantities by Fractional distillation of 58 × 10−4	0. 158
acetone*	3. Acetone (also known as propanone, dimethyl ketone, 2-propanone, propan-2-one and β-ketopropane) is a colorless mobile flammable 06 × 10−4	0. 306
methanol*	5. Methanol, also known as methyl alcohol, carbinol, wood alcohol, wood naphtha or wood spirits, is a Chemical compound 44 × 10−4	0. 544
benzene*	6. Benzene, or benzol, is an organic Chemical compound and a known Carcinogen with the molecular formula C 6 H 6 04 × 10−4	0. 604
water	8. Water is a common Chemical substance that is essential for the survival of all known forms of Life. 94 × 10−4	0. 894
ethanol*	1. 074 × 10−3	1. 074
mercury*	1. Mercury (ˈmɜrkjʊri also called quicksilver or hydrargyrum, is a Chemical element with the symbol Hg ( Latinized hydrargyrum 526 × 10−3	1. 526
nitrobenzene*	1. Nitrobenzene, also known as nitrobenzol or oil of mirbane, is an Organic compound with the Chemical formula C 6 863 × 10−3	1. 863
propanol*	1. Propan-1-ol is a primary Alcohol with the formula CH3CH2CH2OH 945 × 10−3	1. 945
Ethylene glycol	1. Ethylene glycol ( monoethylene glycol ( MEG) 12-ethanediol, IUPAC name: ethane-12-diol) is an Alcohol with two -OH 61 × 10−2	16. 1
sulfuric acid*	2. Sulfuric (or sulphuric acid, H 2 S[[oxygen O]]4 is a strong Mineral acid. 42 × 10−2	24. 2
olive oil	. Olive oil is a fruit oil obtained from the olive ( Olea europaea; family Oleaceae along with Lilacs Jasmine and ash trees 081	81
glycerol*	. 934	934
castor oil*	. The castor oil plant, Ricinus communis, is a Plant Species of the Euphorbiaceae (the evolution of this plant family is relatively unexplored 985	985
corn syrup*	1. Corn syrup is a Syrup, made using Cornstarch as a feedstock and composed mainly of Glucose. 3806	1380. 6
HFO-380	2. Fuel oil is a fraction obtained from Petroleum Distillation, either as a distillate or a residue 022	2022
pitch	2. Pitch is the name for any of a number of highly viscous Liquids which appear Solid. 3 × 108	2. 3 × 1011

* 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. FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code Honey is a sweet and Viscous fluid produced by Honey bees (and some other species and derived from the nectar of Flowers According to the

A more complete table can be found at Transwiki, 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
molten chocolate*	45,000–130,000 [18]
ketchup*	50,000–100,000
peanut butter	~250,000
shortening*	~250,000

* These materials are highly non-Newtonian. Honey is a sweet and Viscous fluid produced by Honey bees (and some other species and derived from the nectar of Flowers According to the Molasses or Treacle is a thick Syrup by-product from the processing of the Sugarcane or Sugar beet into Sugar. Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many Chocolate syrup is a type of condiment that is usually added to food to increase the Chocolate flavor Chocolate ( pronounced or /-ˈələt/ comprises a number of raw and processed foods that are produced from the seed of the tropical Cacao tree Ketchup (also spelled catsup or catchup) also known as tomato ketchup, tomato sauce, red sauce, Tommy sauce, Peanut butter is a food paste made primarily from ground roasted Peanuts with or without added oil Shortening is a semisolid Fat used in food preparation especially baked goods and is so called because it promotes a "short" or crumbly texture (as in Shortbread A non-Newtonian fluid is a Fluid whose flow properties are not described by a single constant value of Viscosity.

Viscosity of solids

On the basis that all solids flow to a small extent in response to shear stress some researchers^[19][20] have contended that substances known as amorphous solids, such as glass and many polymers, may be considered to have viscosity. A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material An amorphous solid is a Solid in which there is no Long-range order of the positions of the Atoms (Solids in which there is long-range atomic order are Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many A polymer is a large Molecule ( Macromolecule) composed of repeating Structural units typically connected by Covalent Chemical bonds This has led some to the view that solids are simply liquids with a very high viscosity, typically greater than 10¹² Pa·s. A solid' object is in the States of matter characterized by resistance to Deformation and changes of Volume. Liquid is one of the principal States of matter. A liquid is a Fluid that has the particles loose and can freely form a distinct surface at the boundaries of This position is often adopted by supporters of the widely held misconception that glass flow can be observed in old buildings. Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many This distortion is more likely the result of glass making process rather than the viscosity of glass. ^[21]

However, others argue that solids are, in general, elastic for small stresses while fluids are not. A solid' object is in the States of matter characterized by resistance to Deformation and changes of Volume. FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code ^[22] Even if solids flow at higher stresses, they are characterized by their low-stress behavior. A solid' object is in the States of matter characterized by resistance to Deformation and changes of Volume. Viscosity may be an appropriate characteristic for solids in a plastic regime. A solid' object is in the States of matter characterized by resistance to Deformation and changes of Volume. 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. A Maxwell material is a Viscoelastic material having the properties both of elasticity and Viscosity.

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. Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. In geology, earth materials that exhibit viscous deformation at least three times greater than their elastic deformation are sometimes called rheids. Geology (from Greek γη gê, "earth" and λόγος Logos, "speech" lit In Geology, a rheid is a Solid material that deforms by viscous flow

Viscosity of amorphous materials

Viscous flow in amorphous materials (e. An amorphous solid is a Solid in which there is no Long-range order of the positions of the Atoms (Solids in which there is long-range atomic order are g. in glasses and melts)^[24][25][26] is a thermally activated process:

where Q is activation energy, T is temperature, R is the molar gas constant and A is approximately a constant. Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many

The viscous flow in amorphous materials is characterized by a deviation from the Arrhenius-type behavior: Q changes from a high value Q_H at low temperatures (in the glassy state) to a low value Q_L at high temperatures (in the liquid state). The Arrhenius equation is a simple but remarkably accurate formula for the temperature dependence of the Rate constant, and therefore rate of a chemical reaction Depending on this change, amorphous materials are classified as either

• strong when: Q_H − Q_L < Q_L or
• fragile when:

The fragility of amorphous materials is numerically characterized by the Doremus’ fragility ratio:

R_D = Q_H / Q_L

and strong material have whereas fragile materials have

The viscosity of amorphous materials is quite exactly described by a two-exponential equation:

with constants A₁,A₂,B,C and D related to thermodynamic parameters of joining bonds of an amorphous material.

Not very far from the glass transition temperature, T_g, this equation can be approximated by a Vogel-Tammann-Fulcher (VTF) equation or a Kohlrausch-type stretched-exponential law. The glass transition temperature, T g is the temperature at which an Amorphous solid, such as Glass or a Polymer, becomes brittle The stretched Exponential function, also known in the field of dielectric relaxation as the Kohlrausch-Williams-Watts (KWW function is a frequently used Empirical

If the temperature is significantly lower than the glass transition temperature, T < T_g, then the two-exponential equation simplifies to an Arrhenius type equation:

with:

Q_H = H_d + H_m

where H_d is the enthalpy of formation of broken bonds (termed configurons) and H_m is the enthalpy of their motion. The standard enthalpy of formation or "standard heat of formation" of a compound is the change of Enthalpy that accompanies the formation of 1 mole of a According to the nature of the Chemical bonds which hold particles together solids can be classified as metals ionic solids molecular solids or covalent network solids In Thermodynamics and molecular chemistry, the enthalpy (denoted as H, h, or rarely as χ) is a quotient or description of When the temperature is less than the glass transition temperature, T < T_g, the activation energy of viscosity is high because the amorphous materials are in the glassy state and most of their joining bonds are intact.

If the temperature is highly above the glass transition temperature, T > T_g, the two-exponential equation also simplifies to an Arrhenius type equation:

with:

Q_L = H_m

When the temperature is higher than the glass transition temperature, T > T_g, the activation energy of viscosity is low because amorphous materials are melt and have most of their joining bonds broken which facilitates flow.

Volume (bulk) viscosity

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

,

which only depends upon the equilibrium state potentials like temperature and density (equation of state). In Linear algebra, the trace of an n -by- n Square matrix A is defined to be the sum of the elements on the Main diagonal Stress is a measure of the average amount of Force exerted per unit Area. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface In Physics and Thermodynamics, an equation of state is a relation between state variables More specifically an equation of state is a thermodynamic 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 volume viscosity. Volume viscosity (also called bulk viscosity or second viscosity) appears in the Navier-Stokes equation if it is written for Compressible fluid

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). In Fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic Stochastic property changes FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός Large eddy simulation (LES is a numerical technique used to solve the Partial differential equations governing turbulent fluid flow. 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. An ocean (from Greek, ''Okeanos'' (Oceanus) is a major body of saline water, and a principal component of the Hydrosphere.

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. A centimetre ( American spelling: centimeter, symbol cm) is a unit of Length in the Metric system, equal to one hundredth The second ( SI symbol s) sometimes abbreviated sec, is the name of a unit of Time, and is the International System of Units For other uses of the words gram or gramme see Gram (disambiguation. Fluidity is seldom used in engineering practice. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and

The concept of fluidity can be used to determine the viscosity of an ideal solution. In Chemistry, an ideal solution or ideal mixture is a Solution in which the Enthalpy of solution is zero the closer to zero the enthalpy of 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. In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the )

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. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant. 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. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually 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. In Linear algebra, a skew-symmetric (or antisymmetric) matrix is a Square matrix A whose Transpose is also its negative In Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T} The velocity field may be approximated as:

where Einstein notation is now being used in which repeated indices in a product are implicitly summed. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational The second term from the right 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. Isotropy is uniformity in all directions Precise definitions depend on the subject area 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. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two The most general linear relationship between the stress tensor and the rate-of-strain tensor is then a linear combination of these two tensors:^[27]

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. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface 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:

See also

• Deborah number
• Dilatant
• Hyperviscosity syndrome
• Rheology
• Thixotropy
• Viscometer
• Viscometry
• Viscoelasticity
• Viscosity index

References

Additional reading

• Massey, B. S. (1983). Mechanics of Fluids, Fifth Edition, Van Nostrand Reinhold (UK). ISBN 0-442-30552-4.  

External links

• SAE-ISO-AGMA comparison chart
• SAE J300 Motor Oil Viscosity Chart
• SAE J306 Automotive Gear Oil Viscosity Chart
• Gas Dynamics Toolbox Calculate coefficient of viscosity for mixtures of gases
• Physical Characteristics of Water A table of water viscosity as a function of temperature
• Glass Viscosity Measurement Viscosity measurement, viscosity units and fixpoints, glass viscosity calculation
• diracdelta.co.uk conversion between kinematic and dynamic viscosity.
• Vogel-Tammann-Fulcher Equation Parameters
• Dispersion Technology
• The effects of viscosity in a car engine

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