Hong Life CalculatorsUnderstanding the power-law fluid viscosity calculator — concept and principle
Many everyday and industrial fluids do not have a single viscosity. Paint, ketchup and polymer melts get thinner the faster they are sheared; some dense suspensions get thicker. The power-law (Ostwald–de Waele) model describes this with two parameters, and this calculator gives the apparent viscosity and shear stress at a chosen shear rate and classifies the fluid.
How it is calculated
- Shear stress: τ = K × γ̇ⁿ (Pa).
- Apparent viscosity: μ_app = τ ÷ γ̇ = K × γ̇ⁿ⁻¹ (Pa·s).
- Type: n < 1 shear-thinning, n = 1 Newtonian, n > 1 shear-thickening.
K is the consistency index in Pa·sⁿ, n the dimensionless flow behaviour index, and γ̇ the shear rate in 1/s. With n = 1, K is the Newtonian viscosity.
The principle
For a Newtonian fluid such as water, stress is proportional to shear rate, so viscosity is constant. In polymer solutions and melts, long molecules are tangled at rest; under shear they untangle and align with the flow, so resistance grows less than proportionally and viscosity falls. Plotting log τ against log γ̇ for such fluids often gives a straight line over a useful range: its slope is n and its value at γ̇ = 1 s⁻¹ is K. The power law is that straight line written as an equation.
Worked example
- Defaults: K = 10 Pa·sⁿ, n = 0.5, γ̇ = 100 s⁻¹.
- Shear stress: τ = 10 × 100^0.5 = 10 × 10 = 100 Pa.
- Apparent viscosity: μ_app = 10 × 100^(0.5 − 1) = 10 × 0.1 = 1 Pa·s.
- Because n = 0.5 < 1, the fluid is shear-thinning.
| Flow index n | Apparent viscosity (Pa·s) |
|---|---|
| 0.125 | 0.177827941 |
| 0.25 | 0.316227766 |
| 0.375 | 0.5623413252 |
| 0.5 | 1 |
| 0.625 | 1.77827941 |
| 0.75 | 3.16227766 |
| 1 | 10 |
How to read the result
With n = 0.5, every hundredfold increase in shear rate cuts the apparent viscosity tenfold: 10 Pa·s at 1 s⁻¹, 1 Pa·s at 100 s⁻¹ and 0.1 Pa·s at 10,000 s⁻¹. That is why a shear-thinning paint stays on the brush and wall (low shear) yet spreads easily when brushed (high shear). Always state the shear rate when quoting the viscosity of a non-Newtonian fluid; a single number is meaningless without it.
Common mistakes
- Comparing K values with different n; the units of K (Pa·sⁿ) depend on n, so only fluids with similar n compare directly.
- Extrapolating outside the measured range; real fluids often show constant-viscosity plateaus at very low and very high shear rates.
- Using the power law for fluids with a yield stress, such as toothpaste, which need models like Herschel–Bulkley or Bingham.
Frequently asked questions
What is the difference between shear-thinning and thixotropy?
Shear-thinning is an immediate dependence on shear rate; thixotropy is a time-dependent drop in viscosity under constant shear that recovers slowly at rest.
What model is used when the power law fails at low shear?
The Carreau or Cross models add a zero-shear viscosity plateau and blend into power-law behaviour at higher rates.
Is blood shear-thinning?
Yes, because red blood cells aggregate at low shear and disperse at high shear.
Sources: Bird, R. B., Stewart, W. E. & Lightfoot, E. N. Transport Phenomena. Wiley. · Chhabra, R. P. & Richardson, J. F. Non-Newtonian Flow and Applied Rheology. Butterworth-Heinemann.