Rheology the most sensitive method for material characterization because flow behavior is responsive to
properties such as molecular weight and molecular weight distribution. Rheology measurements are also
useful in following the course of a chemical reaction. Such measurements can be employed as a quality
check during production or to monitor and/or control a process. Rheological measurements allow the study
of chemical, mechanical, and thermal treatments, the effects of additives, or the course of a curing reaction.

Spindle geometries:

**Disc spindles** produce accurate, reproducible apparent viscosity determinations in most fluids.

Because their defined geometry facilitates mathematical analysis, **cylindrical spindles** are particularly
valuable when measuring **non-Newtonian fluids**.

**Coaxial cylinder** geometry is indicated for applications where extremely well-defined shear rate
and shear stress data is required.

**Cone-plate** geometry offers absolute viscosity determinations with precise shear rate and shear
stress information. The sample volumes required are very small and the sample cup is jacketed for
temperature control. This geometry is very suitable for advanced rheological analysis of **non-Newtonian
liquids**.

**T-bar** spindles make possible the measurement of non-flowing or slow-flowing materials such as pastes,
gels and creams.

Non-flowing or slow-flowing sample materials such as pastes, crams, and gels present problems in viscosity measurements. Conventional rotating spindles tend to push the sample material aside, what results in a continuously decreasing viscosity reading. A solution to that problem is a special stand which slowly raises and lowers the viscometer while a special T-bar spindle rotates in the sample material. The crossbar of the spindle thus continuously cuts into fresh material, describing a helical path through the sample as it rotates.

How do the viscometers work?

Usually, a viscometer measures the torque required to rotate a spindle in a fluid. The spindle is driven by
a synchronous motor through a calibrated spring and the deflection of the spring is displayed by the viscometer.
By changing speeds and spindles, a variety of viscosity ranges can be measured. For a given viscosity, the
viscous drag, or resistance to flow (indicated by the degree to which the spring winds up), is proportional
to the spindle's speed of rotation and is related to the spindle's size and shape. The drag increases with
an increase of spindle's size and/or rotational speed. For a given spindle geometry and speed, an increase
in viscosity will be indicated by an increase in the deflection of the spring. The minimum range is obtained
by using the largest spindle at the highest level; the maximum range by using the smallest spindle at the
slowest speed. Measurements made using the same spindle at different speeds are used to detect and evaluate
the rheological properties of the tested fluid.

To obtain the viscosity of the tested fluid multiply the reading of the viscometer by the spindle factor (given
in the manual of the viscometer).

Viscosity is the measure of the internal friction of a fluid. This friction becomes apparetn when a layer of fluid is made to move relatively to another layer. The greater the friction the greater the amount of force required to cause this movement which is called

If we have two parallel planes of fluid of equal area *A* and they are separted by a distance dx and are moving
in the same direction at different velocities *v _{1}* and

The velocity gradinet, dv/dx, is a measure of the speed at which the intermediate layers move with respec to
each other. It describes the shearing the liquid experiences and is called **shear rate - R** and its unit
of measure is called reciprocal second (sec^{-1}).

The term F/A indicates the force per unit area required to produce the shearing action and it is called
**shear stress - S** and its unit is N/m^{2}. So viscosity can be defined as:

viscosity = shear stress *S*/ shear rate *R*.

The fundamental unit of viscosity is the **poise**. A material requiring a shear stress of one dyne
per square centimeter to produce a shear rate of s^{-1} has a viscosity of one poise or 100 centipoise.
One Pascal-second (Pa*s) is equal 10 poise.

Newtonian fluids

Below graphs show the relationship between shear stress *S* and shear rate *R* and the fluid's viscosity at a
varying shear rate *R*. Typical Newtonian fluids incude water and thin motor oils.

So, at a given temperature the viscosity of a Newtonian fluid remains constant regardless of which viscometer
model, spindle or speed is used to measure it.

The behavior of Newtonian liquids in experiments conducted at constant temperature and pressure has the following features:

- the only stress generated in simple shear flow is the shear stress
*S*, the two normal stress differences are zero - the shear viscosity doesn't vary with shear rate
- the viscosity is constant with respect to the time of shearing and the stress in liquid falls to zero immediately the shearing is stopped
- the viscosities measured in different types of deformation are always in simple proportion to one another.

non-Newtonian fluids - time independent

Generally speaking, a non-Newtonian fluid is defined as one for which the relationship *S/R* is not constant.
The viscosity of non-Newtonian fluids changes as the shear rate is varied. Thus, the parameters of viscometer
model, spindle and rotational speed al have an effect on the measured viscosity. This measured viscosity is
called **apparent viscosity** and is accurate when explicit experimental parameters are adhered to. There
are several types of non-Newtonian flow behavior, characterized by the way a fluid's viscosity changes in
response to variations in shear rate.

**Pseudoplastic**:

fluid displayes a decreasing viscosity with an increasing shear rate, some examples include paints and emulsions. This type of behavior is called**shear-thinning**.**Dilatant**:

is characterized by an increasing viscosity with an increase in shear rate, some examples include clay slurries, candy compounds, corn starch in water, and sand/water mixtures. Dilatancy is also reffered to as**shear-thickening**liquids.**Bingham**:

liquid behaves like solid under static conditions. A certain amount of force must be applied to the fluid before any flow is induced. This force is called**yield value**. Tomato catsup is an example of such fluid. Once the yield value is exceeded and flow begins, plastic fluids may display Newtonian, pseudoplastic or dilatant flow characteristics.

Some fluids display a change in viscosity with time under conditions of constant shear rate.

**Tixotropic**:

fluid undergoes a decrease in viscosity with time, while it is subjected to constant shearing (greases).**Rheopectic**:

fluid's viscosity increases with time as it is sheared at a constant rate.

Diagram summary of non-Newtonian fluids:

where 1-viscoplastic fluid, 2-bingham fluid, 3-pseudoplastic fluid, 4-Newtonian fluid, 5-dilatant fluid.

Laminar and turbulent flow

The definition of viscosity implies the existence of

The increased energy input is manifested as an apparently greater shear stress than would be observed under laminar flow conditions at the same shear rate. This results in a very high viscosity readings.

The point at which laminar flow evolves into turbulent flow depends on other factors besides the velocity at which the layers move. Material's viscosity and specific gravity and geometry of the viscometer spindle and sample container all onfluence the point at which this transition occurs. Turbulent flow is characterized by a relative sudden and substantial incrase in viscosity above a certain shear rate. The higher the viscosity of a fluid the less likely it is to experience turbulence.

Determination of parameters of time independent non-Newtonian fluids

A common method for characterizing non-Newtonian flow is to compute the ratio of the fluid's viscosities
as measured at two different speeds with the same spindle. These measurements are usually made at speeds
that differ by a factor of 10 (2 and 20 RPM, or 10 and 100 RPM). In constructing the ratio, the
viscosity value at the lower is placed in the numerator, and the one at the higher speed in the denominator.
For pseudoplastic fluids the ratio exceeds 1.0 as the degree of pseudoplastic behavior increases. For
dilatant fluids, the ratio is less than 1.0 as the degree of dilatancy increases.

Similar method omitts calculation of viscosity and uses display readings do derive the viscosity ratio:

viscosity ratio: log(Mx/M10x), where Mx is viscometer reading at speed x and M10x is viscometer reading
at speed 10x

Another method of analyzing non-Newtonian flow is constructing a plot of viscosity vs. spindle speed using
the same spindle. Viscosity is plotted along the y-axis and speed (RPM) along the x-axis. The slope and
shape of the curve indicate the type and degree of flow behavior.

Or one can plot viscometer reading on the x-axis as a function of speed on the y-axis. If the graph is drawn
in log-log scale the result is frequently a straight line. Then, the slope of the line (indicating
the type and degree of non-Newtonian flow) and its intercept with the x-axis (indicating its yield value if any)
can be used as empirical constants.

Template method is also used to characterize non-Newtonian fluids. Its use is limited to fluids that flollow a power law, meaning that they display one type of non-Newtonian behavior as shear rate is varied. This method is usable only with data generated with cylindrical spindles or coaxial cylinders. The data is fitted to a template to determine a constant called the STI. Ceratin parameters of the viscometer and the STI are fitted to a second template, which is then used to predict the fluid's viscosity at any selected shear rate.

Some fluids (bingham and viscoplastic) wont' flow until a certain amount of force is applied. This force is
called ** yield stress**. Yield stress values can help determine whether a pump has sufficient power to
start in a flodded system, and often correlate with other properties of suspensions. A simple method for
determining a relative yield value is to plot viscometer readings on the x-axis vs. speed (RPM) on the y-axis.
The obtained line is extrapolated to zero RPM. The corresponding value for the viscometer reading
represents the yield value. Extrapolating the line to zero RPM is easy if the line is fairly straight
(Bingham flow). Once a straight line is obtained the angle this line forms with the y-axis is measured.
The power law index of this fluid can then be calculated from the following equation:

power law index N = tan(the angle between the plot line and y-axis).

If the angle is less than 45 degrees, the fluid is pseudoplastic, if greater than 45 degrees than it is dilatant.

The power law index can be sued to calcualte the effective shear rate at a given speed using the equation:

shear rate S = power law index N / (0.2095 * viscometer speed in RPM).

A method for determining yield value and plastic viscosity when a plot of viscometer reading vs. speed produces
a curved line is to plot the square root of the shear stress vs. the square root of the shear rate. This often
straightens the line and facilitates extrapolation to zero shear rate. This method is most suitable for
pseudoplastic fluids with a yield value conforming to a model of flow behavior - the Casson equation.

Determination of parameters of time dependent non-Newtonian fluids

Usually, the analysis of thixotropic and rheopectic fluids involves plotting changes in viscosity as a function of time. The simplest method is to select a spindle and speed and leave the viscometer running for an extended period, noting the readings at regular intervals. It is essential to have the sample in the same temperature through out the experiment. A change in fluid's viscosity over time indicates time-dependent behavior: a decrease means thixotropy, an increase rheopexy.

Another method is to graph the viscometer reading vs. speed, using a single spindle. Starting at a low speed, note the reading at each successively higher speed until the reading goes off the scale. A graph of such readings is an upward curve. Without stopping the viscometer, reduce the speed incrementally to the starting point, again taking readings at each speed. This is a downward curve. It is good to alow some time before each speed change. If the fluid is time independent the curves will overlap. If they don't, the fluid is time dependent. If the upward curve indicates a higher viscosity than the downward curve, the fluid is thixotropic; if the uwpward curve indicates lower viscosity than the downward curve, the fluid is rheopectic.

Thisxotropic break down coefficient quantifies the degree of thixotropy (or rheopexy). First, one must plot
voscometer reading vs. log time, taking readings at regular intervals. This should produce a straigh line.
Then, use the equation:

T_{b} = ((St1 - St2)/(ln(t2/t1)) * F

where, St1 and St2 are viscometer reading at t1 and t2 minutes respectively, and F is the spindle/speed factor

Plots of thixotropic behavior can be sometimes used to predict the gel point of a fluid. Plot the log viscometer reading vs. time, using a single spindle and speed. If the resulting line has a steep slope, gelling is likely to occur. If the line curves and flattens out, gelation is unlikely.

Yet another technique requires a plot of time vs. the reciprocal of the viscometer reading. The gel point
can be read from the curve intercept at at viscometer reading of 100. Fluids which don't gel will be
asymptotic to the vertical axis.