Viscosity
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Image:Pitch drop experiment.jpg
Viscosity is a measure of the resistance of a fluid to deformation under shear stress. It is commonly perceived as "thickness", or resistance to pouring. 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 low viscosity, while vegetable oil is "thick" having a high viscosity.
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Newton's theory
When a shear stress is applied to a solid body, the body deforms until the deformation results in an opposing force to balance that applied; forming an equilibrium. However, when a shear stress is applied to a fluid, such as a wind blowing over the surface of the ocean, the fluid flows, and continues to flow while the stress is applied. When the stress is removed, in general, the flow decays due to internal dissipation of energy. The "thicker" the fluid, the greater its resistance to shear stress and the more rapid the decay of its flow.
In general, in any flow, layers move at different velocities and the fluid's "thickness" arises from the shear stress between the layers that ultimately opposes any applied force.
Image:Laminar shear.png
Image:Laminar shear flow.PNG
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.
- <math>\tau=\mu \frac{\partial u}{\partial y}</math>.
Here, the constant μ is known as the coefficient of viscosity, viscosity, or 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 t. Assuming that the plates are very large, with a large area A, such that edge effects are neglected and that the lower plate is fixed, let a force F be applied to the upper plate. Incidentely, if this force causes the plate to move, the substance is concluded to be a fluid. The velocity of the moving plate and the top of the fluid must have the same velocity U. Now, by experimentation, 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 = mu(AU/t). Where mu is the proportionality factor called the absoulte viscosity (with units Pa-s or slugs/s-ft). The equation can be expressed in terms of shear stress; rho = F/A = mu(U/t). U/t is the rate of angular deformation and can be written as an angular 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:
- <math>\nu = \frac {\mu} {\rho}</math>.
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.
Viscosity is the principal means by which energy is dissipated in fluid motion, typically as heat.
Measurement of viscosity
Viscosity is measured with various types of viscometer, typically at 25°C (standard state). 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.
Units
Viscosity (dynamic viscosity): <math>{\mu}</math>
The SI physical unit of dynamic viscosity (greek symbol: <math>{\mu}</math>) is the pascal-second (Pa·s), which is identical to 1 kg·m-1·s-1. In France there have been some attempts to establish the poiseuille (Pl) as a name for the Pa·s but without international success. 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 (P) 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 poise = 100 centipoise = 1 g·cm−1·s−1 = 0.1 Pa·s.
- 1 centipoise = 1 mPa·s.
Kinematic viscosity: <math>\nu = {\mu} / {\rho}</math>
Kinematic viscosity (Greek symbol: <math>{\nu}</math>) 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.
Conversion between kinematic and dynamic viscosity, then, is given by <math>\nu \rho = \mu</math>, and so if ν=1 St then
- μ=νρ=0.1 kg·m−1s−1·(ρ/(g/cm3))=0.1 poise·(ρ/(g/cm3)). [1]
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. 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. 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 some common materials
Some dynamic viscosities of Newtonian fluids are listed below:
| viscosity (Pa·s) | |
|---|---|
| hydrogen | 8.4 × 10-6 |
| air | 17.4 × 10-6 |
| xenon | 21.2 × 10-6 |
| viscosity (Pa·s) | |
|---|---|
| ethanol | a 1.074 × 10-3 |
| acetone | a 0.306 × 10-3 |
| methanol | a 0.544 × 10-3 |
| propanol | a 1.945 × 10-3 |
| benzene | a 0.604 × 10-3 |
| water | a 0.890 × 10-3 |
| nitrobenzene | a 1.863 × 10-3 |
| mercury | a 1.526 × 10-3 |
| sulfuric acid | a 24.2 × 10-3 |
| glycerol | a 934 × 10-3 |
| olive oil | 81 × 10-3 |
| castor oil | 0.985 |
| molten polymers | 103 |
| pitch | 1011 |
| glass | 1040 |
a Data from CRC Handbook of Chemistry and Physics, 73rd 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
Can solids have a viscosity?
Amorphous solids, such as glass, may be considered to have viscosity, on the basis that all solids flow, to some small extent, in response to shear stress. 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 idea that apparent glass flow can be observed in old buildings.
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 of one third of the thermodynamic pressure, which only depends upon the equilibrium state potentials like temperature and density. However, 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. 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 symbolised by φ (=1/μ) 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 of a solution of a and b is:
F ≈ [χ(a)F(a)] + [χ(b)F(b)]
which is only slightly simpler than the equivalent equation in terms of viscosity:
η ≈ 1/[χ(a)/η(a) +χ(b)/η(b)]
Where χ = mole fraction of a or b and η = the viscosity of pure a or b
Etymology
The word "viscosity" derives from the Latin word "Template:Lang" for mistletoe. A viscous glue was made from mistletoe berries and used for lime-twigs to catch birds.
See also
External links
- Online Dynamic Viscosity Converter - convert between various units of dynamic viscosity, such as Ppascal second, kilogram-force second per square meter, pound-force second per square inch, poise, and so on
- Interactive Dynamic Viscosity Conversion Table - convert selected unit to all other units of dynamic viscosity
- Online Kinematic Viscosity Converter - convert between various units of kinematic viscosity, such as square meter per second, square foot per second ,stokes, and so on
- Interactive Kinematic Viscosity Conversion Table - convert selected unit to all other units of kinematic viscosity
- Gas Dynamics Toolbox Calculate coefficient of viscosity for mixtures of gases using VHS model
- Viscosity Page A table of items sorted by viscosity in centipoise (cP)
Bibliography
- Massey, B S (1983) Mechanics of Fluids, fifth edition, ISBN 0442305524bg:Вискозитет
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