**Rheology**^{1}

Rheology is the scientific study of deformation and flow of matter. Some important
rheologic characteristics of matter are viscosity and newtonian / non-newtonian
behaviour.

**Dynamic viscosity η**

Between two plates of similar type with an area A and a distance x is filled in
a homogeneous fluid. This fluid can be devided in parallel layers of differential thickness.
If one of the plates is moved parallelly to the other plate with constant velocity the fluid layer
directly adjoining to the plate will be accelerated on this side but also decelerated by the next
fluid layer. So every layer is accelerated by the adjoining faster layer and decelerated by the
slower one next to it.

This leads to a velocity profile between the plates which is linear in simple, idealised cases.
It can be observed, that the force F, thats needed to set the plate in motion, is proportional to plate
area A, to plate velocity v and inverse proportional to plate distance x.

Summarised:

The constant is the ** dynamic viscosity η**
(unit: N s / m² = Pa s). The higher the viscosity is the
higher is the expenditure of energy to get the substance in motion.

The ** shear stress τ** is the relation of force to surface:

The velocity gradient dv/dx is the ** shear rate γ**.
This leads to:

**Kinematic viskosity ν**

If the dynamic viscosity η of a substance is set in relation with its density ρ the
kinematic viskosity ν is gained. It must be considered that both dynamic viscosity and density
are temperature-dependend.

**Newtonian fluids**

If the viscosity of a substance is independent of the shear rate it is a
newtonian fluid. The shear rate is linear dependent of the shear stress.
Movement of fluid particles is described by the Navier-Stokes equation.
Examples for newtonian fluids are water, air and some oils/gases.

**Non-Newtonian fluids**

Many fluids don't show a linear relation of shear rate and shear stress or need a minimum shear
rate before they show fluid behaviour. Depending
on the shown characteristic, fluids can be classified:

__Sheer thinning:__With increasing shear rate the viscosity decreases. Changes of the inner structure of the fluid causes a decreasing inner friction. These changes only depend on the shear rate and are independent on the time.__Dilatant:__Also called "shear thickening". Opposite effect of shear thinning substances, the materials' viscosity increases with increasing shear rate. Structure changes causes higher interactions between fluid particles.__Yield point:__Some substances show elastic behaviour up to a specific shear rate, but are fluid-like up from this yield point (yield strength). If the fluid is Newtonian above the yield point its called__Bingham fluid__, if it is Non-Newtonian its called__Casson fluid__.__Thixotropy:__At constant shear rate a fluid shows a time-depending decreasing of viscosity, the longer the fluid is sheared the lower viscosity becomes. After the end of shearing viscosity gets back to its original value, no permanent change is observable.__Rheopexy:__Opposite effect to thixotropy. Rheopexic fluids show a time-depending increasing of viscosity at constant shear rate. The increased viscosity falls back to its original value after end of shearing.

**Viscosity-temperature relation**

The effects of changes in temperature on the viscosity of a substance is descibed
by the __Andrade equation__:

a,b = specific parameters and T = thermodynamic temperature in K

By raising the fluids temperature its viscosity decreases vice versa by cooling a fluid its viscosity
increases. It must be observed that the specific parameters of the Andrade euqation are only valid in
limited range. In cases of wider temperature-ranges different sets of parameters has to be choosen.

**Couette flow**

A flow caused by the relative motion of two plates is called Couette flow. No pressure
gradient is needed, the flow is induced only by the plate-fluid power transmission.

**General measuring methods rotational viscosimeter:**

If the outer cylinder (sample cup) is rotating and the inner cylinder is fix it is
a __Couette__ system. If the outer cylinder is fix and the inner cylinder is
rotating it is called __Searle__ system. By shearing the the fluid in the gap
between the cylinders the propelled cylinder is decelerated depending on the
viscosity of the fluid. This deceleration gives the measuring signal which can be
evaluated.

The difference between these two systems is the behaviour of the fluid
respectively of the occuring flow in the gap. With a rotating outer cylinder
(Couette system) a velocity gradient between the fluid layers is induced.
The fluid flows laminary inside the gap without any radial velocity component.
If the inner cylinder is rotating (Searle system) a velocity gradient in opposite
direction compared to a Couette system is forming at low rotational speeds. At
higher rotational speeds the flow is decomposing and turbulences (vortices)
perpenticulary to the axis of rotation occur. This is called __Taylor
vortex__ and is induced by centrifugal forces which influence the fluid to
move from the inside to the outside part of the gap. The vortices change the
measurement results and must be compensated.