Coupled Concentric Helical Herschel-Bulkley Flow Calculator

Calculation sequence

1. Supply concentric geometry, pressure gradient, angular speed and an isothermal time-independent HB fit. The inner wall rotates without axial translation and the outer wall is stationary. Use a measured consistency/yield stress on the same shear-rate basis; density enters only the conservative inertia screen, not an assumed turbulent correction.
2. Solve both no-slip integrals with the combined stress magnitude. Refine quadrature from 256 to 512 nodes and require flow and torque changes below 0.2%. Reconstruct both wall velocities with adaptive quadrature. Enforce pressure-work plus shaft-work equality with integrated constitutive dissipation, so a sign or torque-unit error cannot pass as a valid result.
3. Unyielded locations follow the combined invariant, not separate axial and rotational yield radii. The combined screening Reynolds number must stay below one; this protects the creeping approximation but is not a complete Taylor-vortex criterion. Nonconverged high-yield or poorly conditioned cases fail without partial results. Compare the Newtonian and pure-rotation limits before considering more complicated geometry.
4. Calculations remain local in a dedicated WebAssembly worker. The existing self-hosted scientific engine is about 33 MB raw on first use; the envelope additionally loads the pinned CoolProp engine and eccentric flow loads a 178 KB FEM package. Nothing is downloaded until calculation. No composition, heat history or measurements are sent to a calculation server. Save retains canonical input values; file backup can be shared with colleagues, while CSV/PDF retain samples, assumptions and sources. Browser storage alone can be lost.

Worked check

For Ri=0.05 m, Ro=0.10 m, K=2 Pa.s^n, n=0.7, yield stress 5 Pa, omega=0.1 rad/s and G=100 Pa/m, the coupled flow is about 1.56554e-7 m3/s and torque about 0.116757855 N.m per metre. Total input power is about 0.011691441 W per metre, including both pressure and shaft work.

Synthetic examples for reproducibility, not measured field results.

Assumptions and limitations

• Steady, concentric, uniform annulus, 0.2<=Ri/Ro<=0.9. Fully developed creeping flow, no end losses or axial wall motion.
• No eccentricity, cuttings, beds, wall slip, elasticity, thixotropy, inertia, Taylor vortices or turbulence. The separate FEM workflow addresses a bounded eccentric regularized case.
• No temperature coupling or thermal viscosity law. A scalar consistency fit is not a downhole fluid characterization. Numerical power closure is not field validation.

Sources and maintenance

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

• Hussain et al.: coupled helical viscoplastic flow in eccentric annuli
• SciPy: bounded roots, optimization and numerical linear algebra
• SciPy: adaptive quadrature and bounded least squares

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