Guide to the Rheological Model Fitter

Short Intro

Rheology is the study of how materials deform and flow in response to applied force. This Rheological Model Fitter calculator is an educational tool designed to help students, researchers, and formulators analyze experimental viscosity data by fitting it to common mathematical models. Understanding a material's rheological behavior is critical in fields like pharmaceuticals, food science, cosmetics, and materials engineering for predicting stability, processability, and sensory performance.

What This Calculator Does

The tool automates the process of fitting experimental shear stress versus shear rate data to four fundamental rheological models. By inputting raw data, users can quickly determine the key parameters that define the fluid's behavior according to each model. The calculator provides:

• Fitting to Newtonian, Bingham Plastic, Power Law, and Herschel-Bulkley models.
• Calculation of key parameters such as viscosity, yield stress, consistency index, and flow behavior index.
• An R-squared (R²) value to indicate the goodness of fit for the chosen model.

When to Use It

This tool is useful in various scenarios, including:

• Academic Learning: For students learning about fluid mechanics and non-Newtonian behaviors.
• Formulation Screening: In early-stage R&D to quickly compare the rheological profiles of different prototypes (e.g., creams, gels, suspensions).
• Quality Control: As a preliminary check to see if a batch of material conforms to an expected rheological model.
• Data Visualization: To quickly extract model parameters from raw rheometer data for presentations or reports.

Inputs Explained

The tool requires a set of experimental data points, with each point consisting of two values:

• Shear Rate (γ̇): The rate at which a fluid is sheared or "worked." It is measured in inverse seconds (s⁻¹). This is typically the independent variable controlled by the rheometer.
• Shear Stress (τ): The force per unit area required to produce the shear rate. It is measured in Pascals (Pa). This is the resulting stress measured by the rheometer.

Data should be entered as two columns, with each row representing a single data point (e.g., 10, 55). A minimum of three data points is required for a meaningful fit.

Results Explained

The output provides the calculated parameters for the selected model:

• Newtonian Model:
  • Viscosity (η): The constant of proportionality between shear stress and shear rate, measured in Pascal-seconds (Pa·s). Represents the fluid's resistance to flow.
• Power Law (Ostwald-de Waele) Model:
  • Consistency Index (K): A measure of the fluid's average viscosity, in Pa·sⁿ.
  • Flow Behavior Index (n): A dimensionless number indicating the degree of non-Newtonian behavior. If n < 1, the fluid is shear-thinning. If n > 1, it is shear-thickening. If n = 1, it behaves like a Newtonian fluid.
• Bingham Plastic Model:
  • Yield Stress (τ₀): The minimum shear stress required to initiate flow, measured in Pa. Below this stress, the material behaves like a solid.
  • Plastic Viscosity (ηₚ): The viscosity of the fluid once it has started to flow, measured in Pa·s.
• Herschel-Bulkley Model:
  • Combines the Bingham and Power Law models. It includes a Yield Stress (τ₀), a Consistency Index (K), and a Flow Behavior Index (n).
• R-squared (R²): A statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable. A value closer to 1.0 indicates a better fit of the model to the data.

Formula / Method

The calculator uses linear regression on transformed data to fit the models. This is a common and computationally simple approach.

Newtonian

The model is τ = η * γ̇. The tool performs a linear regression on τ vs. γ̇. The slope of the line is the viscosity η.

Power Law (Ostwald-de Waele)

The model is τ = K * (γ̇)ⁿ. To fit this, the equation is linearized by taking the logarithm of both sides: log(τ) = log(K) + n * log(γ̇). The tool performs a linear regression on log(τ) vs. log(γ̇). The slope is the flow behavior index (n), and the antilog of the intercept is the consistency index (K).

Bingham Plastic

The model is τ = τ₀ + ηₚ * γ̇ for τ > τ₀. The tool performs a linear regression on the high-shear-rate portion of the data, where the relationship is linear. The slope of this line is the plastic viscosity (ηₚ), and the y-intercept is the yield stress (τ₀).

Herschel-Bulkley

The model is τ = τ₀ + K * (γ̇)ⁿ. The tool uses a simplified fitting method where the yield stress (τ₀) is estimated from the lowest shear stress data point. This value is then subtracted from all stress data (τ - τ₀), and a Power Law fit is performed on the remaining data to find K and n.

Step-by-Step Example

Let's analyze a shear-thinning fluid without a yield stress using the Power Law model.

1. Input Data: Suppose we have the following experimental data (Shear Rate, Shear Stress):

0.5, 3.5
1, 6.0
5, 21.2
10, 35.7
50, 126.2
100, 212.1
            

2. Select Model: Choose "Power Law (Ostwald-de Waele)" from the dropdown menu.

3. Fit Model: After clicking "Fit Model," the tool linearizes the data (log-log plot) and performs a regression.

4. Interpret Results: The output will be similar to this:

The R² value is very close to 1, indicating an excellent fit. The flow behavior index (n) is 0.850, which is less than 1, confirming the fluid is shear-thinning (its viscosity decreases as shear rate increases).

Tips + Common Errors

• Data Quality is Key: Ensure your data is clean and free from artifacts. Outliers, especially at low shear rates, can significantly skew model parameters.
• Sufficient Data Points: Use at least 5-10 data points spanning several decades of shear rate for a robust fit. The minimum is 3, but more is better.
• Model Selection: Visually inspect a plot of your data first. If it looks like a straight line through the origin, try Newtonian. If it's a curve starting from the origin, try Power Law. If it intercepts the y-axis, try Bingham or Herschel-Bulkley.
• Log-Log Fit Failures: The Power Law and Herschel-Bulkley models require log transformations. This will fail if your data contains zero or negative values for shear rate or stress. Ensure all data points are positive.
• Simplified Fits: Be aware that the methods used here (especially for Bingham and Herschel-Bulkley) are simplified. For publication-quality results, non-linear regression algorithms are often preferred.

Frequently Asked Questions

1. What is the difference between the Bingham Plastic and Herschel-Bulkley models?
   The Bingham model assumes that once the yield stress is overcome, the fluid has a constant viscosity (plastic viscosity). The Herschel-Bulkley model is more general; it assumes that after yielding, the fluid's behavior follows a Power Law, meaning it can be shear-thinning or shear-thickening.
2. What does a "good" R-squared (R²) value mean?
   An R² value close to 1.0 (e.g., >0.98) generally indicates that the chosen model is a good mathematical description of the data. However, a good fit does not guarantee the model is physically correct for the material.
3. Why did I get an error saying "not enough data points"?
   The tool requires at least three valid data points to perform a regression. For some models, like the simplified Herschel-Bulkley fit, it needs at least three points remaining after accounting for the yield stress. Ensure your input contains enough valid numerical pairs.
4. Can I use this tool for viscoelastic materials?
   No. This tool is for steady-state shear viscosity measurements only. It does not analyze viscoelastic properties like storage modulus (G') or loss modulus (G'').
5. What are typical units for rheological parameters?
   The standard SI units used by the tool are: Shear Rate (s⁻¹), Shear Stress (Pa), Viscosity (Pa·s), Yield Stress (Pa), and Consistency Index (Pa·sⁿ).
6. My material is shear-thickening. Which model should I use?
   The Power Law or Herschel-Bulkley models can describe shear-thickening behavior. This occurs when the Flow Behavior Index (n) is greater than 1.
7. What is the physical meaning of the Consistency Index (K)?
   K is analogous to viscosity. For a Power Law fluid, K is numerically equal to the shear stress when the shear rate is exactly 1.0 s⁻¹. Its units (Pa·sⁿ) depend on n, making it tricky to compare K values for fluids with different n values.
8. Why does the Bingham model only use high-shear data for its fit?
   At low shear rates, near the yield stress, the flow can be unstable or non-linear. The Bingham model's linear relationship is most accurately observed at higher shear rates where flow is fully developed. The tool fits this linear region to find the plastic viscosity and extrapolates back to find the yield stress.
9. Can a fluid have a negative yield stress?
   Physically, no. A negative yield stress from a model fit is usually a mathematical artifact, indicating that the Bingham or Herschel-Bulkley model is likely inappropriate for the data. The tool correctly caps the yield stress at zero.
10. How does this tool differ from professional rheology software?
    This is a simplified educational tool using basic linear regression. Professional software often employs more sophisticated non-linear regression algorithms, offers more models, and provides advanced statistical analysis and graphical plotting capabilities.

References

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