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Cross Model Polymer Melt Viscosity Calculator (Shear-Thinning)

Molten thermoplastics exhibit non-Newtonian pseudoplastic (shear-thinning) flow, where apparent viscosity drops by orders of magnitude at high processing shear rates.

Operating shear rate in processing machinery (e.g. 100-1,000 s⁻¹ for extrusion, 1,000-10,000 s⁻¹ for injection molding gate).

Low-shear limiting viscosity at rest (typically 1,000 to 50,000 Pa·s).

Limiting viscosity at extreme shear rates (often assumed 0 or small constant).

Time constant marking the onset of shear thinning (typically 0.01 to 0.5 s).

Power-law exponent (m = 1 - n; typically 0.6 to 0.85 for commercial plastics).

Calculated Result
262.0 Pa·s

Apparent Melt Viscosity (η)

Shear Stress (τ)

261.98 kPa

Viscosity Drop Ratio (η₀/η)

19.1 × lower than zero-shear

Relaxation Dimensionless Rate

50.0

Calculation Breakdown

  1. Cross Formulationη = η∞ + (η₀ - η∞) / [1 + (λ·γ̇)^m] = 262.0 Pa·s
  2. Pseudoplastic Flow Stressτ = η · γ̇ = 261.98 kPa

What Is the Cross Model Polymer Melt Viscosity Calculator (Shear-Thinning)?

The Cross model is one of the most widely used rheological equations in polymer processing and mold-filling software (such as Moldflow and Moldex3D).

It mathematically describes how entangled macromolecular polymer chains disentangle and align parallel to the flow direction under shear, reducing hydrodynamic flow resistance.

How Does the Cross Model Polymer Melt Viscosity Calculator (Shear-Thinning) Work?

At low shear rates (γ̇ << 1/λ), thermal Brownian motion re-entangles chains faster than flow can orient them, yielding a constant zero-shear viscosity η₀.

At high shear rates (γ̇ >> 1/λ), alignment dominates, and viscosity decreases with slope -m.

Cross Model Polymer Melt Viscosity Calculator (Shear-Thinning) Formula & Variables

The core mathematical equation utilized by this calculator is expressed as:

η(γ̇) = η_∞ + (η_0 - η_∞) / [ 1 + (λ · γ̇)^m ], τ = η(γ̇) · γ̇

Four-parameter Cross model unifying the low-shear Newtonian plateau with high-shear power-law pseudoplasticity.

How to Use the Cross Model Polymer Melt Viscosity Calculator (Shear-Thinning)

  1. Enter the processing shear rate corresponding to your die or mold runner geometry.
  2. Input the fitted Cross model constants for the specific polymer resin grade.
  3. Use the resulting apparent viscosity to calculate pressure drop and pumping power.

Step-by-Step Example Calculation

HDPE Melt at 1000 s⁻¹ Injection Rate

Input Values:

shearRatePerSec:1000
zeroShearViscosityPaS:5000
infiniteShearViscosityPaS:10
crossTimeConstantLambdaSec:0.05
crossPowerLawIndexM:0.75
Worked Steps: Predicts viscosity drops from 5000 Pa·s at rest down to ~276 Pa·s at 1000 s⁻¹.

Understanding Your Result

Shear-thinning factor indicates how many times thinner the melt becomes compared to its rest state.

Factors That Affect the Result

  • Temperature: Higher melt temperatures shift the entire viscosity curve downward according to the WLF or Arrhenius equation.
  • Molecular weight: Higher molecular weight elevates the zero-shear plateau η₀ (scaling as Mw^3.4).

When Should You Use This Calculator?

  • Mold filling and gate pressure drop simulation.
  • Sizing extrusion dies and hot runner manifolds.

Assumptions & Limitations

  • Applies to steady shear flow; does not capture viscoelastic extensional thickening or normal stress differences.

Frequently Asked Questions

Calculation Accuracy & Reference Note

Standard industry polymer rheology model.

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