Polymer Melt Rheology, Viscosity & Shear Flow Fundamentals

Polymer melt rheology governs every major plastic processing method, including injection molding, extrusion, film blowing, and blow molding. Because polymer melts exhibit non-Newtonian, pseudoplastic (shear-thinning) behavior, their apparent viscosity drops by several orders of magnitude under high shear rates in runner gates and extrusion dies. Accurate rheological characterization prevents mold filling defects, die swell, melt fracture, and thermal degradation during high-speed compounding.

By completing this lesson, you will be able to:

• Explain pseudoplastic shear-thinning behavior and the Cross/Carreau viscosity models.
• Calculate uncorrected vs Rabinowitsch-corrected wall shear rate ((\dot{\gamma}_w)) and apparent viscosity ((\eta)) in capillary flow.
• Compare Newtonian vs Non-Newtonian polymer fluid dynamics.
• Diagnose die swell and sharkskin surface melt fracture incorporating Bagley end corrections.

3.1 Power-Law (Ostwald-de Waele) Model

The relationship between shear stress ((\tau)) and shear rate ((\dot{\gamma})) is expressed as:

$$\tau = K \dot{\gamma}^n$$ $$\eta = \frac{\tau}{\dot{\gamma}} = K \dot{\gamma}^{n-1}$$

Where:

• (K) = Consistency index ((\text{Pa}\cdot\text{s}^n))
• (n) = Flow behavior index (dimensionless). For shear-thinning polymer melts, (n < 1).

graph LR
    A["Zero Shear Viscosity (η₀)"] --> B["Shear Thinning Power-Law Region (Slope = n-1)"]
    B --> C["Infinite Shear Viscosity (η_∞)"]

3.2 Capillary Flow Corrections: Rabinowitsch & Bagley

$$\dot{\gamma}_w = \left( \frac{3n + 1}{4n} \right) \dot{\gamma}_{app}$$ $$\tau_w = \frac{\Delta P - \Delta P_e}{2(L/R + e)}$$

Worked Numerical Example:

<div className="problem-statement">

Problem: Polypropylene melt with power-law index (n = 0.35) flows through a capillary die of radius (R = 1.0\text{ mm}) (0.001 m) and length (L = 30\text{ mm}) (0.03 m). Pressure drop is (\Delta P = 15\text{ MPa}) ((15 \times 10^6\text{ Pa})), and volumetric flow rate is (Q = 1.2 \times 10^{-6}\text{ m}^3/\text{s}). Calculate uncorrected wall shear stress, apparent wall shear rate, Rabinowitsch-corrected true wall shear rate, and true viscosity.

</div> <div className="solution-step">

Solution:

1. Uncorrected Wall Shear Stress:

$$\tau_w = \frac{15 \times 10^6 \times 0.001}{2 \times 0.03} = 250,000\text{ Pa} = 250\text{ kPa}$$

1. Apparent Wall Shear Rate:

$$\dot{\gamma}_{app} = \frac{4Q}{\pi R^3} = \frac{4 \times (1.2 \times 10^{-6})}{3.14159 \times (0.001)^3} = 1527.88\text{ s}^{-1}$$

1. Rabinowitsch Corrected True Wall Shear Rate (for $n=0.35$):

$$\text{Correction Factor} = \frac{3(0.35) + 1}{4(0.35)} = \frac{2.05}{1.40} = 1.4643$$ $$\dot{\gamma}_{true} = 1.4643 \times 1527.88 = 2237.28\text{ s}^{-1}$$

1. True Viscosity:

$$\eta_{true} = \frac{250,000}{2237.28} = 111.74\text{ Pa}\cdot\text{s} \quad (\text{vs Apparent Viscosity of } 163.62\text{ Pa}\cdot\text{s})$$

• Injection Molding Gate Design: High shear rates ((10^4 - 10^5\text{ s}^{-1})) reduce viscosity dramatically.
• Extrusion Die Swell Management: Polymer chains orientation relaxes upon exiting the die. (Illustrative Indian industry scenario based on standard polyolefin compounding practices).

• Polymer melts are shear-thinning ((n < 1)).
• Apparent viscosity overestimates melt flow resistance without Rabinowitsch correction.
• Rabinowitsch Correction: Adjusts apparent shear rate for non-Newtonian velocity profiles.
• Bagley Correction: Accounts for extensional pressure drops at capillary entry/exit.

1. Macosko, C. W. (1994). Rheology: Principles, Measurements, and Applications, VCH Publishers.
2. ASTM D3835 — Standard Test Method for Determination of Properties of Polymeric Materials by Capillary Rheometer.