Calculation of Shear Rate and Shear Stress form factors.

Rheometers and viscometers work with torque and angular velocity. Since we normally work with shear stress and shear rates a method is needed to convert from 'instrument numbers' to 'rheology numbers'.

Each measuring system used in an instrument will have its associated 'form factors' to convert torque to shear stress and to convert angular velocity to shear rate.

We will call the shear stress form factor C1 and the shear rate factor C2

Shear Stress = C1 x Torque
Shear Rate = C2 x angular velocity *
Viscosity = Shear Stress / Shear Rate

* For some measuring systems such as parallel plates, the gap between the measuring systems can be set by the user. In this case the equation used is :
Shear Rate = C2 x angular velocity / gap.

The following section shows how the form factors are calculated for each measuring system.

Cone and plate

C1 =     1                C2 = 1 / θ
        2/3 π r3

Where r = radius of cone θ = cone angle in radians

Parallel plates

C1 =     1                    C2 = 3r / 4
        2/3 π r3

Where r = radius of cone

NOTE: The shear stress varies across the radius for a parallel plate. The above formula refers to the 3/4 radius position if the test sample is Newtonian.

Coaxial cylinders

C1 =         1                  C2 =     2  ri2 ro2        
        2 π ra2   H                      ra2 (ro2 - ri2)

Where ra = (ri + ro) / 2
ri = inner radius
ro = outer radius
H = height of cylinder

NOTE: C1 takes the shear stress as that occurring at an average radius position ra