Herschel-Bulkley Rotational Couette Torque Calculator

Solve low-inertia concentric-cylinder HB rotation, torque, yielded radius and dissipation with exact stress balance and no-slip integral closure.

Results

Calculate to view linked results. Default values are examples, not recommended operating inputs.

Calculation sequence

Enter measured HB consistency, exponent and yield stress with concentric radii, wetted length and inner angular speed. The outer cylinder is fixed and there is no axial flow or pressure gradient. This is a rotational torque/velocity problem, not an eccentric drilling annulus pressure-loss correction.
Integrate the exact azimuthal stress C/r squared and the HB inverse flow curve over the yielded region. Solve the no-slip angular-speed integral for C. When stress reaches yield before the outer cylinder, the remaining outer annulus is an unyielded stationary plug, not a moving liquid layer or a negative shear rate.
Convert C into drive torque and power. Independently integrate stress times shear rate over the annular volume and require dissipation to match shaft power. In the zero-yield Newtonian limit recover the analytical circular Couette torque, rather than a narrow-gap approximation.
Use a deliberately conservative rotational Reynolds screening value based on inner-wall apparent viscosity and require it <=1. This excludes inertial/Taylor-vortex effects but is not a complete stability theorem. The cited paper uses an outer rotating cup; relative rotation gives the same creeping shear/torque law, while velocity here is referenced to the fixed outer wall.
Calculations run locally in a dedicated WebAssembly worker. The advanced Python/scientific engine is a roughly 33 MB first download, separate from ordinary calculators; it is self-hosted and version pinned. No composition or measurements are sent to a calculation server. Save and file backups retain canonical inputs, while CSV/PDF include returned tables and source notes. The worker is released after idle time and failures never return partial results.

Worked check

The synthetic case Ri=0.05 m, Ro=0.10 m, K=2 Pa.s^n, n=0.8, yield=5 Pa and Omega=0.1 rad/s describes slow inner-cylinder rotation over 1 m. The outer region can remain unyielded while the inner annulus shears. Setting yield=0 and n=1 recovers the exact Newtonian circular Couette solution; no drilling-flow prediction is implied.

Synthetic examples for reproducibility, not measured field results.

Assumptions and limitations

Concentric steady creeping azimuthal HB flow only. No axial/helical flow, eccentricity, end effects, wall slip, cuttings, elasticity, thixotropy or temperature variation. Do not superpose this torque with axial HB pressure loss; nonlinear helical shear components are coupled.
Radius ratio Ri/Ro=0.1-0.95, exponent=0.4-1, nonnegative yield stress and positive consistency/speed. Require rotational Reynolds number<=1, no-slip closure and power/dissipation closure. These are software gates, not Taylor-instability or field-accuracy guarantees.
Torque excludes bearings, seals, end drag and instrument calibration. Fitted HB parameters and yield stress must match the actual liquid; synthetic defaults are not a mud/rheometer calibration. The model does not predict drilling torque or hole cleaning.

Implemented equations

Rotational stress balance and yielded shear

\begin{aligned}\tau(r)&=C/r^2\\\dot\gamma(r)&=\left[\frac{\max(0,C/r^2-\tau_y)}K\right]^{1/n}\\\Omega&=\int_{R_i}^{R_y}\dot\gamma(r)\frac{dr}{r}\\\mathcal T&=2\pi LC,\quad P=\mathcal T\Omega\end{aligned}

Sources and maintenance

Published by PetroCalcHub Team. Implementation and source checks: 2026-10-22. This does not claim independent expert review.

Report a correction

Related tools and references

Fit measured Herschel-Bulkley rheology Exact axial HB annulus (no rotation)Yield-stress annular friction correlation Drilling hydraulics method guide