# Viscosity

Clear liquid above has lower viscosity than the substance below.

Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress. In everyday terms (and for fluids only), viscosity is "thickness". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity. Put simply, the less viscous the fluid is, the greater its ease of movement (fluidity).(1)

Viscosity describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction. For example, high-viscosity felsic magma will create a tall, steep stratovolcano, because it cannot flow far before it cools, while low-viscosity mafic lava will create a wide, shallow-sloped shield volcano. All real fluids (except superfluids) have some resistance to stress and therefore are viscous, but a fluid which has no resistance to shear stress is known as an ideal fluid or inviscid fluid.

The study of flowing matter is known as rheology, which includes viscosity and related concepts.

## Properties and behavior

### Overview

Laminar shear of fluid between two plates. Friction between the fluid and the moving boundaries causes the fluid to shear. The force required for this action is a measure of the fluid's viscosity. This type of full is known as a Couette flow.
Laminar shear, the non-constant gradient, is a result of the geometry the fluid is flowing through (e.g. a pipe).

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.

The relationship between the shear stress and the velocity gradient can be obtained by considering two plates closely spaced 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 at velocity u (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

$F=\mu A \frac{u}{y}$,

where μ is the proportionality factor called the viscosity.

This equation can be expressed in terms of shear stress $\tau=\frac{F}{A}$ Thus as expressed in differential form by Isaac Newton for straight, parallel and uniform flow, the shear stress between layers is proportional to the velocity gradient in the direction perpendicular to the layers:

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

Hence, through this method, the relation between the shear stress and the velocity gradient can be obtained.

Note that the rate of shear deformation is $\frac{u} {y}$ which can be also written as a shear velocity, $\frac{du} {dy}$.

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.

### Types of viscosity

Viscosity, the slope of each line, varies among materials

Newton's law of viscosity, given above, is a constitutive equation (like Hooke's law, Fick's law, Ohm's law). It is not a fundamental law of nature but an approximation that holds in some materials and fails in others. Non-Newtonian fluids exhibit a more complicated relationship between shear stress and velocity gradient than simple linearity. Thus there exist a number of forms of viscosity:

• Newtonian: fluids, such as water and most gases which have a constant viscosity.
• Shear thickening: viscosity increases with the rate of shear.
• Shear thinning: viscosity decreases with the rate of shear.
• Thixotropic: materials who become less viscous over time when shaken, agitated, or otherwise stressed.
• Rheopectic: materials who become more viscous over time when shaken, agitated, or otherwise stressed.
• A Bingham plastic is a material that behaves as a solid at low stresses but flows as a viscous fluid at high stresses.

## Viscosity coefficients

Viscosity coefficients can be defined in two ways:

• Dynamic viscosity, also absolute viscosity, the more usual one;
• Kinematic viscosity is the dynamic viscosity divided by the density.

Viscosity is a tensorial quantity that can be decomposed in different ways into two independent components. The most usual decomposition yields the following viscosity coefficients:

• Shear viscosity, the most important one, often referred to as simply viscosity, describing the reaction to applied shear stress; simply put, it 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).
• Volume viscosity or bulk viscosity, describes the reaction to compression, essential for acoustics in fluids, see Stokes' law (sound attenuation).

Alternatively,

• Extensional viscosity, a linear combination of shear and bulk viscosity, describes the reaction to elongation, widely used for characterizing polymers.

For example, at room temperature, water has a dynamic shear viscosity of about 1.0×10−3 Pa·s and motor oil of about 250×10−3 Pa·s.(3)

## Viscosity measurement

Viscosity is measured with various types of rheometers. 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 (centistokes, cSt) through the conversion equations.

A Ford viscosity cup measures the rate of flow of a liquid. This, under ideal conditions, is proportional to the kinematic viscosity.

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 such as the Dynatrol 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

### Dynamic viscosity

The usual symbol for dynamic viscosity used by mechanical and chemical engineers — as well as fluid dynamicists — is the Greek letter mu (μ)(4, 5, 6, 7). The symbol η is also used by chemists, physicists, and the IUPAC(8).

The SI physical unit of dynamic viscosity is the pascal-second (Pa·s), (equivalent to N·s/m2, or kg/ms). 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 cgs physical unit for dynamic viscosity is the poise(9) (P), named after Jean Louis Marie Poiseuille. It is more commonly expressed, particularly in ASTM standards, as centipoise (cP). Water at 20 °C has a viscosity of 1.0020 cP or 0.001002 kilogram/meter second.

1 P = 1 g·cm−1·s−1.

The relation to the SI unit is

1 P = 0.1 Pa·s,
1 cP = 1 mPa·s = 0.001 Pa·s.

### Kinematic viscosity

In many situations, we are concerned with the ratio of the viscous force to the inertial force (i.e. the Reynolds number, Re = VD / ν) , the latter characterised by the fluid density ρ. This ratio is characterised by the kinematic viscosity (Greek letter nu, ν), defined as follows:

$\nu = \frac {\mu} {\rho},$

The SI unit of ν is m2/s. The SI unit of ρ is kg/m3.

The cgs physical unit for kinematic viscosity is the stokes (St), named after George Gabriel Stokes. It is sometimes expressed in terms of centistokes (cSt or ctsk). In U.S. usage, stoke is sometimes used as the singular form.

1 St = 1 cm2·s−1 = 10−4 m2·s−1.
1 cSt = 1 mm2·s−1 = 10−6m2·s−1.

Water at 20 °C has a kinematic viscosity of about 1 cSt.

The kinematic viscosity is sometimes referred to as diffusivity of momentum, because it has the same unit as and is comparable to diffusivity of heat and diffusivity of mass. It is therefore used in dimensionless numbers which compare the ratio of the diffusivities.

## References

• (1) Symon, Keith (1971). Mechanics (Third ed.). Addison-Wesley. ISBN 0-201-07392-7.
• (2) The Online Etymology Dictionary
• (3) Raymond A. Serway (1996). Physics for Scientists & Engineers (4th ed.). Saunders College Publishing. ISBN 0-03-005932-1.
• (4) ASHRAE handbook, 1989 edition
• (5) Streeter & Wylie Fluid Mechanics, McGraw-Hill, 1981
• (6) Holman Heat Transfer, McGraw-Hill, 2002
• (7) Incropera & DeWitt, Fundamentals of Heat and Mass Transfer, Wiley, 1996
• (8) IUPAC Gold Book, Definition of (dynamic) viscosity
• (9) IUPAC definition of the Poise

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