Model non‑Newtonian flow with the Ellis viscosity relation. Estimate viscosity, shear stress, and shear rate. Use trusted units, then export results in seconds today.
The Ellis model is commonly written using shear stress. The apparent viscosity is:
μ(τ) = μ₀ / [ 1 + (|τ|/τ½)^(α−1) ]
The corresponding shear-rate relation is:
γ̇ = (τ/μ₀) [ 1 + (|τ|/τ½)^(α−1) ]
τ and then computes μ = τ/γ̇.
| μ₀ (Pa·s) | τ½ (Pa) | α | τ (Pa) | γ̇ (1/s) | μ(τ) (Pa·s) |
|---|---|---|---|---|---|
| 2.00 | 25 | 2.0 | 10 | 7.00 | 1.43 |
| 2.00 | 25 | 2.0 | 25 | 25.00 | 1.00 |
| 2.00 | 25 | 2.0 | 50 | 150.00 | 0.67 |
| 5.00 | 40 | 2.8 | 20 | 6.71 | 2.98 |
| 5.00 | 40 | 2.8 | 80 | 107.03 | 0.75 |
The Ellis model describes shear‑thinning liquids whose viscosity decreases smoothly with increasing shear stress. It is useful when you need a simple, stable curve that matches many polymer solutions, paints, and suspensions without a yield stress term. At very low stress it approaches μ≈μ₀.
The zero‑shear viscosity μ₀ represents the low‑stress plateau, while τ½ is the shear stress where viscosity drops to about half of μ₀. The exponent α controls how quickly thinning develops; larger α produces a sharper transition. These parameters are typically obtained by fitting a flow curve.
Datasheets often report Ellis parameters using shear stress, but many experiments are recorded as shear rate. When shear rate is provided, this calculator solves for τ that satisfies the Ellis relation and then computes μ = τ/γ̇ for your selected output units. This avoids manual iteration and reduces unit mistakes.
For moderately viscous solutions, μ₀ may range from 0.05 to 50 Pa·s, while τ½ can vary from a few pascals to several hundred pascals depending on structure and concentration. Values of α commonly fall between about 1.5 and 4 in practical fits. Use the example table as a sanity check, not a specification.
Ellis equations require consistent units because τ½ and τ must share the same stress unit and μ₀ must match the viscosity unit used in τ/μ₀. This tool offers Pa, kPa, psi, and bar plus common viscosity units such as Pa·s, mPa·s, and cP. Keep track of absolute versus gauge pressure separately; the model uses shear stress.
The apparent viscosity is the effective resistance to flow at the chosen operating point. Lower μ generally implies easier pumping and lower pressure drop, while higher μ indicates stronger resistance. Evaluate several τ or γ̇ points to see how rapidly viscosity decreases and where the curve flattens.
Engineers apply Ellis fits in pipe‑flow estimates, mixer sizing, coating and extrusion calculations, and CFD material models where a smooth viscosity function is preferred. A continuous μ(τ) curve can improve numerical stability compared with piecewise rules and simplifies documentation in process sheets.
Ellis does not capture yield stress, strong time dependence, viscoelasticity, or thixotropy. Fit parameters at the correct temperature and formulation, and validate predictions with measured flow‑curve data. If a yield is evident, consider Herschel–Bulkley or Bingham alternatives.
μ₀ is the zero‑shear viscosity, the low‑stress plateau value. It approximates the viscosity you would measure at very small shear stress or very small shear rate before shear‑thinning becomes significant.
τ½ is the characteristic shear stress at which the viscosity falls to roughly half of μ₀. It sets the stress scale for the onset of thinning and helps position the transition region on the curve.
α is usually obtained by fitting experimental data. Higher α means a steeper drop in viscosity with stress, while α near 1 approaches Newtonian behavior. Start with your rheometer fit rather than guessing.
When shear rate is the input, the Ellis relation is implicit in τ. The calculator uses a stable numerical solver to find τ that matches your γ̇ and then computes μ = τ/γ̇ consistently.
Not reliably. The Ellis model does not include a yield term, so it may under‑predict resistance at very low stresses for materials that behave like they have a yield. Consider yield‑stress models if your data show a clear intercept.
Fit μ₀, τ½, and α at the same temperature you will operate. If temperature varies, build a separate temperature correlation for μ₀ (and sometimes τ½) and use parameters appropriate to each condition.
Use the unit your process and standards expect. Pa·s is SI; mPa·s and cP are convenient for low viscosities and are numerically equal. The calculator converts units so you can report consistently.
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.