Encyclopedia
Viscosity is a measure of the resistance of a fluid to deform under
shear stress. It is commonly perceived as "thickness", or resistance to pouring. Viscosity describes the property of a fluid that resists the force,friction, causing the fluid to flow . Thus,
water is "thin", having a lower viscosity, while vegetable oil is "thick" having a higher viscosity. All real fluids have some resistance to shear stress, but an idealized fluid which has no resistance to shear stress is known as an
ideal fluid .
Newton's theory
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.
Isaac Newton postulated that, for straight, parallel and uniform flow, the shear stress, t, 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 µ 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 , 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 = µ, where µ is the proportionality factor called the
absolute viscosity . The equation can be expressed in terms of shear stress; t = F/A = µ. u/y is the rate of shear deformation and can be 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 ?. 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.
Measurement of viscosity
Viscosity is measured with various types of viscometer, typically at 25 °C . 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 through conversion equations.
Also used in paint, a Stormer viscometer uses load-based rotation in order to determine viscosity. It uses units, Krebs units , unique to this viscometer.
Units
Viscosity :
The SI
physical unit of dynamic viscosity is the pascal-second , which is identical to 1
kg·m
-1·s
-1. In
France there have been some attempts to establish the
poiseuille 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 named after Jean Louis Marie Poiseuille. It is more commonly expressed, particularly in ASTM standards, as
centipoise . The centipoise is commonly used because water has a viscosity of 1.0020 cP .
- 1 poise = 100 centipoise = 1 g·cm-1·s-1 = 0.1 Pa·s.
- 1 centipoise = 0.001 Pa·s.
Kinematic viscosity:
Kinematic viscosity has SI units . The cgs physical unit for kinematic viscosity is the
stokes , named after
George Gabriel Stokes . It is sometimes expressed in terms of
centistokes . 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 , and so if ?=1 St then
- µ=??=0.1 kg·m-1s-1·=0.1 poise·.
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 .
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 ; and
- Viscosity tends to fall as temperature increases ; 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
Viscosity of water
The viscosity of water is 0.890 × 10
-3 Pa•s at about 25°C.
Viscosity of various materials
Some dynamic viscosities of Newtonian fluids are listed below:
Gases :
Liquids :
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
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 10
12 Pa·s. This position is often adopted by supporters of the widely held misconception that
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 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 5x10
4 to 10
6 Pa·s depending upon the resolution of the numerical grid.
Fluidity
The reciprocal of viscosity is
fluidity, usually symbolised by f or F , depending on the convention used, measured in
reciprocal poise , 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 , the fluidity of a solution of a and b is:
F ˜ [?F] + [?F]
which is only slightly simpler than the equivalent equation in terms of viscosity:
? ˜ 1/[?/? +?/?]
Where ? = mole fraction of a or b
and ? = the viscosity of pure a or b
The linear viscous stress tensor
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 breakdown 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 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 , then the most general way that the symmetric term can be broken down in a coordinate-independent way is as the sum of a constant tensor and a traceless symmetric tensor :
where 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 :
where is the coefficient of bulk viscosity and is the coefficient of 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 must be added to the viscous stress tensor to obtain the total stress tensor for the fluid.
The infinitesimal force on an infinitesimal area is then given by the usual relationship:
Etymology
The word "viscosity" derives from the
Latin word "" for
mistletoe. A viscous glue was made from mistletoe berries and used for lime-twigs to catch birds.
See also
- Deborah number
- Dilatant
- Thixotropy
- Viscosity index
- Hyperviscosity syndrome
References
External links
- Calculate coefficient of viscosity for mixtures of gases using VHS model
- A table of items sorted by viscosity in centipoise
- A table of water viscosity as a function of temperature
