Rheometry: the science of rheological measurements
What is a rheometer? An instrument that creates a flow and enforces stress / deformation and II: measurements: rheometry measures deformation /stress on a material with unknown constitutive behavior
- shear versus extension - small strain versus large strain - transient versus steady
rheometry rheometry
Homogeneous nonhomogeneous, indexers (complex flow fields) material functions and classification of rheometers Introduction
Drag flow: shear flow generated by a moving and a fixed solid surface
Pressure driven flow: shear generated by a presure 5: shear rheometry: drag flows difference over a closed channel (versus pressure driven flows) Measure the:
- relaxation modulus
- transient shear viscosity
- steady shear viscosity
- first normal stress difference coefficient
- second normal stress difference coefficient
Drag flow: sliding plates
Working equations
Shear flow geometries Critical time for homogeneous simple shear flow Drag flow: concentric cylinders (Couette flow) Drag flow: concentric cylinders (Couette flow)
Assumptions: Shear stress: - Steady, laminair, isothermal flow
- vθ = r Ω only and vr = vz = 0 - Negligible gravity and end effects - Symmetry in θ, ∂/∂θ = 0 Torque measured at the inner cylinder
Equations of motion, cyl. coordinates →
Outer cylinder Boundary conditions →
Drag flow: concentric cylinders (Couette flow) Drag flow: concentric cylinders (Couette flow)
Strain and strain rate Change of variable: Dropping subscripts:
Very narrow gap (Ri/Ro>0.99):
Integrating (rotating inner cyl.):
Ri/Ro<0.99, cylinder coord.: Differentiating with respect to stress:
Shear rate varies across the gap With: It follows: Drag flow: concentric cylinders (Couette flow)
For 0.5 < κ < 1.0 expand:
In Mclaurin series:
Where n, the power law index, in terms of torque and rotation rate :
For κ > 0.5, n = const. :
Errors Remarks
End effects The deriviation shown is typical for nonhomogeneous flows:
Secondary flows (Taylor vortices) - the derivative of the velocity with position is difficult to measure - transform to derivative of stress with respect to rotation / flow rate - see parallel plate and pressure driven rheometers
Shear heating Drag flow: cone and plate rheometer
Assumptions: - Steady, laminair, isothermal flow
-vΦ (r, θ) only; vr = vz = 0 - β < 0.10 rad (≈ 6o) - Negligible bodyforces - Spherical liquid boundary
Equations of motion, spherical coordinates Boundary conditions Drag flow: cone and plate rheometer
Drag flow: cone and plate rheometer Drag flow: plate-plate rheometer
Edge failure Assumptions: - Steady, laminair, isothermal flow
-vΦ (r, θ) only; vr = vz = 0 - Negligible bodyforces - Cylinderical edge
Equations of motion, cyl. coordinates
Shear heating: Drag flow: plate-plate rheometer
Velocity, shear rate and shear:
Torque:
Change of variables:
Normal stress