Calculator
Choose a mode, enter parameters, and compute viscosity at one or many shear rates.
Formula Used
The Cross model describes shear-thinning viscosity as a smooth transition from a high, low-shear plateau to a lower, high-shear plateau:
- η(γ̇): viscosity at shear rate γ̇ (Pa·s)
- η0: zero-shear viscosity (Pa·s)
- η∞: infinite-shear viscosity (Pa·s)
- k: time constant (s)
- m: dimensionless exponent controlling transition sharpness
How to Use This Calculator
- Select a calculation mode: single value, list, or range.
- Pick the shear rate unit that matches your dataset.
- Enter η0, η∞, k, and m from fitting or literature.
- Provide shear rate inputs, then press Calculate.
- Review the table and export results using CSV or PDF.
Tip: When fitting parameters, keep units consistent across experiments.
Example Data Table
Sample values for a shear-thinning fluid (illustrative only).
| γ̇ (1/s) | η0 (Pa·s) | η∞ (Pa·s) | k (s) | m | η(γ̇) (Pa·s) |
|---|---|---|---|---|---|
| 1 | 12 | 0.9 | 0.15 | 0.9 | ~11.14 |
| 10 | 12 | 0.9 | 0.15 | 0.9 | ~7.28 |
| 50 | 12 | 0.9 | 0.15 | 0.9 | ~3.04 |
| 100 | 12 | 0.9 | 0.15 | 0.9 | ~2.20 |
1) Why the Cross model matters
Many polymer melts, paints, food gels, and drilling fluids thin as shear rate increases. The Cross model captures this behavior with a smooth transition from a low-shear plateau to a high-shear plateau. It is widely used because it is stable, differentiable, and easy to fit across several decades of shear rate.
2) What the parameters represent
The model uses four parameters: η0 is the zero-shear viscosity, η∞ is the infinite-shear viscosity, k is a characteristic time scale (seconds), and m controls how sharp the transition is. When k is larger, shear thinning begins at lower shear rates. When m increases, the drop in viscosity becomes steeper.
3) Typical data ranges in practice
In rotational rheometry, shear rates commonly span about 0.01 to 1000 s−1. Fluids may show η0 values from 0.01 Pa·s (thin oils) to 103 Pa·s (highly structured materials). For many engineered fluids, η∞ is nonzero and often falls between 1% and 30% of η0, depending on the formulation and temperature.
4) How to estimate starting values
A practical starting point is to read η0 from the lowest stable shear rate region and η∞ from the highest shear rate plateau, if it exists. The transition shear rate is often near 1/k. For example, k = 0.2 s suggests thinning becomes noticeable around 5 s−1. Choose m between 0.5 and 2 for many shear-thinning systems.
5) Interpreting the viscosity table
This calculator reports viscosity in Pa·s and cP, and also shows (k·γ̇)m and the denominator 1 + (k·γ̇)m. When (k·γ̇)m is much smaller than 1, viscosity approaches η0. When it is much larger than 1, viscosity approaches η∞. The range mode is useful for generating evenly spaced shear-rate sweeps for reports.
6) Comparing Cross and power-law fits
A simple power-law model can match the mid-range slope but does not provide plateau limits, so it can overpredict at very low or very high shear rates. The Cross model includes both plateaus, which makes it better for process simulation where equipment may operate outside a narrow measurement window.
7) Quality checks for reliable results
Ensure η∞ ≤ η0 and keep unit consistency for shear rate. If your data are in min−1 or hr−1, convert them before fitting, or select the matching unit here. If the calculated curve seems shifted, verify the time constant k is in seconds and not in minutes. Also review repeatability at low shear where measurements are sensitive to slip.
8) Using results in engineering decisions
Cross parameters can feed CFD and pressure-drop calculations in pipelines, mixing tanks, and coating flows. Because k and m determine the thinning onset and slope, they are helpful for comparing formulations at the same temperature. Export the CSV for quick plotting, or generate a PDF report for design reviews and documentation.
1) What is the Cross model used for?
It models shear-thinning viscosity with realistic low-shear and high-shear plateaus. This is useful for polymers, paints,
suspensions, and many process fluids that change viscosity with shear rate.
2) What units should I use for viscosity?
Enter η0 and η∞ in Pa·s. The calculator also reports cP for convenience, where 1 Pa·s equals 1000 cP.
3) How do I choose k?
k is a time scale in seconds. A rough guide is that thinning starts near γ̇ ≈ 1/k. If thinning begins around 10 s−1,
a starting k is about 0.1 s.
4) What does the exponent m control?
m controls the sharpness of the transition. Larger m makes the viscosity drop more rapidly with shear rate, while smaller
m creates a gentler change across the transition region.
5) Why is η∞ sometimes set to zero?
Some fluids approach very low viscosity at high shear, so η∞ is small compared with η0. Setting η∞ to zero is an
approximation when the high-shear plateau is not clearly measurable.
6) Can I use a list of shear rates from my experiment?
Yes. Choose “List of shear rates” and paste your values separated by commas or spaces. The output table will compute
viscosity for each point and can be exported as CSV or PDF.
7) What if my curve looks wrong?
Check that η∞ is not larger than η0, confirm k is in seconds, and verify your shear-rate unit selection. If needed,
reduce the range points to avoid extremely dense tables.