Estimate flow behavior using practical rheology inputs. Check logarithm limits and unit choices. Download clean reports for labs and field work.
The Herschel–Bulkley model relates shear stress and shear rate: τ = τ₀ + K · γ̇ⁿ.
Solving for n when τ, τ₀, K, and γ̇ are known: n = ln((τ − τ₀)/K) / ln(γ̇).
Use consistent stress units for τ, τ₀, and K. The exponent n is dimensionless.
| τ (Pa) | τ₀ (Pa) | K (Pa·s^n) | γ̇ (1/s) | n (computed) |
|---|---|---|---|---|
| 120 | 20 | 45 | 100 | 0.173238 |
| 260 | 40 | 60 | 50 | 0.420102 |
| 95 | 10 | 35 | 200 | 0.111914 |
The Herschel–Bulkley model describes materials that resist motion until a yield stress is exceeded, then flow with a nonlinear stress–rate relationship. In this calculator, the flow index n is computed from measured shear stress τ, yield stress τ₀, consistency K, and shear rate γ̇. Use it to classify flow behavior and compare formulations under consistent conditions.
The index n controls curvature on a log–log plot of (τ−τ₀) versus γ̇. When n is less than 1, apparent viscosity typically decreases as shear rate rises (shear‑thinning). When n is greater than 1, apparent viscosity tends to increase with shear rate (shear‑thickening). If n is close to 1, behavior approaches a Bingham-like linear response after yield.
Many suspensions, polymer melts, and food products show n values broadly between about 0.2 and 1.5, depending on microstructure and temperature. For screening work, treat n≈0.4–0.8 as strongly shear‑thinning, n≈0.8–1.1 as near‑linear, and n>1.1 as shear‑thickening. Always report the test protocol alongside n.
Because the calculation uses a logarithm, τ must be greater than τ₀ and γ̇ must be positive. Small differences between τ and τ₀ can amplify noise, especially at low shear rates. If you see unstable results, repeat measurements, increase averaging time, or fit multiple data points rather than a single pair.
Stress units must match for τ, τ₀, and K. If K is expressed in Pa·sⁿ, then γ̇ should be entered in s⁻¹ so that K·γ̇ⁿ has units of Pa. If you switch to kPa or psi, convert all stress terms together to keep the exponent calculation meaningful.
Rheology tests often span shear rates from roughly 0.1 to 1000 s⁻¹ to capture both low‑rate structure and high‑rate alignment. Choose γ̇ values that reflect your process: pumping, coating, mixing, extrusion, or injection. Comparing n across products is most reliable when the shear‑rate window is similar.
n is sensitive to τ₀ and K. Overestimating τ₀ typically pushes (τ−τ₀) smaller and can bias n downward, while underestimating τ₀ can inflate n. Likewise, K errors shift the log ratio. For rigorous reporting, compute n from a regression over many points and quote confidence intervals.
Once n is known, you can estimate shear stress at new shear rates for the same material state, support pump and pipe calculations, and compare shear response across batches. In QC, a drifting n may indicate particle flocculation, polymer degradation, or formulation changes that alter flow structure.
Enter measured shear stress, yield stress, consistency index, and shear rate. Values must be positive, and shear stress must exceed yield stress so the logarithm is defined.
n less than 1 indicates shear‑thinning behavior: the material’s apparent viscosity tends to drop as shear rate increases, after yield is exceeded.
n greater than 1 indicates shear‑thickening behavior: resistance to flow tends to rise with increasing shear rate, after yield is exceeded.
Most issues come from τ ≤ τ₀, zero or negative shear rate, or inconsistent units. Make sure all stress terms use the same unit and γ̇ is greater than zero.
For quick checks, one point can help. For reliable material characterization, fit multiple τ–γ̇ pairs and estimate τ₀, K, and n together using regression.
Yes. Many non‑Newtonian materials change structure with temperature, which can shift both K and n. Record temperature and measurement geometry when comparing results.
Use the CSV button for a spreadsheet row, or the PDF button for a printable report. Exports include your inputs, the computed n value, and key unit notes.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.