This guide explains the principles behind the Shear Rate vs. Shear Stress Plotter, a fundamental tool in rheology for characterizing the flow behavior of fluids. Understanding this relationship is crucial for predicting how materials will behave during processing, application, and storage.

What This Calculator Does

The tool is designed to process and visualize rheological data. Its primary functions are:

• Data Visualization: It plots shear stress (τ) as a function of shear rate (γ̇), providing an immediate visual representation of a fluid's behavior.
• Viscosity Calculation: It calculates apparent viscosity (η = τ / γ̇) at each data point and can plot viscosity against shear rate.
• Rheological Modeling: It fits experimental data to several common mathematical models (e.g., Newtonian, Bingham Plastic, Power Law) to quantify fluid properties.
• Parameter Extraction: From the model fits, it extracts key rheological parameters like yield stress, plastic viscosity, consistency index, and flow behavior index.

When to Use It

This tool is invaluable in various scientific and industrial fields for analyzing fluids that do not exhibit simple (Newtonian) behavior. Common applications include:

• Product Development: Formulating products like paints, cosmetics, and food items (e.g., ketchup, yogurt) to achieve desired texture and flow properties.
• Quality Control: Ensuring batch-to-batch consistency in manufacturing processes by verifying that a product's rheological profile meets specifications.
• Process Engineering: Designing pumping and mixing systems by understanding how a fluid's viscosity changes under different flow conditions.
• Research: Studying the fundamental properties of complex fluids, such as polymer solutions, suspensions, and emulsions.

Inputs Explained

The tool requires two columns of experimental data measured by a rheometer:

• Shear Rate (γ̇ or D): This represents the rate at which a fluid is deformed or sheared. It is essentially a velocity gradient within the fluid. Common units are reciprocal seconds (1/s or s⁻¹) or revolutions per minute (rpm), which the tool converts to s⁻¹.
• Shear Stress (τ): This is the force per unit area required to produce the shearing flow. It is a measure of the internal resistance of the fluid to deformation. Common units are Pascals (Pa) or dyne per square centimeter (dyn/cm²).

Results Explained

The primary outputs are the plot and the fitted model parameters:

• The Plot: A graph showing the relationship between shear rate and shear stress. The shape of this "flow curve" identifies the fluid type (e.g., Newtonian, shear-thinning, shear-thickening).
• Fitted Parameters: These numbers quantify the fluid's behavior. For example, in a Power Law model (τ = Kγ̇ⁿ):
  • K (Consistency Index): A measure of the fluid's average viscosity. Higher K means a more viscous fluid.
  • n (Flow Behavior Index): Describes the degree of non-Newtonian behavior.
    • n = 1: Newtonian (viscosity is constant)
    • n < 1: Shear-thinning (viscosity decreases with increasing shear rate)
    • n > 1: Shear-thickening (viscosity increases with increasing shear rate)
  • τ₀ (Yield Stress): A parameter from models like Bingham and Herschel-Bulkley. It's the minimum stress required to initiate flow. Below this stress, the material behaves like a solid.
• R² (Coefficient of Determination): Indicates how well the chosen model fits the experimental data. A value closer to 1.0 suggests a better fit.

Formula / Method

The tool applies non-linear regression (Levenberg-Marquardt algorithm) to fit the data to one of several standard rheological models:

Common Rheological Models

• Newtonian: Describes ideal fluids like water and oils.
  τ = η ⋅ γ̇
• Bingham Plastic: Models materials that behave as a rigid solid at low stress but flow with a constant viscosity above a yield stress (e.g., toothpaste).
  τ = τ₀ + ηₚ ⋅ γ̇
• Power Law (Ostwald-de Waele): Widely used for many non-Newtonian fluids without a yield stress (e.g., polymer solutions, paints).
  τ = K ⋅ (γ̇)ⁿ
• Herschel-Bulkley: A general model that combines Bingham and Power Law behaviors, describing fluids with both a yield stress and shear-dependent viscosity (e.g., ketchup, drilling mud).
  τ = τ₀ + K ⋅ (γ̇)ⁿ

Here, τ is shear stress, γ̇ is shear rate, η is Newtonian viscosity, τ₀ is yield stress, ηₚ is plastic viscosity, K is the consistency index, and n is the flow behavior index.

Step-by-Step Example

Let's analyze a shear-thinning fluid like a polymer solution.

1. Collect Data: Suppose a rheometer provides the following data (Shear Rate in 1/s, Shear Stress in Pa):
   1, 2.5
10, 15.8
50, 56.2
100, 100.0
2. Input Data: Paste these values into the tool's data input area and ensure the units are set to "1/s" and "Pa".
3. Visualize: The tool will plot these four points. You'll observe a curve that is not a straight line through the origin, indicating non-Newtonian behavior. The slope decreases as shear rate increases.
4. Select a Model: Since there's no apparent yield stress and the viscosity clearly changes, the Power Law model is a good starting point. Select it from the dropdown.
5. Fit and Interpret: Click "Fit Model." The tool might return parameters like K ≈ 2.5 Pa·sⁿ and n ≈ 0.8. Since n < 1, this confirms the fluid is shear-thinning. An R² value close to 1.0 (e.g., 0.999) would indicate an excellent fit. The tool will also overlay the fitted Power Law curve on the data points.

Tips + Common Errors

• Correct Units: Always double-check that the selected units in the tool match the units of your raw data. Mismatches are a common source of error.
• Data Quality: Ensure your data is clean. Erroneous points, especially at very low shear rates, can significantly skew model fitting. Consider removing outliers if they are known measurement artifacts.
• Choosing a Model: Visually inspect the plot first. If the data appears to intercept the y-axis above zero, a model with a yield stress (Bingham, Herschel-Bulkley) is appropriate. If it goes through the origin, Power Law or Newtonian models are better choices.
• Log vs. Linear Scale: Use the log-log scale plotting option to better visualize data that spans several orders of magnitude, which is common in rheology. A Power Law fluid will appear as a straight line on a log-log plot.
• Over-interpretation: A high R² value is good, but physical relevance is key. Ensure the chosen model makes sense for the type of material you are studying.

Frequently Asked Questions (FAQs)

1. What's the difference between shear rate and shear stress?
   Shear rate is the speed of deformation (how fast you're stirring or spreading), while shear stress is the force required to cause that deformation. Viscosity is the ratio of stress to rate.
2. What is a non-Newtonian fluid?
   A non-Newtonian fluid is one whose viscosity changes depending on the applied shear rate. Most complex fluids, from ketchup to blood, are non-Newtonian.
3. How do I know if my fluid is shear-thinning or shear-thickening from the plot?
   On a stress vs. rate plot, a shear-thinning fluid's curve bends towards the x-axis (slope decreases), while a shear-thickening fluid's curve bends towards the y-axis (slope increases). On a viscosity vs. rate plot, the trend is more direct: viscosity decreases for shear-thinning and increases for shear-thickening.
4. What does yield stress (τ₀) physically represent?
   It's the minimum force needed to make the material flow. Think of toothpaste: it sits on your brush (behaving like a solid) until you apply enough force (stress) to make it spread.
5. Why is my viscosity plot noisy at low shear rates?
   At very low shear rates, the measured shear stress is also very low and often close to the rheometer's resolution limit. Since viscosity is calculated by dividing stress by rate, any small error in the stress measurement gets magnified when divided by a tiny shear rate, leading to noise.
6. How should I handle a zero shear rate data point?
   A data point at zero shear rate should theoretically have zero shear stress (unless there is residual stress). This point cannot be displayed on a logarithmic scale. It's often used to anchor a model but is typically excluded from log plots.
7. What is a good R² value for model fitting?
   For clean experimental data, an R² value above 0.98 is generally considered a good fit. However, a good fit does not guarantee the model is physically correct for the material.
8. Can this tool analyze viscoelastic materials?
   No. This tool is for steady-state shear rheology, which measures how a material flows. It does not analyze viscoelastic properties like elasticity, which are typically measured with oscillatory tests (G', G'').
9. What is the difference between the Bingham and Herschel-Bulkley models?
   The Bingham model assumes that once the yield stress is overcome, the fluid has a constant viscosity (it's a straight line). The Herschel-Bulkley model is more general, allowing the viscosity to change with shear rate (like a Power Law fluid) after the yield stress is overcome.

References

For further reading and a deeper understanding of the principles of rheology, consult these authoritative sources:

1. Mezger, T. G. (2020). The Rheology Handbook: For Users of Rotational and Oscillatory Rheometers. Vincentz Network GmbH & Co. KG.
2. Barnes, H. A., Hutton, J. F., & Walters, K. (1989). An Introduction to Rheology. Elsevier Science Publishers B.V. View on ScienceDirect
3. Schramm, G. (2000). A Practical Approach to Rheology and Rheometry. Gebrueder HAAKE GmbH.
4. Macosko, C. W. (1994). Rheology: Principles, Measurements, and Applications. VCH Publishers. View on Wiley Online Library

Disclaimer

This content is for informational and educational purposes only. It is not intended to be a substitute for professional engineering advice, quality control protocols, or scientific validation. The formulas and methods described are based on established principles of rheology, but their application to specific materials or processes requires expert judgment and experimental verification. Always consult with a qualified professional for specific applications.