Apparent Viscosity from Rotational Rheometer Calculator

What “apparent viscosity” means in rotational testing

Apparent viscosity (η_app) is the ratio of shear stress to shear rate at the test point. For Newtonian fluids it equals the true viscosity, independent of rate. For shear‑thinning or shear‑thickening materials it is a snapshot value that depends on speed, geometry, and temperature. This calculator reports η_app from measured torque and rotation.

Core measurements: torque and angular velocity

A rotational rheometer typically measures torque in N·m and rotational speed in rad/s or rpm. Torque reflects the resistance to deformation, while angular velocity sets the deformation rate. Many lab runs operate between 0.01 and 200 rpm, with torques from µN·m to mN·m. Converting these into stress and shear rate requires the fixture dimensions.

Converting rpm to rad/s and why it matters

Calculations use angular velocity ω in rad/s, where ω = 2π·(rpm)/60. Reporting in rad/s avoids unit ambiguity and supports consistent shear‑rate formulas. A change from 10 rpm to 100 rpm increases ω by ten times. For many fluids, η_app can drop strongly across that decade.

Cone‑and‑plate: near‑uniform shear rate

With a small cone angle, the shear rate is approximately constant across the radius. That makes cone‑and‑plate popular for quick flow curves and limited sample volumes. Small angles like 1° to 4° are common, with radii from 10 to 30 mm. The calculator uses the standard cone relation γ̇ ≈ ω/θ (θ in radians).

Parallel plate: shear rate varies with radius

For parallel plates, shear rate increases linearly with radius and peaks at the rim. Many labs report the edge shear rate γ̇_R = ωR/h, using plate radius R and gap h. Typical gaps range from 0.2 to 2.0 mm depending on particle size and roughness. Consistent reporting is essential when comparing datasets.

Concentric cylinder: narrow‑gap approximation

Couette (concentric cylinder) tests are useful for low viscosity liquids and suspensions. When the gap is small compared with the inner radius, the shear rate can be approximated as γ̇ ≈ ωR_i/(R_o−R_i). Length L and radii set the stress conversion from torque. Large gaps may require more advanced corrections.

Temperature control and time dependence

Viscosity often changes by 2–10% per °C for many liquids. Always record setpoint temperature, soak time, and any thermal ramp rate. For thixotropic materials, η_app also depends on test history and dwell time at each speed. Use consistent pre‑shear and rest periods for comparable results.

Reporting checklist for reproducible results

Include geometry type, radius, gap, cone angle, cylinder radii, and active length. Report torque, speed, calculated shear rate, stress, and η_app with units. Note surface finish to reduce slip, and mention any corrections applied. A clear method section makes your apparent‑viscosity numbers defensible and reusable.

FAQs

1) Is apparent viscosity the same as true viscosity?

Only for Newtonian fluids. For non‑Newtonian materials, apparent viscosity depends on shear rate, geometry, and temperature, so it is a test‑condition value rather than a single constant.

2) Which geometry should I choose for my sample?

Cone‑and‑plate is good for small volumes and uniform shear rate. Parallel plates handle larger particles with bigger gaps. Concentric cylinders suit low‑viscosity liquids and reduce edge effects.

3) Parallel plate shear rate changes with radius. What should I report?

Most reports use the edge shear rate at the outer radius. State the radius and gap you used so others can reproduce the same shear‑rate definition and compare η_app fairly.

4) Why does my viscosity look negative?

It is usually a sign convention issue: torque or rotation direction may be recorded with a negative sign. Use absolute values for magnitude, and verify instrument settings before final reporting.

5) Can I calculate uncertainty for η_app?

Yes. Combine relative uncertainties from torque, speed, radius, gap, and angle using standard propagation rules. Torque resolution and gap setting often dominate at low speeds and small gaps.

6) How do I use this for shear‑thinning fluids?

Run multiple speeds to build a flow curve. The calculator will give η_app at each point. Plot η_app versus shear rate to visualize shear‑thinning or thickening behavior.

7) What corrections might be needed beyond this calculator?

Depending on the setup, you may need end‑effect, wide‑gap, inertia, or slip corrections. Some standards also apply Rabinowitsch‑type corrections for non‑Newtonian plate and capillary analyses.