---
title: "Rotational Rheometer & Couette–Poiseuille Flow Interactive Lab | Sandbox Physics"
description: "Why can torque rise as a liquid becomes easier to shear? Torque measures the total resistance at a chosen rotation speed."
canonical_url: "https://sandboxphysics.com/labs/couette-poiseuille/"
language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/couette-poiseuille/"
last_modified: "2026-10-29"
---

M096 · IN06 rotational rheometry

# Rotational Rheometer & Couette–Poiseuille Flow

**Why can torque rise as a liquid becomes easier to shear?** Torque measures the total resistance at a chosen rotation speed. A faster turn deforms the liquid faster, so torque can rise even when the resistance per unit deformation rate falls. A calibrated speed scan, together with the cylinder geometry, separates these effects.

Calibrate cylinder and cone-plate rheometers, save torque scans and compare Newtonian and shear-thinning flow. The original planar workshop stays available.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

Steady, isothermal no-slip flow uses a finite-gap Searle cylinder or a truncated small-angle cone with a local parallel-gap approximation. The cylinder omits end torque; the cone omits finite-angle corrections and edge flow. An inelastic power law applies over a finite speed interval. Elastic memory, yield stress, wall slip, secondary flow and heating are omitted. Newtonian geometric conversions are nominal for non-Newtonian samples; fitted corrections are geometry-specific. A separately acquired Newtonian reference calibrates sensor counts. Temperature is prescribed and recorded; material parameters have no temperature law. The original planar Couette–Poiseuille workshop remains available as a separate mode.

### A flow curve from saved sensor readings

Default simulated speed sweep, separately calibrated

$\dot\gamma_{\mathrm{nom}}/\mathrm{s}^{-1},\quad |\tau_{\mathrm{nom}}|/\mathrm{Pa}$

Turquoise: increasing speed. Gold: decreasing speed. The two sweeps retrace for this time-independent material. The line is fitted only to saved observations.

Apparent viscosity and the finite-gap correction

$\dot\gamma_{\mathrm{nom}}/\mathrm{s}^{-1},\quad \eta/\mathrm{Pa\,s}$

Turquoise uses Newtonian geometric conversions. Pink corrects the cylinder wall rate or the truncated-cone stress with the fitted power-law index.

Raw observations and calibration

Saved encoder speed and torque sensor counts

| Time (s) | Speed (rad/s) | Counts | Temperature (°C) | Sweep |
| --- | --- | --- | --- | --- |

The CSV and JSON include geometry, standard-liquid readings, zero readings, calibration identifier, version and seed. Preview controls leave these records unchanged.

Physics tutorial

## Separate turning speed from resistance to shear

**Background** The motor turns an inner cylinder inside a stationary cup. Fluid transfers torque from the bob to the cup; a sensor in the shaft measures the required drive.

**Why it matters** The same liquid resists a fast turn more than a slow one. A useful viscosity measurement must account for both the turning speed and the gap geometry.

Start with the essentials

- **Focus question**: Why can torque rise as a liquid becomes easier to shear?
- **One-sentence intuition**: A rising torque can accompany a falling viscosity when shear rate rises faster than stress.

### Core mathematical model

#### Finite-gap Newtonian wall rate and torque

$$
\dot\gamma_i=\frac{2|\Omega|R_o^2}{R_o^2-R_i^2},\qquad M=\frac{4\pi\eta L\Omega R_i^2R_o^2}{R_o^2-R_i^2}
$$

The inner bob rotates and the outer cup is stationary. The length is the wetted side length; end-face torque is omitted. The exact gap factor matters as the annulus widens.

#### Power-law correction from the measured scan

$$
n=\frac{d\log|M|}{d\log|\Omega|},\qquad \dot\gamma_i=\frac{2|\Omega|}{n[1-(R_i/R_o)^{2/n}]}
$$

The fitted slope comes from the calibrated saved torque observations. This corrected rate assumes an instantaneous power-law liquid that is sheared throughout the annulus.

#### Motor power becomes viscous heat

$$
M\Omega=\int_{R_i}^{R_o}|\tau(r)|\dot\gamma(r)\,2\pi rL\,dr\ge0
$$

Reversing rotation reverses the supplied torque. Their product stays positive. The isothermal model assumes an external bath removes this heat.

#### Include the truncated cone centre

$$
\begin{aligned}h(r)&=\max(h_0,r\tan\alpha)\\ |M|&=2\pi\int_0^R K\left(\frac{|\Omega| r}{h(r)}\right)^n r^2\,dr\end{aligned}
$$

Each circular ring contributes its stress magnitude times its area and lever arm. The signed motor torque follows the rotation. Cone angle and truncation clearance define the same local gap used in the scene. This thin-gap model assumes a completely filled radius and neglects finite-angle corrections and edge flow.

### Common difficulties

#### A sweep can retrace

**Typical misconception** A shear-thinning liquid must draw a hysteresis loop.

**Better mental model** This power law has no memory. Its upward and downward sweeps follow the same curve apart from detector noise. Time-dependent rebuilding needs a different constitutive model.

### Run the experiment

1. 01

   #### Change the measuring geometry

   Choose A cone makes shear nearly uniform and inspect Sample gap.

   **What to observe:** A real cone and lower plate replace the cup and bob. Gap height is magnified twelve times; the saved scan carries its own cone dimensions and reference calibration.

2. 02

   #### Compare two Newtonian speeds

   Choose Newtonian standard, then Faster, same viscosity.

   **What to observe:** The ideal torque grows with speed. A newly retained scan still gives a flow index close to one.

3. 03

   #### Widen the gap

   Choose A wider annulus and inspect both viscosity curves.

   **What to observe:** The nominal and corrected wall viscosities separate for a thinning sample. The geometry and fitted index determine the correction.

4. 04

   #### Keep and preserve a measurement

   Keep a speed sweep, then drag the gold collar or move the gap slider. Export the record.

   **What to observe:** The apparatus previews the new settings. The saved observations and exports retain the earlier geometry and calibration.

Physical conclusion

Calibrate torque, record several speeds and distinguish the nominal Newtonian conversion from a material-dependent wall rate.

## Related experiments and articles

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Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Does a harder turn mean a thicker liquid?**](https://sandboxphysics.com/stories/instruments/rotational-rheometer/)

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Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Stress relaxation, creep and the Weissenberg number in a Maxwell liquid**](https://sandboxphysics.com/stories/complex-fluids/stress-relaxation-and-creep/)
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