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Viscosity is a property of a fluid that characterises its perceived "thickness" or resistance to pouring. It describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction. Thus, methanol 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.

Figure to follow

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 inertial to viscous forces, the latter characterised by the fluid density ρ. This ratio is characterised by the kinematic viscosity:

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

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 principle 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).

Units

Dynamic viscosity

The SI physical unit of dynamic viscosity if the Pascal-second (Pa·s), which is identical to 1 N·s/m 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!

The cgs physical unit for dynamic viscosity is the poise (P) named for Jean Louis Marie Poiseuille. It is more commonly expressed, particularly in ASTM standards, as centipoise (cP).

1 poise = 100 centipoise = 1 g/cm·s = 0.1 Pa·s.

Kinematic viscosity

The SI physical unit of kinematic viscosity is the (m/s). The cgs physical unit for kinematic viscosity is the stokes (abbreviated S or St), named for George Gabriel Stokes . It is sometimes expressed in terms of centistokes (cS). US usage is the stoke.

1 stokes = 100 centistokes = 1 cm/s = 0.0001 m/s.

Molecular origins

It seems natural to see the origin of viscosity in terms of the attractive and repulsive forces between molecules. However, gases have substantial viscosity even though their inter-molecular forces are weak suggesting some other mechanism.

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
• Viscosty increases with temperature.

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 with temperature.

Viscosity of some common materials

Some dynamic viscosities of Newtonian fluids are listed below:

Gases (at 0 °C):

Liquids (at 20 °C):

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 miniscule extent, in response to shear stress. Advocates of such a view hold that the distinction bewteen solids and liquids is unlcear and that solids are simply liquids with a very high viscosity, typically greater than 10 Pa·s. This position is often adopted by supporters of the widely held urban myth 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 characterised by their low-stress behaviour. Viscosity may be an appropriate characteristic for solids in a plastic regime.

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 characterises the transport and dissipation of energy in the smaller-scale flow. Typical values of eddy viscosity used are in excess of 10 Pa·s.

Bibliography

• Massey, B S (1983) Mechanics of Fluids, fifth edition, ISBN 0442305524

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