# Viscosity

File:Pitch drop experiment.jpg
The pitch drop experiment at the University of Queensland. The viscosity of pitch is 100 billion times that of water.
For the image-editing utility, see: Viscosity (software)

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.

## 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, 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.

File:Laminar shear.png
Laminar shear of fluid between two plates. Friction between the fluid and the moving boundaries cause the fluid to shear. The force required for this action is a measure of the fluid's viscosity.
File:Laminar shear flow.PNG
Laminar shear, the non-linear gradient is a result of the geometry the fluid is flowing through (a pipe).

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.

$\displaystyle \tau=\mu \frac{\partial u}{\partial y}$ .

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.

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:

$\displaystyle \nu = \frac {\mu} {\rho}$ .

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 the most of the fluids is a constant in a wide range of shear rates. The fluids without a constant viscosity are called Non-Newtonian fluids.

### Units

#### Viscosity (dynamic viscosity)

The SI physical unit of dynamic viscosity is the pascal-second (Pa·s), which is identical to 1 N·s/m2 or 1 kg/(m·s). 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·s) = 0.1 Pa·s.

1 centipoise = 1 mPa·s.

#### Kinematic viscosity

The SI physical unit of kinematic viscosity is the (m2/s). 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 = 0.0001 m2/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. 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 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:

Gases (at 0 °C):

 viscosity (Pa·s) hydrogen 8.4 × 10-6 air 17.4 × 10-6 xenon 21.2 × 10-6

Liquids (at 20 °C):

 viscosity (Pa·s) ethyl alcohol 0.248 × 10-3 acetone 0.326 × 10-3 methanol 0.59 × 10-3 benzene 0.64 × 10-3 water 1.0030 × 10-3 nitrobenzene 2.0 × 10-3 mercury 17.0 × 10-3 sulfuric acid 30 × 10-3 olive oil 81 × 10-3 castor oil 0.985 glycerol 1.485 molten polymers 103 pitch 107 glass 1040

Many fluids such as honey have a wide range of viscosities.

## Can solids have a viscosity?

It is commonly asserted that amorphous solids, such as glass, have viscosity, arguing on the basis that all solids flow, to some possibly minuscule extent, in response to shear stress. Advocates of such a view hold that the distinction between solids and liquids is unclear and 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 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. Typical values of eddy viscosity used in modeling ocean circulation are in excess of 107 Pa·s.

## Fluidity

The reciprocal of viscosity is fluidity, usually symbolised by φ (=1/μ), measured in reciprocal poise (cm·s/g), sometimes called the rhe. Fluidity is seldom used in engineering practice.

F = 1/η

Fluidity can be used to determine the viscosity of an ideal solution.

F ≈ [χ(a)F(a)] + [χ(b)F(b)] F ≈ 1/[χ(a)/η(a) +χ(b)/η(b)]

ln η = χ(a)*ln η(a) + χ(b)*ln η(b)

Where χ = mole fration of a/b and η = the viscosity coefficent of pure a/b

## Etymology

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