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.

How hard does the motor have to turn?

Ideal live torque $110\,\mathrm{\mu N\,m}$

Live nominal shear rate $28.0\,\mathrm{s}^{-1}$

Index from saved scan $1.00$

Saved apparent viscosity near the reference speed $0.100\,\mathrm{Pa\,s}$

Measuring geometryAngular speed

\Omega/(\mathrm{rad}\,\mathrm{s}^{-1})

2

Cylinder gap

h/\mathrm{mm}

1

Drag the gold drive collar to change speed. The cup or lower plate stays fixed; turquoise tracers follow the solved fluid velocity.

Sample, calibration and model

Consistency

K/(\mathrm{Pa}\,\mathrm{s}^{n})

0.1

Flow index

n

1

Recorded temperature

T/{}^\circ\mathrm C

25

Sensor noise bound

a/\mathrm{count}

5

Sampling seed

s_0

179

A separate Newtonian reference at the recorded temperature calibrates sensor counts. Recalibrate after changing geometry or temperature. Temperature is a recorded setpoint; this model has no temperature dependence.

Steady no-slip cylinder flow or a local thin-gap cone model; end torque, heating, yield stress and elastic memory are omitted. The power law applies only within the finite scan range. Motor and tracers replay at four times physical speed; their sizes are schematic. The cone gap height is drawn twelve times larger; sample fills the defined gap to its edge.

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.

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 misconceptionA shear-thinning liquid must draw a hysteresis loop.

Better mental modelThis 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

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.
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.
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.
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.

Start with the essentials

Focus question: Can measuring shear stress reveal an unknown viscosity?
One-sentence intuition: Check first that the approximation applies, then infer parameters from independent raw readings.

Core mathematical model

Velocity and shear stress across the gap

\begin{aligned}\eta&=y/H,\quad G=-dp/dx\\u&=U\eta+\frac{GH^2}{2\mu}\eta(1-\eta)\\\tau&=\mu U/H+GH(1/2-\eta)\end{aligned}

The lower wall is fixed. The shear stress is the stress on a fluid plane whose normal points upward.

Flow rate per unit width

q=\int_0^H u\,dy=\frac{UH}{2}+\frac{GH^3}{12\mu}

Layers moving forward and backward can cancel in the total flow, even while the fluid dissipates energy.

Power balance per unit plate area

\begin{aligned}Gq+U\tau(H)&=\mathcal D\\\mathcal D&=\int_0^H\mu(u_y)^2\,dy\end{aligned}

The moving wall can take mechanical energy out of the fluid. The work done by pressure plus the work done by the wall equals the viscous heating.

Estimates from the measured profiles

\begin{aligned}\widehat u&=c+a\eta+b\eta(1-\eta)\\\widehat\mu&=\frac{\sum_i\widehat u_{y,i}\tau_i}{\sum_i\widehat u_{y,i}^2}\\\widehat G&=2\widehat\mu b/H^2\end{aligned}

The velocity fit includes a free offset. The stress fit assumes the sensor has no bias, and the velocity derivative must be nonzero.

Checking that inertia stays small

\mathrm{Re}_{\mathrm{peak}}=\rho H\max_y|u|/\mu

A teaching threshold of 200 blocks acquisition outside the chosen low-inertia range. Staying below it does not prove the flow is stable.

Common difficulties

A reference curve is not an estimate

Typical misconceptionYou can read off the estimate without recording any data.

Better mental modelClear the recorded data and the estimate disappears. Sensor perturbations can produce unphysical fits; the estimates are neither clipped nor independently calibrated.

Run the experiment

01
Separate the two drives
Compare Wall only, Pressure only and Combined drive.
What to observe: The linear and parabolic parts add, and each matches the no-slip values at the walls.
02
Find counterflow
Select Counterflow and move the gold probe across the gap.
What to observe: Even while the upper wall moves forward, fluid in the interior can flow backward.
03
Check the energy balance
Change the wall speed and look at the wall power and its sign.
What to observe: Pressure work plus wall work equals the viscous heating, which is never negative.
04
Measure and infer
Acquire the gap profile, increase perturbations and export CSV.
What to observe: The fit uses only the recorded velocity and shear readings; it does not use the reference viscosity or gradient.

Rotational Rheometer & Couette–Poiseuille Flow

A flow curve from saved sensor readings

Separate turning speed from resistance to shear

Core mathematical model

Finite-gap Newtonian wall rate and torque

Power-law correction from the measured scan

Motor power becomes viscous heat

Include the truncated cone centre

Common difficulties

A sweep can retrace

Run the experiment

Change the measuring geometry

Compare two Newtonian speeds

Widen the gap

Keep and preserve a measurement

Velocity, stress and dissipation

Core mathematical model

Velocity and shear stress across the gap

Flow rate per unit width

Power balance per unit plate area

Estimates from the measured profiles

Checking that inertia stays small

Common difficulties

A reference curve is not an estimate

Run the experiment

Separate the two drives

Find counterflow

Check the energy balance

Measure and infer

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