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## Viscosity
## Additional recommended knowledge
## EtymologyThe 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. ## Viscosity coefficientsWhen 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 **Dynamic viscosity**is the viscosity coefficient that determines the dynamics of incompressible Newtonian fluid;**Kinematic viscosity**is the*dynamic viscosity*divided by the density for Newtonian fluid;**Volume viscosity**is the viscosity coefficient that determines the dynamics of compressible Newtonian fluid;**Bulk viscosity**is the same as*volume viscosity***Shear viscosity**is the viscosity coefficient when applied stress is a shear stress, valid for non-Newtonian fluids;**Extensional viscosity**is the viscosity coefficient when applied stress is extensional stress; valid for non-Newtonian fluids.
*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. Isaac Newton postulated that, for straight, parallel and uniform flow, the shear stress, τ, between layers is proportional to the velocity gradient, ∂ - .
Here, the constant η is known as the The relationship between the shear stress and the velocity gradient can also be obtained by considering two plates closely spaced apart at a distance James Clerk Maxwell called viscosity ## Viscosity MeasurementDynamic 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 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. The fluids without a constant viscosity are called Non-Newtonian fluids. 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. 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.
Volume viscosity can be measured with acoustic rheometer. ## 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 SI physical unit of dynamic viscosity is the pascal-second (Pa·s), which is identical to 1 kg·m 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 - 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 viscosityIn 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 - .
where μ is the (dynamic) viscosity, and ρ is the density. Kinematic viscosity (Greek symbol: ν) has SI units (m - 1 stokes = 100 centistokes = 1 cm
^{2}·s^{−1}= 0.0001 m^{2}·s^{−1}. - 1 centistokes = 1 mm
^{2}/s
## Dynamic versus kinematic viscosityConversion between kinematic and dynamic viscosity, is given by νρ = η. For example, - if ν = 0.0001 m
^{2}·s^{-1}and ρ = 1000 kg m^{-3}then η = νρ = 0.1 kg·m^{−1}·s^{−1}= 0.1 Pa·s - if ν = 1 St (= 1 cm
^{2}·s^{−1}) and ρ = 1 g cm^{-3}then η = 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. ## Example: viscosity of waterBecause of its density of ρ = 1 g/cm 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 originsThe 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. ## GasesViscosity 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. Within the regime where the theory is applicable: - Viscosity is independent of pressure and
- Viscosity increases as temperature increases.
## Effect of temperature on the viscosity of a gasSutherland'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* - η
_{0}= reference viscosity in (Pa·s) at reference temperature*T*_{0} *T*= input temperature in kelvin*T*_{0}= reference temperature in kelvin*C*= Sutherland's constant for the gasous material in question
Valid for temperatures between 0 < Sutherland's constant and reference temperature for some gases
## Viscosity of a dilute gasThe Chapman-Enskog equation - ; T*=κT/ε
- η
_{0}= viscosity for dilute gas (uP) *M*= molecular weight (kg/m^3)*T*= temperature (K)- σ = the collision diameter (Å)
- ε / κ = the maximum energy of attraction divided by the Boltzman constant (K)
- ω
_{η}= the collision integral *T** = reduced temperature (K)
## LiquidsIn 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. - 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 blends of liquidsThe viscosity of the blend of two or more liquids can be estimated using the Refutas equation 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 The next step is to calculate the VBN of the blend, using this equation: - (2)
where Once the viscosity blending number of a blend has been calculated using equation (2), the final step is to determine the viscosity of the blend by solving equation (1) for - (3)
where ## Viscosity of materialsThe 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 airThe viscosity of air depends mostly on the temperature.
At 15.0 °C, the viscosity of air is 1.78 × 10 ## Viscosity of waterThe viscosity of water is 8.90 × 10 ## Viscosity of various materials
Some dynamic viscosities of Newtonian fluids are listed below:
* Data from CRC Handbook of Chemistry and Physics, 73 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:
* These materials are highly non-Newtonian. ## Viscosity of solidsOn the basis that all solids flow to a small extent in response to shear stress some researchers However, others argue that solids are, in general, elastic for small stresses while fluids are not. 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 ## Viscosity of amorphous materials
Viscous flow in amorphous materials (e.g. in glasses and melts)
where The viscous flow in amorphous materials is characterised by a deviation from the Arrhenius-type behaviour: - strong when:
*Q*_{H}−*Q*_{L}<*Q*_{L}or - fragile when:
The fragility of amorphous materials is numerically characterized by the Doremus’ fragility ratio:
and strong material have whereas fragile materials have The viscosity of amorphous materials is quite exactly described by a two-exponential equation:
with constants Not very far from the glass transition temperature, If the temperature is significantly lower than the glass transition temperature, , then the two-exponential equation simplifies to an Arrhenius type equation:
with:
where When the temperature is less than the glass transition temperature, If the temperature is highly above the glass transition temperature, , the two-exponential equation also simplifies to an Arrhenius type equation:
with:
When the temperature is higher than the glass transition temperature, ## Volume (Bulk) viscosityThe 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 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. ## Eddy viscosityIn the study of turbulence in fluids, a common practical strategy for calculation is to ignore the small-scale ## FluidityThe reciprocal of viscosity is The concept of fluidity can be used to determine the viscosity of an ideal solution. For two components which is only slightly simpler than the equivalent equation in terms of viscosity: where χ ## The linear viscous stress tensor(See 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 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 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.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 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 ( − The infinitesimal force ## See also- Deborah number
- Dilatant
- Hyperviscosity syndrome
- Rheology
- Thixotropy
- Viscometer
- Viscometry
- Viscosity index
## References**^**Symon, Keith (1971).*Mechanics*, Third Edition, Addison-Wesley.__ISBN 0-201-07392-7__.**^**The Online Etymology Dictionary**^**Happel, J. and Brenner , H. "Low Reynolds number hydrodynamics",*Prentice-Hall*, (1965)**^**Landau, L.D. and Lifshitz, E.M. "Fluid mechanics",*Pergamon Press*,(1959)**^**Barnes, H.A. "A Handbook of Elementary Rheology", Institute of Non-Newtonian Fluid mechanics, UK (2000)**^**Raymond A. Serway (1996).*Physics for Scientists & Engineers*, 4th Edition, Saunders College Publishing.__ISBN 0-03-005932-1__.**^**Dukhin, A.S. and Goetz, P.J. "Ultrasound for characterizing colloids", Elsevier, (2002)**^**IUPAC definition of the Poise**^**James Ierardi's Fire Protection Engineering Site**^**J.O. Hirshfelder, C.F. Curtis and R.B. Bird (1964).*Molecular theory of gases and liquids*, First Edition, Wiley.__ISBN 0-471-40065-3__.**^**Robert E. Maples (2000).*Petroleum Refinery Process Economics*, 2nd Edition, Pennwell Books.__ISBN 0-87814-779-9__.**^**C.T. Baird (1989),*Guide to Petroleum Product Blending*, HPI Consultants, Inc. HPI website**^**Chocolate Processing.*Brookfield Engineering website*. Retrieved on 2007-12-03.**^**Elert, Glenn. Viscosity.*The Physics Hypertextbook*.**^**The Properties of Glass , page 6, retrieved on August 1, 2007**^**"Antique windowpanes and the flow of supercooled liquids", by Robert C. Plumb, (Worcester Polytech. Inst., Worcester, MA, 01609, USA), J. Chem. Educ. (1989), 66 (12), 994-6**^**Gibbs, Philip. Is Glass a Liquid or a Solid?. Retrieved on 2007-07-31.**^**Viscosity calculation of glasses**^**R.H.Doremus (2002). "Viscosity of silica".*J. Appl. Phys.***92**(12): 7619-7629. ISSN 0021-8979.**^**M.I. Ojovan and W.E. Lee (2004). "Viscosity of network liquids within Doremus approach".*J. Appl. Phys.***95**(7): 3803-3810. ISSN 0021-8979.**^**M.I. Ojovan, K.P. Travis and R.J. Hand (2000). "Thermodynamic parameters of bonds in glassy materials from viscosity-temperature relationships".*J. Phys.: Condensed matter***19**(41): 415107. ISSN 0953-8984.**^**L.D. Landau and E.M. Lifshitz (translated from Russian by J.B. Sykes and W.H. Reid) (1997).*Fluid Mechanics*, 2nd Edition, Butterworth Heinemann.__ISBN 0-7506-2767-0__.
## Additional reading- Massey, B. S. (1983).
*Mechanics of Fluids*, Fifth Edition, Van Nostrand Reinhold (UK).__ISBN 0-442-30552-4__.
Categories: Continuum mechanics | Viscosity |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Viscosity". A list of authors is available in Wikipedia. |