## Rheology

**Introduction**

Rheology is the science of flow and deformation
of matter and describes the interrelation between force,
deformation and time. The term comes from Greek *rheos *meaning
to flow. Rheology is applicable to all materials, from gases to
solids.

The science of rheology is only about 90 years of age. It was founded by two scientists meeting in the late '20s and finding out having the same need for describing fluid flow properties. The scientists were Professor Marcus Reiner and Professor Eugene Bingham.

The Greek philosopher Heraclitus described
rheology as *panta rei* - *everything flows*.
Translated into rheological terms by Marcus Reiner this means
everything will flow if you just wait long enough.

Fluid rheology is used to describe the consistency of different products, normally by the two components viscosity and elasticity. By viscosity is usually meant resistance to flow or thickness and by elasticity usually stickiness or structure.

Fluids are normally divided into three different groups according to their flow behaviour: Flow curves are normally use for the graphical description of flow behaviour. They are either manually drawn or, being standard today, produced by software coming with the measuring instrument.

**Kinematic and dynamic viscosity**

Dynamic viscosity takes into account the effect of shear rate and time and is therefore the only type of viscosity relevant for non-Newtonian design purposes. Dynamic viscosity is measured with dynamic instruments, either rotating (shearing) or oscillating.

An instrument only capable of measuring shearing viscosities is called a

*viscometer*and the oscillating type is called a

*rheometer*. Various models for approximation of rheological data have been presented. One of the most widely spread models is the so-called power law for approximation of viscosity data. The main reason for the power law being so popular is that the shearing rheological behaviour of a fluid is represented simply by a straight line in a log-log shear rate/shear stress graph. Another reason is that the shearing behaviour of most fluids lends itself to a good approximation applying the power law.

The basic equations for calculation of pressure drop and shear rate in various geometries, for Newtonian fluids and for power law non-Newtonian fluids, appear as follows:

- Circular ducts, Newtonian fluids
- Circular ducts, power law fluids
- Rectangular ducts, Newtonian fluids
- Rectangular ducts, power law fluids

- Viscous materials: in a purely viscous material all energy added is dissipated into heat
- Elastic materials: in a purely elastic material all energy added is stored in the material
- Viscoelastic materials: a viscoelastic material exhibits viscous as well as viscoelastic behaviour

Typical examples of viscoelastic materials are bread dough, polymer melts and
artificial or natural gels.

*Note:* in the rheological sense water is a "viscous" fluid. Normally,
however, the term "viscous" is used for fluids with high viscosity.

In most cases of viscoelastic behaviour the time factor has a significant impact
on the flow properties observed. A measure of the influence of time is the so-called
*Deborah Number*, D:

D = (response time) / (observation time)

An example of a system having a large Deborah Number is a normal glass window. If
old enough, *e.g.* an old church window, a difference in thickness at the top and
at the bottom can be easily measured. Although the viscosity of glass is high, about
10^{40} Pas, it is still a liquid and consequently it flows. However, the
observation time has to be long, perhaps a couple of centuries, to observe the
movement.

When shearing a viscoelastic fluid so-called normal stresses will appear. These
normal stresses can result in flow behaviour quite
different from that of Newtonian fluids.

**Viscosity and elasticity measurements**

*dynamic instruments*in order to get

*quantitative results*useful for design and development of products and process equipment. For design of products,

*e.g.*in the food, cosmetic or paint industry, rheometric measurements are often performed to establish the elastic properties, such as

*gel strength*and

*yield value*, both important parameters affecting

*e.g.*particle carrying ability and spreadability. For design of process equipment the properties during shearing of the product is of prime interest. Those properties are established in a normal viscosity measurement.

Typical

*measuring geometries*are the concentric cylinder, the cone-and-plate and the plate-and-plate geometries. Simpler viscometers often work with a so-called

*dip-in geometry (spindle)*thus not being able to apply a defined and uniform shear field. Hence these types of viscometers are not suitable for measurements on Non-Newtonian fluids. Instead, they are preferably used for production control purposes.

A rheometric measurement normally consists of a

*strain (deformation)*or a

*stress*analysis at a constant frequency (normally 1 Hz) combined with a

*frequency*analysis,

*e.g.*between 0.1 and 100 Hz. The strain sweep gives information of the

*elastic modulus G'*, the

*viscous modulus G''*and the

*phase angle δ*. A large value of G ' in comparison of G '' indicates pronounced elastic (gel) properties of the product being analysed. For such a product the phase angle is also small,

*e.g.*20º (a phase angle of 0º means a perfectly elastic material and a phase angle of 90º means a perfectly viscous material). The frequency sweep gives information about the

*gel strength*where a large slope of the G ' curve indicates low strength and a small slope indicates high strength.

A viscometric measurement normally consists of a

*shear rate*analysis. The shear rate sweep should preferably cover the range applied in the intended equipment. For liquid foods a shear rate range from around 1 to 1,000 s

^{-1}covers the needs for a low-viscous product,

*e.g.*milk or juice, and a shear rate range from around 1 to 100s

^{-1}covers the needs for a high-viscous product,

*e.g.*tomato paste or quark.

Below a number of examples from measurements on some fermented dairy products are found. The fermented cream has a fat content of about 35%, the fermented milk "type 1" has a fat content of 0.5% and the fermented milk "type 2" has a fat content of 1.5%. Note that despite the difference in the elastic modulus G ' between the two fermented milk types being significant, the viscosity curves are nearly identical. The practical implication of this is that when the two products are sitting in a cup the "type 2" fermented milk seems to have appreciably higher viscosity than the "type 1" milk, but when subjected to shear,

*e.g.*when being pumped through a pipe, the pressure drop will be much the same for both products. What is observed in the "cup analysis" is instead the more pronounced elastic properties of the "type 2" milk giving the impression of a higher viscosity.

For the youghurt a significant degree of thixotropy can be seen in that the "up curve",

*i.e.*the curve obtained when increasing the shear rate from zero and upwards, is appearing above the "down curve",

*i.e.*the curve obtained when going back in shear rate to zero. For comparison of the degree of thixotropic behaviour the distance or area between the two curves can be calculated, applied to either the shear stress curves or the apparent viscosity curves.

- Strain sweep, fermented cream
- Frequency sweep, fermented cream
- Strain sweep, fermented milk type 1
- Shear rate sweep, fermented milk type 1
- Strain sweep, fermented milk type 2
- Shear rate sweep, fermented milk type 2
- Shear rate sweep, youghurt

**In-line and on-line instruments**

In some cases there is a need for instruments capable of doing continuous viscosity measurements in a process line. For this purpose there is a possibility to use an in-line or an on-line viscometer, the main difference being that the in-line instrument is mounted directly in the process line and the on-line instrument is mounted in a by-pass line. In principle the instruments used are the same for both ways of installation.

The instruments are mainly of two types; the vibrational type and the rotational type.

The vibrational type works with a vibrating rod sensing the resistance from the fluid. In principle this type of instrument is less suitable for measurements on shear thinning and elastic fluids. In all cases of Non-Newtonian fluid behaviour it gives only a relative value.

The rotational type in principle works as a normal bob-and-cup instrument with the main difference that the measuring cylinder is a flow-through cell and hence the measurement is influenced by shear forces from the rotation as well as from the flow.

In practise both types of instruments have to be calibrated for a specific product and a specific flow rate.

#### Rheology literature

A vast number of books within various aspects of rheology is available in normal or specialized book-shops and libraries. The below list just shows some examples (note that a few titles appear in more than one list).

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