# Dictionary Definition

viscous adj

1 having a relatively high resistance to flow
[syn: syrupy]

# User Contributed Dictionary

## English

### Etymology

First attested in 1605. From viscous and viscosus, from viscum.### Pronunciation

- /ˈvɪs.kəs/, /"vIs.k@s/
- Rhymes with: -ɪskəs

### Adjective

- Having a thick, sticky consistency between solid and liquid; having a high viscosity.

#### Synonyms

#### Related terms

#### Translations

having a thick, sticky consistency

# Extensive Definition

Viscosity is a measure of the resistance
of a fluid which is being
deformed by either shear stress
or extensional
stress. It is commonly perceived as "thickness", or resistance
to flow. 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 lower
viscosity, while vegetable
oil is "thick" having a higher viscosity. All real fluids
(except superfluids)
have some resistance to stress,
but a fluid which has no resistance to shear stress is known as an
ideal fluid or inviscid fluid. For example a high viscosity magma
will create a tall volcano, because it cannot spread fast enough,
low viscosity lava will create a shield volcano, which is large and
wide. The study of viscosity is known as rheology.

## 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.

## Viscosity coefficients

When 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 and rheology.- Dynamic viscosity determines the dynamics of an incompressible Newtonian fluid;
- Kinematic viscosity is the dynamic viscosity divided by the density for a Newtonian fluid;
- Volume viscosity determines the dynamics of a compressible Newtonian fluid;
- Bulk viscosity is the same as volume viscosity
- Shear viscosity is the viscosity coefficient when the applied stress is a shear stress (valid for non-Newtonian fluids);
- Extensional viscosity is the viscosity coefficient when the applied stress is an extensional stress (valid for non-Newtonian fluids).

Shear viscosity and dynamic viscosity are much
better known than the others. That is why they are often referred
to as simply viscosity. Simply put, this quantity is the ratio
between the pressure exerted on the surface of a fluid, in the
lateral or horizontal direction, to the change in velocity of the
fluid as you move down in the fluid (this is what is referred to as
a velocity gradient).
For example, at room temperature, water has a nominal viscosity of
1.0 × 10-3 Pa∙s and motor oil has a nominal apparent viscosity of
250 × 10-3 Pa∙s.

- Extensional viscosity is widely used for characterizing polymers.
- Volume viscosity is essential for Acoustics in fluids, see Stokes' law (sound attenuation)

## 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, ∂u/∂y, in the
direction perpendicular to the
layers.

- \tau=\eta \frac.

Here, the constant η is known as the coefficient
of viscosity, the viscosity, the dynamic viscosity, or the
Newtonian 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 (as
opposed to just shearing
elastically until the shear stress in the substance balances
the applied force), 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 =
η(Au/y), where η is the proportionality factor called the absolute
viscosity (with units Pa·s = kg/(m·s) or slugs/(ft·s)). The
absolute viscosity is also known as the dynamic viscosity, and is
often shortened to simply viscosity. The equation can be expressed
in terms of shear stress; τ = F/A = η(u/y). The rate of shear
deformation is u/y and can be also written as a shear velocity,
du/dy. Hence, through this method, the relation between the shear
stress and the velocity gradient can be obtained.

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 measurement

Dynamic 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.

Extensional viscosity can be measured with
various rheometers
that apply extensional
stress

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 (), and dynamic viscosity is also
commonly referred to using the Greek symbol mu (). The SI physical
unit of dynamic viscosity is the pascal-second
(Pa·s), which is identical to kg·m−1·s−1. If a fluid with a viscosity of one Pa·s
is placed between two plates, and one plate is pushed sideways with
a shear
stress of one pascal, it
moves a distance equal to the thickness of the layer between the
plates in one second.

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 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 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 viscosity

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 (\nu ), defined as
follows:

- \nu = \frac .

where \mu is the (dynamic or absolute) viscosity
(in centipoise cP), and \rho is the density (in grams/cm^3), and
\nu is the kinematic viscosity (in centistokes cSt ).

Kinematic viscosity (Greek symbol: ) has SI units
Pa.s/(kg/m3) = 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.
- 1 centistokes = 1 mm2·s-1 = 10-6m2·s−1

#### Saybolt Universal Viscosity

At one time the petroleum industry relied on
measuring kinematic viscosity by means of the Saybolt viscometer,
and expressing kinematic viscosity in units of Saybolt Universal
Seconds (SUS). Kinematic viscosity in centistoke can be converted
from SUS according to the arithmetic and the reference tabel
provided in ASTM D 2161. It can
also be converted in computerized method, or vice versa.

#### Relation to Mean Free Path of Diffusing Particles

In relation to diffusion, the kinematic viscosity provides a better understanding of the behavior of mass transport of a dilute species. Viscosity is related to shear stress and the rate of shear in a fluid, which illustrates its dependence on the mean free path, \lambda , of the diffusing particles.From fluid
mechanics, shear
stress, \tau , is the rate of change of velocity with distance
perpendicular to the direction of movement.

- \tau = \mu \frac.

Interpreting shear stress as the time rate of
change of momentum,p,
per unit area (rate of momentum flux) of an arbitrary control
surface gives

- \tau = \frac = \frac.

Further manipulation will show

- \frac = \dot = \rho \bar A \; \; \Rightarrow \; \; \tau = \underbrace_ \cdot \frac \; \; \Rightarrow \; \; \nu = \frac = 2 \bar \lambda

where

- \dot is the rate of change of mass
- \rho is the density of the fluid
- \bar is the average molecular speed
- \mu is the dynamic viscosity.

#### Dynamic versus kinematic viscosity

Conversion between kinematic and dynamic viscosity is given by \nu \rho = \mu.For example,

- if \nu = 0.0001 m2·s-1 and \rho = 1000 kg m-3 then \mu = \nu \rho = 0.1 kg·m−1·s−1 = 0.1 Pa·s
- if \nu = 1 St (= 1 cm2·s−1) and \rho = 1 g cm-3 then \mu = \nu \rho = 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 water

Because of its density of \rho = 1 g/cm3 (varies slightly with temperature), and its dynamic viscosity is near 1 mPa·s (see #Viscosity of water section), the viscosity values of water are, to rough precision, all powers of ten:Dynamic viscosity:

- = 1 mPa·s = 10-3 Pa·s = 1 cP = 10-2 poise

Kinematic viscosity:

- = 1 cSt = 10-2 stokes = 1 mm²/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
behavior of gaseous viscosity.

Within the regime where the theory is applicable:

- Viscosity is independent of pressure and
- Viscosity increases as temperature increases.

James
Clerk Maxwell published a famous paper in 1866 using the
kinetic theory of gases to study gaseous viscosity. (Reference:
J.C. Maxwell, "On the viscosity or internal friction of air and
other gases", Philosophical Transactions of the Royal Society of
London, vol. 156 (1866), pp. 249-268.)

#### Effect of temperature on the viscosity of a gas

Sutherland's
formula can be used to derive the dynamic viscosity of an
ideal
gas as a function of the temperature:

- = _0 \frac \left (\frac \right )^

where:

- = viscosity in (Pa·s) at input temperature T
- _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 gaseous material in question

Valid for temperatures between 0 T < 555 K
with an error due to pressure less than 10% below 3.45 MPa

Sutherland's constant and reference temperature
for some gases (also see: )

#### Viscosity of a dilute gas

The Chapman-Enskog
equation may be used to estimate viscosity for a dilute gas.
This equation is based on semi-theorethical assumption by Chapman
and Enskoq. The equation requires three empirically determined
parameters: the collision diameter (σ), the maximum energy of
attraction divided by the Boltzmann
constant (є/к) and the collision integral (ω(T*)).

- _0 \times 10^7 = \frac

- T*=κT/ε Reduced temperature (dimensionless)
- _0 = viscosity for dilute gas (uP)
- M = molecular mass (g/mol)
- T = temperature (K)
- = the collision diameter (Å)
- / = the maximum energy of attraction divided by the Boltzmann constant (K)
- _ = the collision integral

### 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 blends of liquids

The viscosity of the blend of two or more liquids can be estimated using the Refutas equation. The calculation is carried out in three steps.The first step is to calculate the Viscosity
Blending Number (VBN) (also called the Viscosity Blending Index) of
each component of the blend:

- (1) \mbox = 14.534 \times ln[ln(v + 0.8)] + 10.975\,

where v is the kinematic viscosity in centistokes
(cSt). It is important that the kinematic viscosity of each
component of the blend be obtained at the same temperature.

The next step is to calculate the VBN of the
blend, using this equation:

- (2) \mbox_\mbox = [x_A \times \mbox_A] + [x_B \times \mbox_B] + ... + [x_N \times \mbox_N]\,

where x_X is the mass
fraction of each component of the blend.

Once the viscosity blending number of a blend has
been calculated using equation (2), the final step is to determine
the kinematic viscosity of the blend by solving equation (1) for
v:

- (3) v = e^ - 0.8

where VBN_ is the viscosity blending number of
the blend.

## Viscosity of selected substances

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) or 1.78 × 10−4 P. One can get the viscosity of air as a function of temperature from the Gas Viscosity Calculator### Viscosity of water

The viscosity of water is 8.90 × 10−4 Pa·s or 8.90 × 10−3 dyn·s/cm2 or 0.890 cP at about 25 °C. As a function of temperature T (K): μ(Pa·s) = A × 10B/(T−C) where A=2.414 × 10−5 Pa·s ; B = 247.8 K ; and C = 140 K.Viscosity of water at different temperatures is
listed below.

### Viscosity of various materials

Some dynamic viscosities of Newtonian fluids are
listed below:

- Data from CRC Handbook of Chemistry and Physics, 73rd edition, 1992-1993.

A more complete table can be found at Transwiki,
including the following:

- These materials are highly non-Newtonian.

## Viscosity of solids

On the basis that all solids flow to a small
extent in response to shear stress
some researchers have contended that substances known as amorphous
solids, such as glass
and many polymers, may
be considered to have viscosity. 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 misconception that
glass flow can be observed in old buildings. This distortion is
more likely the result of glass making process rather than the
viscosity of glass.

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.

## Viscosity of amorphous materials

Viscous flow in amorphous
materials (e.g. in glasses and melts) is a thermally
activated process:

\eta = A \cdot e^

where Q is activation energy, T is temperature, R
is the molar gas constant and A is approximately a constant.

The viscous flow in amorphous materials is
characterized by a deviation from the Arrhenius-type
behavior: Q changes from a high value Q_H at low temperatures (in
the glassy state) to a low value Q_L at high temperatures (in the
liquid state). Depending on this change, amorphous materials are
classified as either

- strong when: Q_H - Q_L or
- fragile when: Q_H - Q_L \ge Q_L

The fragility of amorphous materials is
numerically characterized by the Doremus’ fragility ratio:

R_D = Q_H/Q_L

and strong material have R_D whereas fragile
materials have R_D \ge 2

The viscosity of amorphous materials is quite
exactly described by a two-exponential equation:

\eta = A_1 \cdot T \cdot [1 + A_2 \cdot e^] \cdot
[1 + C \cdot e^]

with constants A_1 , A_2 , B, C and D related to
thermodynamic parameters of joining bonds of an amorphous
material.

Not very far from the
glass transition temperature, T_g, this equation can be
approximated by a
Vogel-Tammann-Fulcher (VTF) equation or a
Kohlrausch-type stretched-exponential law.

If the temperature is significantly lower than
the glass transition temperature, T , then the two-exponential
equation simplifies to an Arrhenius type equation:

\eta = A_LT \cdot e^

with:

Q_H = H_d + H_m

where H_d is the enthalpy
of formation of broken bonds (termed configurons) and H_m is the
enthalpy of their
motion. When the temperature is less than the glass transition
temperature, T , the activation energy of viscosity is high because
the amorphous materials are in the glassy state and most of their
joining bonds are intact.

If the temperature is highly above the glass
transition temperature, T > T_g, the two-exponential equation
also simplifies to an Arrhenius type equation:

\eta = A_HT\cdot e^

with:

Q_L = H_m

When the temperature is higher than the glass
transition temperature, T > T_g, the activation energy of
viscosity is low because amorphous materials are melt and have most
of their joining bonds broken which facilitates flow.

## Volume (bulk) viscosity

The negative-one-third of the trace of the stress tensor is often identified with the thermodynamic pressure,-T_a^a = p,

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 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 (see large
eddy simulation). 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 symbolized by \phi = 1/\eta or F=1/\eta,
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 when a and b
are mixed is

- F \approx \chi_a F_a + \chi_b F_b

which is only slightly simpler than the
equivalent equation in terms of viscosity:

- \eta \approx \frac

where \chi_a and \chi_b is the mole fraction of
component a and b respectively, and \eta_a and \eta_b are the
components pure viscosities.

## The linear viscous stress tensor

(See Hooke's law
and strain
tensor for an analogous development for linearly elastic
materials.)

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 \mathbf is specified by the velocity field
\mathbf(\mathbf). The velocity at a small distance d\mathbf from
point \mathbf may be written as a Taylor
series:

- \mathbf(\mathbf+d\mathbf) = \mathbf(\mathbf)+\fracd\mathbf+\ldots

where \frac is shorthand for the dyadic product
of the del operator and the velocity:

\frac = \begin \frac & \frac & \frac\\
\frac & \frac & \frac\\ \frac & \frac&\frac
\end

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
\mathbf are a function of \mathbf(\mathbf) and all derivatives of
\mathbf(\mathbf) 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
\partial_i v_j where \partial_i is shorthand for \partial /\partial
x_i. 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 decomposition is independent of coordinate
system, and so has physical significance. The velocity field may be
approximated as:

- v_i(\mathbf+d\mathbf) = v_i(\mathbf)+\frac\left(\partial_i v_j-\partial_j v_i\right)dr_i + \frac\left(\partial_i v_j+\partial_j v_i\right)dr_i

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 \mathbf with angular velocity
\omega where:

- \omega=\frac12 \mathbf\times \mathbf=\frac\begin

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):

\frac\left(\partial_i v_j+\partial_j v_i\right) =
\underbrace_ + \underbrace_

where \delta_ is the unit
tensor. The most general linear relationship between the stress
tensor \mathbf and the rate-of-strain tensor is then a linear
combination of these two tensors:

- \sigma_ = \zeta\partial_k v_k \delta_+

where \zeta is the coefficient of bulk viscosity
(or "second viscosity") and \eta 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 (-p\delta_) must be added to the viscous stress
tensor to obtain the total stress tensor for the fluid.

- \sigma_ = -p\delta_+\sigma_\,

The infinitesimal force dF_i on an infinitesimal
area dA_i is then given by the usual relationship:

- dF_i=\sigma_dA_j\,