Non-Newtonian Fluid Rheology: Power Law and Bingham Plastic Models

Typical Scale

Lab: 0.1–10 mL; Pilot: 10–100 L; Plant: 1–100 m³/hr

Key Standards

ASTM D3236, ISO 3219, DIN 53019

Industry Impact

Rheology errors cause ~12% of batch failures in FDA-regulated pharma manufacturing (ISPE 2022 Data)

Measurement Time

Full τ–γ̇ sweep: 2–15 min; yield stress determination: 10–60 min (creep recovery)

⚠️ Why It Matters

1

Incorrect rheological model selection

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2

Underestimation of pumping pressure drop

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3

Overdesign of pipe diameter and pump power

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4

Premature pipeline erosion or seal failure

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5

Batch-to-batch inconsistency in mixing or extrusion

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6

Product quality defects (e.g., uneven coating, phase separation)

📘 Definition

Non-Newtonian fluids exhibit shear stress that is not linearly proportional to shear rate, violating Newton’s law of viscosity. The Power Law model describes shear-thinning (n < 1) or shear-thickening (n > 1) behavior via τ = K·γ̇ⁿ, while the Bingham Plastic model captures yield-stress behavior (τ = τ₀ + μₚ·γ̇) where flow only initiates after exceeding a critical stress τ₀.

🎨 Concept Diagram

AI-generated illustration for visual understanding

💡 Engineering Insight

Never assume τ₀ = 0 just because a fluid flows under gravity—it may still require significant torque to restart in a static pipe loop. Field validation must always include a 'hold-and-restart' test at operating temperature, as yield stress can increase 3–10× upon gelation during dwell time.

📖 Detailed Explanation

All fluids resist flow—but Newtonian fluids (like water or mineral oil) do so consistently: double the stirring force, and flow rate doubles. Non-Newtonian fluids break this rule. Ketchup stays put until shaken hard; then it suddenly gushes—this is yield stress behavior. Toothpaste holds its shape on the brush but spreads easily when squeezed—another sign of τ₀.

The Power Law model (τ = K·γ̇ⁿ) is purely empirical but powerful for design: it collapses complex microstructural effects (particle alignment, polymer chain uncoiling) into two numbers—K and n. However, it predicts infinite viscosity at zero shear, which is unphysical. That’s why the Bingham model is preferred for materials with true solid-like response below τ₀—such as drilling muds that suspend cuttings when circulation stops.

Advanced applications demand hybrid models: the Herschel-Bulkley (τ = τ₀ + K·γ̇ⁿ) generalizes both Power Law and Bingham, while the Casson model better fits blood or chocolate. In CFD, convergence fails if τ₀ is misestimated by >15%, and transient simulations of start-up flow require adaptive time-stepping to resolve the yield surface propagation—often solved via augmented Lagrangian or Papanastasiou regularization.

🔄 Engineering Workflow

Step 1

Step 1: Obtain representative sample under process-relevant temperature and aging conditions

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Step 2

Step 2: Perform controlled-shear-rate rheometry (e.g., vane or concentric cylinder geometry)

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Step 3

Step 3: Fit experimental τ–γ̇ data to Power Law and Bingham models using nonlinear regression

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Step 4

Step 4: Validate model selection via residual analysis and goodness-of-fit (R² > 0.98, RMSE < 5% of max τ)

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Step 5

Step 5: Scale rheological parameters to process equipment using dimensionless similarity (Reₕₑᵣᵣ, Fr, We)

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Step 6

Step 6: Simulate flow fields (CFD) with non-Newtonian constitutive equations

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Step 7

Step 7: Commission with inline viscometry and pressure-drop monitoring during ramp-up

📋 Decision Guide

Rock/Field Condition	Recommended Design Action
τ₀ > 100 Pa and μₚ > 200 cP (e.g., fresh cement slurry)	Use positive displacement pumps; design for full yield stress overcoming at startup; include sweep flow in piping layout
n < 0.4 and K > 10 Pa·sⁿ (e.g., polymer melt in extrusion)	Select low-shear, high-torque extruder screws; avoid sharp bends; validate residence time distribution to prevent thermal degradation
τ₀ ≈ 0–20 Pa and n ≈ 0.6–0.7 (e.g., latex paint)	Use centrifugal pumps with wide-vane impellers; specify shear-thinning-compatible valves; validate spray atomization performance across shear rates

📊 Key Properties & Parameters

Consistency Index (K)

0.1–50 Pa·sⁿ for industrial suspensions and polymer melts

Empirical coefficient in the Power Law model representing apparent viscosity at unit shear rate

⚡ Engineering Impact:

Directly scales pressure drop in laminar flow; errors >20% in K cause >35% error in required pump head

Flow Behavior Index (n)

0.2–0.8 for most food, pharmaceutical, and paint formulations

Dimensionless exponent in the Power Law model indicating degree of shear thinning (n < 1) or thickening (n > 1)

⚡ Engineering Impact:

Determines whether flow becomes easier (n < 1) or harder (n > 1) with increasing agitation—critical for impeller selection and scale-up

Yield Stress (τ₀)

10–2000 Pa for drilling muds, pastes, and gels

Minimum shear stress required to initiate flow in Bingham plastics, below which material behaves as a solid

⚡ Engineering Impact:

Dictates minimum agitator torque and determines whether settled solids will resuspend during restart—key for tank cleaning and startup safety

Plastic Viscosity (μₚ)

10–500 cP for cement slurries and bentonite-based drilling fluids

Slope of the linear region in the Bingham model, representing resistance to flow *after* yield is exceeded

⚡ Engineering Impact:

Controls pressure gradient during steady-state circulation; high μₚ increases frictional losses and heat generation in long pipelines

📐 Key Formulas

Power Law Shear Stress

τ = K · γ̇ⁿ

Calculates shear stress for a given shear rate in Power Law fluids

Variables:

Symbol	Name	Unit	Description
τ	Shear Stress	Pa	Shear stress in Power Law fluid
K	Consistency Index	Pa·sⁿ	Material constant representing fluid consistency
γ̇	Shear Rate	s⁻¹	Rate of shear deformation
n	Flow Behavior Index	dimensionless	Exponent indicating flow behavior (n < 1: pseudoplastic, n > 1: dilatant, n = 1: Newtonian

Typical Ranges:

Polymer melt extrusion

10⁴–10⁶ Pa

Food sauce pumping

10–500 Pa

⚠️ γ̇ < 1000 s⁻¹ to avoid degradation in thermoplastics

🔧 Open Calculator

Bingham Yield Criterion

τ = τ₀ + μₚ · γ̇

Linear relationship between shear stress and shear rate above yield

Variables:

Symbol	Name	Unit	Description
τ	Shear Stress	Pa	Applied shear stress
τ₀	Yield Stress	Pa	Minimum stress required to initiate flow
μₚ	Plastic Viscosity	Pa·s	Slope of the linear relationship between shear stress and shear rate
γ̇	Shear Rate	s⁻¹	Rate of strain in the fluid