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II: Measurements: Rheometry Measures Deformation /Stress on a Material with Unknown Constitutive Behavior

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II: Measurements: Rheometry Measures Deformation /Stress on a Material with Unknown Constitutive Behavior rheometry 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.
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