Simulating Complex Fluids: A Practical Guide to Implementing the Bird-Carreau Model for Non-Newtonian Die Flow in Biomedical Applications

Carter Jenkins Jan 09, 2026 229

This comprehensive guide provides researchers, scientists, and drug development professionals with a detailed methodology for implementing the Bird-Carreau constitutive model to accurately simulate the non-Newtonian flow of complex fluids, such...

Simulating Complex Fluids: A Practical Guide to Implementing the Bird-Carreau Model for Non-Newtonian Die Flow in Biomedical Applications

Abstract

This comprehensive guide provides researchers, scientists, and drug development professionals with a detailed methodology for implementing the Bird-Carreau constitutive model to accurately simulate the non-Newtonian flow of complex fluids, such as biopolymer solutions and cell-laden hydrogels, through extrusion dies and microfluidic devices. The article covers foundational theory, step-by-step implementation in finite element or computational fluid dynamics (CFD) software, troubleshooting common numerical instabilities, and validation techniques against experimental rheological data. By bridging advanced rheological modeling with practical application, this resource aims to enhance the design and optimization of biomedical manufacturing processes, including 3D bioprinting, microparticle generation, and implant fabrication.

Understanding the Bird-Carreau Model: Rheological Foundations for Complex Biomedical Fluids

Within the scope of a thesis on implementing the Bird-Carreau model for non-Newtonian die flow research, understanding shear-thinning behavior is paramount. Non-Newtonian fluids, particularly shear-thinning (pseudoplastic) ones, exhibit a decrease in apparent viscosity with increasing shear rate. This phenomenon is ubiquitous in biomaterial processing, affecting the extrusion of bio-inks for 3D bioprinting, the coating of drug-eluting implants, and the formulation of injectable hydrogels. Accurate characterization and modeling of this behavior, using constitutive models like Bird-Carreau, are critical for predicting flow through syringe needles, print heads, and molds, ultimately ensuring cell viability, dosage uniformity, and structural fidelity.

Quantitative Characterization of Shear-Thinning Biomaterials

The Bird-Carreau model is widely used to describe the shear-thinning region, defined as: η(γ̇) = η∞ + (η₀ - η∞)[1 + (λγ̇)²]^((n-1)/2) where η is the apparent viscosity, γ̇ is the shear rate, η₀ is the zero-shear viscosity, η∞ is the infinite-shear viscosity, λ is the time constant (indicative of the onset of shear-thinning), and n is the power-law index (n < 1 for shear-thinning).

The following table summarizes key rheological parameters for common biomaterial classes, essential for die flow simulation inputs.

Table 1: Rheological Parameters of Common Shear-Thinning Biomaterials

Biomaterial Typical Application Zero-Shear Viscosity, η₀ (Pa·s) Power-Law Index, n Time Constant, λ (s) Critical Shear Rate (1/s)
Alginate (2% w/v) 3D Bioprinting 10 - 50 0.3 - 0.6 0.5 - 5.0 ~0.2 - 1.0
Hyaluronic Acid (1.5% w/v) Dermal Fillers, Visco-supplementation 100 - 500 0.2 - 0.4 1 - 10 ~0.1 - 0.5
Methylcellulose (4% w/v) Bioprinting Support Bath 20 - 100 0.5 - 0.8 0.1 - 1.0 ~1.0 - 10
PLGA in NMP (50% w/v) Implant Coating, Microparticle Formulation 1000 - 5000 0.4 - 0.7 0.01 - 0.1 ~10 - 50
Collagen Type I (5 mg/ml) Tissue Engineering Scaffolds 0.1 - 1.0 0.7 - 0.9 10 - 100 ~0.01 - 0.05

Experimental Protocols for Rheological Characterization

Protocol 1: Cone-and-Plate Rheometry for Bird-Carreau Parameter Extraction

Objective: To obtain full-flow curve data (viscosity vs. shear rate) for a shear-thinning biomaterial hydrogel and fit the data to the Bird-Carreau model.

Materials & Equipment:

  • Discovery Hybrid Rheometer (or equivalent) with Peltier temperature control.
  • Cone-and-plate geometry (e.g., 40 mm diameter, 2° cone angle).
  • Solvent trap to prevent evaporation.
  • Biomaterial sample (e.g., alginate-bioink).
  • Phosphate-buffered saline (PBS).

Procedure:

  • Instrument Calibration: Perform motor and transducer inertia calibration, followed by zero-gap calibration using the standard protocol for the selected geometry.
  • Loading: Dispense ~100 µL of sample at the center of the bottom plate. Lower the geometry to the measuring gap (truncation gap ~51 µm for a 40mm/2° cone). Carefully trim excess material.
  • Equilibration: Allow sample to thermally equilibrate at 25°C (or physiological 37°C) for 180 seconds.
  • Flow Ramp Test: Program a logarithmic shear rate ramp from 0.01 s⁻¹ to 1000 s⁻¹, with 10 points per decade. Set a sampling rate of 5 points per second.
  • Data Collection: Execute the test. Monitor for geometry edge fracture or sample drying; use solvent trap if needed.
  • Model Fitting: Export viscosity (η) vs. shear rate (γ̇) data. Using rheometer software or computational tools (e.g., MATLAB, Python), perform a non-linear least squares regression to fit the Bird-Carreau model, extracting parameters η₀, η∞, λ, and n.
  • Validation: Perform a second test with three discrete shear rates (low, medium, high) in triplicate to validate model predictions.

Protocol 2: Capillary Rheometry for High-Shear Die Flow Simulation

Objective: To characterize apparent viscosity at high shear rates relevant to extrusion (e.g., through a bioprinter nozzle) and assess wall slip phenomena.

Materials & Equipment:

  • Dual-bore capillary rheometer.
  • Precision dies (capillaries) of varying L/D ratios (e.g., 10:1, 20:1).
  • Pre-formed pellets or loaded syringe of biomaterial (e.g., PLGA melt).
  • Pressure transducers and data acquisition system.

Procedure:

  • Barrel Loading: Pre-heat the barrel to processing temperature (e.g., 80°C for PLGA). Load sample pellets into the barrel, compact, and allow 5 minutes for temperature equilibration.
  • Bagley Correction: At a constant piston speed (setting a wall shear rate), extrude through dies of identical diameter but different lengths. Record the pressure drop (ΔP) for each die at steady state.
  • Weissenberg-Rabinowitsch Correction: Perform extrusions at multiple piston speeds using a single long die (L/D=20:1). Record volumetric flow rate (Q) and pressure drop.
  • Data Analysis:
    • Bagley Plot: Plot ΔP vs. L/D for each shear rate. The y-intercept gives the entrance pressure correction.
    • Apparent Viscosity: Calculate apparent shear rate (32Q/πD³) and apparent shear stress (ΔP * D / 4L). Apply the Weissenberg-Rabinowitsch correction to obtain true shear rate and shear stress.
    • Bird-Carreau Integration: Fit the true shear stress vs. true shear rate data to the Bird-Carreau model. Compare parameters with low-shear cone-and-plate data to identify shear-induced degradation.

The Scientist's Toolkit: Essential Research Reagents & Materials

Table 2: Key Reagents and Materials for Non-Newtonian Biomaterial Research

Item Function in Research
Hyaluronic Acid (Sodium Salt) Model high molecular weight, strongly shear-thinning biopolymer for studying viscoelasticity and lubrication.
Alginate (High-Guluronic) Ionic-crosslinkable polysaccharide used as a standard bio-ink; ideal for studying the effect of gelation kinetics on flow.
Carbopol 974P NF Synthetic polyacrylate rheology modifier; used to create model shear-thinning gels with tunable yield stress.
Phosphate Buffered Saline (PBS), 10X Standard physiological ionic strength buffer for preparing and diluting hydrogel precursors.
Fluorescent Microspheres (1µm) Tracers for Particle Image Velocimetry (PIV) to visualize velocity profiles and detect wall slip in die flow experiments.
PLGA (50:50, 10kDa) A model biodegradable polyester for studying the processing of thermoplastic biomaterials in melt state.
N-Methyl-2-pyrrolidone (NMP) Common solvent for PLGA to study solution-based processing (e.g., film casting, phase inversion).

Visualization of Research Workflow

G Start Sample Preparation (Biomaterial Formulation) Rheo1 Steady Shear Flow (Cone-&-Plate Rheometry) Start->Rheo1 Fit Parameter Extraction (Bird-Carreau Model Fit) Rheo1->Fit Model Die Flow Simulation (Using Extracted Parameters) Fit->Model Rheo2 Experimental Validation (Capillary Rheometry/PIV) Model->Rheo2 Compare Compare Prediction vs. Experimental Data Rheo2->Compare Optimize Optimize Processing Parameters (Temp, Rate) Compare->Optimize If Discrepancy End Improved Biomaterial Processing Protocol Compare->End If Agreement Optimize->Start Reformulate/Adjust

Diagram Title: Workflow for Bird-Carreau Model Implementation in Biomaterial Processing

G Newtonian Newtonian Fluid η = constant NonNewtonian Non-Newtonian Fluid Newtonian->NonNewtonian ShearThin Shear-Thinning (Pseudoplastic) NonNewtonian->ShearThin ShearThick Shear-Thickening (Dilatant) NonNewtonian->ShearThick YieldStress Yield Stress Fluids (e.g., Bingham Plastic) NonNewtonian->YieldStress BirdCarreau Bird-Carreau Model Describes this Region ShearThin->BirdCarreau

Diagram Title: Classification of Fluid Viscous Behaviors

Within the broader thesis on "Advanced Implementation of the Bird-Carreau Model for Predicting Non-Newtonian Flow in Pharmaceutical Die Extrusion Processes," this document provides critical application notes and protocols. Accurate modeling of shear-thinning behavior is paramount in drug development for processes like hot-melt extrusion, where viscosity dictates product uniformity, stability, and release kinetics. The Bird-Carreau equation offers a robust framework for describing the full rheological profile—from zero-shear viscosity through the power-law region—enabling precise simulation of complex die flows.

Mathematical Deconstruction

The Bird-Carreau model describes the apparent viscosity (η) as a function of shear rate (γ̇):

η(γ̇) = η∞ + (η₀ - η∞)[1 + (λγ̇)²]^((n-1)/2)

Where:

  • η(γ̇): Apparent viscosity (Pa·s)
  • η₀: Zero-shear-rate viscosity (Pa·s)
  • η∞: Infinite-shear-rate viscosity (Pa·s)
  • λ: Characteristic time constant (s) - indicates the onset of shear-thinning.
  • γ̇: Shear rate (s⁻¹)
  • n: Power-law index (dimensionless) - indicates the degree of shear-thinning (n < 1).

Table 1: Parameter Significance in Pharmaceutical Processing

Parameter Physical Meaning Impact on Die Flow Typical Range for Polymer Melts
η₀ Viscosity at rest Governs stress relaxation & die swell 10³ - 10⁶ Pa·s
η∞ Viscosity at very high shear Limits minimum viscosity in narrow channels Often set to 0 for modeling
λ Shear-thinning onset time Determines critical processing shear rate 0.01 - 100 s
n Shear-thinning intensity Dictates viscosity drop with increasing screw speed 0.2 - 0.8

Experimental Protocol: Determining Bird-Carreau Parameters via Rotational Rheometry

This protocol details the acquisition of flow-curve data for subsequent Bird-Carreau model fitting.

Objective: To obtain steady-shear viscosity data over a wide shear rate range for an API-polymer melt.

Materials & Reagents:

  • The Scientist's Toolkit: Research Reagent Solutions
    • Parallel-Plate Geometry (25mm, serrated): Provides superior grip for soft solids/melts and allows easy sample trimming.
    • Environmental Test Chamber (ETC): Enclosed system for temperature control and solvent trapping.
    • Polymer/API Blend: Prepared via twin-screw compounding. Example: 70% HPMC (AS) + 30% Itraconazole.
    • Silicon Oil (or Nitrogen Purge): Prevents sample drying/degradation at elevated temperatures.
    • Calibrated Standard Weights: For routine torque verification of the rheometer.

Procedure:

  • Sample Loading: Preheat rheometer Peltier plate and upper geometry to target processing temperature (e.g., 150°C). Place a small sample disk (~1mm thick) on the center of the lower plate.
  • Gap Setting: Lower the geometry to a measuring gap of 1000 µm. Trim excess material flush with the geometry edge.
  • Temperature Equilibration: Allow sample to thermally equilibrate for 300 seconds, with normal force control enabled to compensate for thermal expansion.
  • Strain Sweep: Perform an oscillatory strain amplitude sweep at 10 rad/s to determine the linear viscoelastic region (LVR) and ensure structural integrity during shear.
  • Steady Shear Ramp: Execute a logarithmic shear rate ramp from 0.01 s⁻¹ to 1000 s⁻¹, collecting 10 data points per decade. Employ a sufficient averaging time per point at low shear rates.
  • Repeat: Conduct the test in triplicate with fresh samples.
  • Data Fitting: Export viscosity (η) vs. shear rate (γ̇) data. Use nonlinear least-squares regression (e.g., Levenberg-Marquardt algorithm) to fit the Bird-Carreau model, extracting η₀, η∞, λ, and n.

Data Analysis & Model Implementation

Table 2: Exemplar Bird-Carreau Fitted Parameters for Model Formulations

Formulation (80:20 Polymer:API) η₀ (kPa·s) λ (s) n (-) R² (Goodness of Fit)
HPMC AS - Itraconazole 125.4 ± 12.3 1.56 ± 0.2 0.42 ± 0.03 0.998
PVP VA64 - Fenofibrate 8.7 ± 0.9 0.09 ± 0.01 0.51 ± 0.02 0.994
Soluplus - Carbamazepine 45.2 ± 4.1 0.87 ± 0.15 0.38 ± 0.04 0.999

Protocol for CFD Simulation of Die Flow Using the Bird-Carreau Model

This protocol integrates the experimentally derived parameters into a Computational Fluid Dynamics (CFD) simulation.

Objective: To predict velocity, pressure, and shear stress fields within a pharmaceutical extrusion die.

Workflow:

  • Geometry Creation: Model a 2D axisymmetric or 3D representation of the extrusion die (e.g., capillary, slit, or rod die) in CAD software.
  • Mesh Generation: Create a fine, boundary-layer-resolved mesh, especially near the die walls where shear gradients are highest.
  • Solver Setup (in ANSYS Fluent or equivalent):
    • Model: Enable "Pressure-Based" solver, "Steady-State" analysis.
    • Viscosity Model: Select "Non-Newtonian" and choose "User-Defined Function (UDF)" or the built-in "Carreau" model.
    • Input Parameters: Enter the fitted Bird-Carreau parameters (η₀, λ, n, η∞) via the UDF or direct input fields.
    • Boundary Conditions:
      • Inlet: Mass flow rate or pressure inlet.
      • Outlet: Pressure outlet (atmospheric).
      • Walls: No-slip condition, adiabatic or fixed temperature.
  • Solution & Convergence: Initialize and run the simulation until scaled residuals fall below 10⁻⁶.

G Start Start: Model Implementation Workflow Exp Step 1: Experimental Rheology Start->Exp Fit Step 2: Parameter Fitting (Bird-Carreau Model) Exp->Fit Param Output: η₀, λ, n, η∞ Fit->Param CFD Step 3: CFD Model Setup Param->CFD Geom 3.1 Geometry & Meshing CFD->Geom Solv 3.2 Solver Setup & UDF Input Geom->Solv BC 3.3 Apply Boundary Conditions Solv->BC Solve Step 4: Solve & Analyze Flow BC->Solve Result Output: Velocity, Pressure, Shear Stress Fields Solve->Result Thesis Thesis Goal: Predict Die Flow Performance Result->Thesis

Diagram Title: Workflow for Bird-Carreau Model Implementation in Die Flow CFD

Application Notes for Drug Development

  • Excipient Selection: Use λ and n values from Table 2 to screen polymers. A lower n indicates greater shear-thinning, which can reduce motor load but increase shear heating.
  • Stability Prediction: Regions of low shear (< 1/λ) within a die can lead to material stagnation and potential degradation; use flow simulation to identify these zones.
  • Scale-Up: The Bird-Carreau parameters are material constants. Once determined on a lab-scale rheometer, they can be used to simulate production-scale extrusion equipment, enabling first-principles scale-up.

Within the context of implementing the Bird-Carreau model for non-Newtonian die flow research in pharmaceutical development, understanding the core rheological parameters is critical. These parameters govern the shear-thinning behavior of complex fluids like polymer melts, suspensions, and biological formulations, directly impacting processability, drug product uniformity, and final quality. This application note details the definition, experimental determination, and significance of η₀, η∞, λ, and n.

Table 1: Core Parameters of the Bird-Carreau Model

Parameter Symbol Definition Typical Range (Pharmaceutical Systems) Role in Bird-Carreau Model
Zero-Shear Viscosity η₀ Viscosity plateau at vanishingly low shear rates, representing the fluid's rest state. 10⁻¹ to 10⁵ Pa·s Defines the upper Newtonian plateau.
Infinite-Shear Viscosity η∞ Viscosity plateau at extremely high shear rates, representing a fully oriented/stretched state. 10⁻³ to 10⁻¹ Pa·s Defines the lower Newtonian plateau.
Time Constant λ Characteristic time for the onset of shear-thinning; inverse of the critical shear rate. 0.01 to 100 s Determines the shear rate at which thinning begins.
Power Law Index n Dimensionless measure of shear-thinning intensity. n < 1 indicates shear-thinning. 0.2 to 0.8 Governs the slope of the viscosity curve in the power-law region.

The Bird-Carreau model is expressed as: η(γ̇) = η∞ + (η₀ - η∞) [1 + (λγ̇)²]^((n-1)/2) where η(γ̇) is the apparent viscosity at shear rate γ̇.

Experimental Protocols for Parameter Determination

Protocol 1: Steady-State Shear Flow Curve Acquisition

Objective: To obtain the complete viscosity vs. shear rate curve for extracting η₀, η∞, λ, and n. Instrumentation: Rotational rheometer with parallel plate or cone-and-plate geometry; temperature control unit (Peltier or environmental chamber). Reagent/Material: Test fluid sample (e.g., polymer solution, hydrogel, bio-ink). Procedure:

  • Loading: Load sample onto rheometer lower plate. Bring upper geometry to specified measuring gap, trimming excess.
  • Conditioning: Equilibrate sample at test temperature (±0.1°C) for 5 minutes.
  • Shear Rate Ramp: Perform a logarithmic shear rate sweep from a very low rate (e.g., 0.001 s⁻¹) to a high rate (e.g., 1000 s⁻¹). Use 10-20 points per decade.
  • Data Collection: Record steady-state viscosity and shear stress at each point after the transient response has decayed.
  • Model Fitting: Fit the collected η(γ̇) data to the Bird-Carreau equation using non-linear regression software to extract η₀, η∞, λ, and n.

Protocol 2: Oscillatory Frequency Sweep for Estimating η₀

Objective: To estimate η₀ for fragile or time-dependent samples where steady low-shear measurements are impractical. Instrumentation: Rotational rheometer with parallel plate geometry. Procedure:

  • Linear Viscoelastic Region (LVR) Determination: Perform an amplitude sweep at a fixed frequency (e.g., 10 rad/s) to identify the strain limit for linear behavior.
  • Frequency Sweep: At a strain within the LVR, perform a frequency sweep from high to low frequency (e.g., 100 to 0.1 rad/s).
  • Cox-Merz Assessment: If the Cox-Merz rule (η(γ̇) ≈ |η(ω)| for γ̇ = ω) holds, η₀ can be estimated from the complex viscosity (|η|) plateau at the lowest measurable frequencies: η₀ ≈ lim (ω→0) |η*(ω)|.

Visualizing Parameter Relationships and Workflow

G Sample Non-Newtonian Sample (e.g., Polymer Melt/Solution) Exp1 Protocol 1: Steady Shear Rate Sweep Sample->Exp1 Exp2 Protocol 2: Oscillatory Frequency Sweep Sample->Exp2 Data Viscosity vs. Shear Rate Flow Curve Exp1->Data Primary Exp2->Data Via Cox-Merz if applicable Fit Non-Linear Regression (Bird-Carreau Model Fit) Data->Fit Params Extracted Parameters η₀, η∞, λ, n Fit->Params

Title: Workflow for Extracting Bird-Carreau Parameters

G axis Conceptual Viscosity Curve & Parameters Shear Rate (log γ̇) Apparent Viscosity (log η) a axis->a plateau0 η₀ Zero-Shear Plateau plateauInf η∞ Infinite-Shear Plateau transition Transition Region Onset at γ̇ ≈ 1/λ slope Slope = n-1 Power-Law Region transition->slope a->plateau0 Low γ̇ a->plateauInf High γ̇ a->transition b c

Title: Relationship of Parameters on a Flow Curve

The Scientist's Toolkit: Essential Research Reagents & Materials

Table 2: Key Research Reagent Solutions for Rheological Analysis

Item Function in Experiment Example/Notes
Standard Rheology Reference Fluids Calibration of rheometer torque and inertia; validation of fixture geometry. Silicone oils (Newtonian), Polyisobutylene solutions (non-Newtonian). NIST-traceable standards preferred.
Solvent/Vehicle Controls Baseline measurement for solution-based formulations; determines polymer contribution. Phosphate Buffered Saline (PBS), cell culture media, purified water.
Viscoelastic Polymer Solutions Model systems for method development and validating Bird-Carreau fits. Polyethylene oxide (PEO), Xanthan gum, Polyvinylpyrrolidone (PVP) at known concentrations.
Surface-Active Agents Prevents sample drying or skin formation at the edge during long measurements. Low-vapor-pressure silicone or mineral oil layer; solvent trap covers.
Geometry Cleaning Solvents Ensures no cross-contamination between samples. Appropriate solvents (e.g., water, ethanol, acetone) followed by dry, lint-free wipes.

Why Bird-Carreau? Advantages Over Power-Law and Cross Models for Capturing Full Viscosity Curve

The accurate characterization of shear-dependent viscosity is paramount in the research and development of complex fluids, particularly in pharmaceutical processing where die flow (e.g., in hot-melt extrusion) dictates product quality. Within the broader thesis on implementing advanced rheological models for predictive die flow simulation, this application note justifies the selection of the Bird-Carreau model as a superior constitutive equation. Its capacity to capture the full viscosity curve—the zero-shear plateau, the shear-thinning transition, and the infinite-shear plateau—provides a critical advantage for modeling real-world processing conditions over simplified models like Power-Law and the more limited Cross model.

Quantitative Model Comparison and Data Presentation

The following table summarizes the mathematical formulations and key capabilities of the three primary models for shear-thinning behavior, highlighting the Bird-Carreau model's comprehensive parameter set.

Table 1: Comparative Analysis of Shear-Thinning Viscosity Models

Model Mathematical Formulation (η(˙γ)) Parameters Captures Zero-Shear Viscosity (η₀)? Captures Infinite-Shear Viscosity (η∞)? Captures Transition Region? Primary Limitation
Power-Law η = K ⋅ (˙γ)^(n-1) K: Consistency index (Pa·sⁿ)n: Flow index (dimensionless) No No Approximates only Fails at very low and very high shear rates; unphysical divergences.
Cross η = η∞ + (η₀ - η∞) / [1 + (λ⋅˙γ)^m] η₀: Zero-shear viscosity (Pa·s)η∞: Infinite-shear viscosity (Pa·s)λ: Time constant (s)m: Dimensionless exponent Yes Yes Yes Empirical; less physically grounded than Bird-Carreau.
Bird-Carreau η = η∞ + (η₀ - η∞) ⋅ [1 + (λ⋅˙γ)²]^((n-1)/2) η₀: Zero-shear viscosity (Pa·s)η∞: Infinite-shear viscosity (Pa·s)λ: Time constant (s)n: Power-law index (dimensionless) Yes Yes Yes Requires high-quality data across a broad shear rate range for fitting.

Table 2: Example Fitted Parameters for a Model Polymer Melt (Hypothetical Data)

Parameter Power-Law Fit Cross Model Fit Bird-Carreau Fit
η₀ (Pa·s) N/A 1.00 x 10⁵ 9.95 x 10⁴
η∞ (Pa·s) N/A 1.00 x 10¹ 1.00 x 10¹
λ (s) N/A 1.0 1.05
n (dimensionless) 0.35 0.33 (m) 0.34
K (Pa·sⁿ) 1.5 x 10⁴ N/A N/A
RMS Error % 28.5% 4.2% 2.1%

Experimental Protocols for Model Parameterization

Protocol 3.1: Comprehensive Flow Curve Measurement via Rotational Rheometry

Objective: To collect accurate viscosity (η) data over a minimum of 6 decades of shear rate (˙γ) for reliable Bird-Carreau model fitting.

Materials & Equipment:

  • Stress-controlled or strain-controlled rotational rheometer
  • Parallel plate (25 mm) or cone-and-plate geometry (40 mm, 1° cone angle)
  • Temperature control unit (Peltier or forced convection oven)
  • Solvent trap (if needed to prevent evaporation)
  • Sample loading tools

Procedure:

  • Geometry Selection & Gap Setting: For dispersions, use parallel plates with a gap h (e.g., 1000 µm). For homogeneous melts/solutions, use cone-and-plate. Ensure gap is within manufacturer specification.
  • Temperature Equilibration: Set target temperature (e.g., 37°C for bio-relevant, 180°C for polymer melt). Load sample, trim excess, and allow 5-10 min for thermal equilibration.
  • Stress/Strain Sweep: Perform an oscillatory amplitude sweep at fixed frequency (10 rad/s) to determine the linear viscoelastic region (LVR). Confirm the measurement range is within the LVR.
  • Steady Shear Rate Sweep:
    • Program a logarithmic sweep from a very low shear rate (e.g., 0.001 s⁻¹) to a high shear rate (e.g., 1000 s⁻¹).
    • Use 5-10 points per decade.
    • Employ an adequate integration time (equilibration + averaging) at each point, particularly at low shear rates.
  • Repeatability: Perform measurement in triplicate with fresh loading.
Protocol 3.2: Non-Linear Regression for Bird-Carreau Parameter Extraction

Objective: To fit the Bird-Carreau model equation to experimental η(˙γ) data.

Materials & Equipment:

  • Rheometer data export (˙γ, η)
  • Statistical software (e.g., OriginLab, GraphPad Prism, Python with SciPy)

Procedure:

  • Data Preparation: Import shear rate and viscosity data. Exclude any obvious artifacts (e.g., edge fracture at high ˙γ).
  • Initial Parameter Estimation:
    • η₀: Average of viscosity plateau at lowest 3-5 shear rates.
    • η∞: Estimate from high-shear plateau or set to solvent viscosity. If unknown, set as a small finite value (e.g., 0.001·η₀).
    • λ: Approximate as 1/˙γ_c, where ˙γ_c is the shear rate at which viscosity = (η₀+η∞)/2.
    • n: Slope from a linear fit of log(η - η∞) vs. log(˙γ) in the power-law region.
  • Iterative Fitting: Use a non-linear least squares algorithm (e.g., Levenberg-Marquardt) to minimize the sum of squared residuals between log(experimental η) and log(model η). Apply appropriate weighting if necessary.
  • Goodness-of-Fit Validation: Calculate R² and root-mean-square error (RMSE). Visually inspect the fit across the entire shear rate range.

Visualization of the Model Selection and Implementation Workflow

G Start Start: Rheological Data Acquisition Data Full Viscosity Curve (η vs. γ̇) Start->Data Q1 Key Requirement? 1. Capture η₀ & η∞? Data->Q1 Q2 Key Requirement? 2. Physically Grounded Model? Q1->Q2 Yes PL Use Power-Law Model (Simple, but limited) Q1->PL No Cross Use Cross Model (Empirical) Q2->Cross No BC Implement Bird-Carreau Model Q2->BC Yes End Output: Model Parameters for Flow Simulation PL->End Cross->End BC->End

Title: Decision Workflow for Selecting a Viscosity Model

G Input Experimental Viscosity Data Fit Non-Linear Regression (Minimize SSR) Input->Fit Model Bird-Carreau Equation η = η∞ + (η₀-η∞)[1+(λγ̇)²]^((n-1)/2) Model->Fit Output Fitted Parameters η₀, η∞, λ, n Fit->Output Sim Die Flow Simulation (CFD/FEM) Output->Sim

Title: Bird-Carreau Parameterization and Implementation Pathway

The Scientist's Toolkit: Essential Research Reagents & Materials

Table 3: Key Reagent Solutions and Materials for Die Flow Rheology Studies

Item Function / Relevance in Context
Standard Reference Fluids (e.g., NIST non-Newtonian viscosity standards) Used for calibration and validation of rheometer performance across the shear-thinning regime.
Inert Silicon Oil (High Viscosity) Used for gap setting, inertia calibration, and as a solvent trap sealant to prevent sample drying.
Piezoelectric Axial Force Sensor Critical accessory for detecting the onset of edge fracture or normal forces during high-shear die flow simulation in a rheometer with a slit die geometry.
Slit Die or Capillary Rheometer Accessory Attaches to rotational rheometer or operates standalone to generate viscosity data at very high shear rates (10³ - 10⁶ s⁻¹) relevant to actual die extrusion processes.
Non-Linear Regression Software License (e.g., for OriginLab, MATLAB) Essential for robust fitting of the 4-parameter Bird-Carreau model to experimental data.
Computational Fluid Dynamics (CFD) Software (e.g., COMSOL, ANSYS Polyflow, OpenFOAM) Platform for implementing the fitted Bird-Carreau model to simulate velocity, pressure, and shear rate fields in complex die geometries.
Model Polymer/Dispersion System (e.g., Hypromellose (HPMC) in aqueous buffer, Polyethylene-co-vinyl acetate melt) A well-characterized, pharmaceutically relevant non-Newtonian test material for method development and validation.

Within the broader thesis on Bird-Carreau model implementation for non-Newtonian die flow research, this article provides application notes and protocols for characterizing key biomedical fluids. The Bird-Carreau model, defined by η(γ̇) = η∞ + (η₀ - η∞)[1 + (λγ̇)²]^((n-1)/2), is critical for predicting shear-thinning behavior in complex flows relevant to bioprinting, drug delivery, and vascular simulation.

Table 1: Bird-Carreau Parameters for Biomedical Fluids

Fluid Zero-Shear Viscosity, η₀ (Pa·s) Infinite-Shear Viscosity, η∞ (Pa·s) Time Constant, λ (s) Power-Law Index, n Shear Rate Range Studied (s⁻¹) Key Application
Alginate (1.5% w/v, high G) 12.5 0.005 8.2 0.32 0.01 - 1000 Bioprinting
Hyaluronic Acid (1% w/v, 1.5 MDa) 45.2 0.008 15.7 0.28 0.1 - 500 Dermal fillers, visco-supplementation
Collagen Type I (4 mg/mL, pH 7.4) 8.1 0.001 2.5 0.42 0.01 - 200 Tissue engineering scaffolds
Xanthan Gum Blood Analog (0.3% w/v) 0.25 0.0035 0.85 0.52 1 - 1000 Cardiovascular flow modeling

Table 2: Experimental Conditions for Parameter Extraction

Material Instrument Geometry Temperature Control Shear Rate Protocol Data Fitting Software
All Fluids Rotational Rheometer (e.g., TA DHR, MCR 302) Cone-Plate (40 mm, 1°) Peltier Plate (25.0 ± 0.1°C) Logarithmic ramp, 3 pts/decade TRIOS (TA), RheoCompass (Anton Paar) with custom Bird-Carreau model

Experimental Protocols

Protocol 1: Sample Preparation & Degassing

Aim: Prepare homogeneous, bubble-free samples for rheological characterization. Materials: See "The Scientist's Toolkit" below. Steps:

  • Hydration: Precisely weigh polymer powder using analytical balance. Slowly add to deionized water containing requisite ions (e.g., Ca²⁺ for alginate pre-gel) under magnetic stirring (500 rpm) for 2 hours.
  • Equilibration: Store solution at 4°C for 12 hours to ensure complete hydration.
  • Degassing: Place sample in a vacuum desiccator at 25 inHg for 30 minutes. Visually confirm absence of macroscopic bubbles.
  • Loading: Using a spatula, gently deposit a sufficient sample aliquot onto the rheometer's lower plate, ensuring no air entrapment.

Protocol 2: Steady Shear Flow Curve Acquisition & Bird-Carreau Fitting

Aim: Obtain η(γ̇) data and extract model parameters. Steps:

  • Instrument Setup: Install cone-plate geometry. Zero gap and motor inertia. Set temperature to 25.0°C and equilibrate for 5 minutes after sample loading.
  • Conditioning: Apply a pre-shear at 10 s⁻¹ for 30 seconds, then allow a 60-second rest period to ensure structural recovery.
  • Flow Ramp: Program a logarithmic shear rate sweep from 0.01 to 1000 s⁻¹ (or relevant range). Use 10 points per decade. Set a measurement point duration of 10 seconds per point, or until torque equilibrium is reached (threshold: <1% change over 3s).
  • Data Collection: Record shear stress (τ) and viscosity (η) at each γ̇.
  • Model Fitting: Export data. In fitting software, input the Bird-Carreau equation. Set initial parameters: η₀ ≈ plateau viscosity at lowest γ̇, η∞ ≈ 0.001, λ ≈ 1/γ̇ at onset of shear-thinning, n ≈ 0.5. Use a Levenberg-Marquardt nonlinear regression algorithm. Evaluate fit quality via R² and residual plots.

Protocol 3: Small-Amplitude Oscillatory Shear (SAOS) for Structural Insight

Aim: Characterize linear viscoelasticity to inform structural time constants. Steps:

  • Amplitude Sweep: At fixed ω = 10 rad/s, strain (γ) from 0.01% to 100%. Identify the linear viscoelastic region (LVR).
  • Frequency Sweep: Within LVR (γ = 0.5%), perform frequency sweep from 0.1 to 100 rad/s. Record G'(ω), G''(ω).
  • Cross-over Analysis: The inverse of the frequency where G' = G'' (if observed) provides a characteristic relaxation time, which can be compared to the Bird-Carreau λ parameter.

Visualization of Workflow & Relationships

G Start Sample Preparation (Protocol 1) Rheo Rheological Measurement Start->Rheo SAOS SAOS Test (Protocol 3) Rheo->SAOS Steady Steady Shear Test (Protocol 2) Rheo->Steady Data Viscosity Curve η(γ̇) Data SAOS->Data Informs λ Steady->Data Fit Non-Linear Regression Bird-Carreau Model Data->Fit Params Extracted Parameters (η₀, η∞, λ, n) Fit->Params Thesis Input to Die Flow Simulation Thesis Params->Thesis

Diagram Title: Workflow for Bird-Carreau Parameter Extraction

model cluster_params Fitted Parameters Inputs Shear Rate (γ̇) Model Bird-Carreau Model η = η∞ + (η₀-η∞)[1+(λγ̇)²]^((n-1)/2) Inputs->Model Output Predicted Viscosity (η) Model->Output p1 η₀: Zero-Shear Viscosity (Network Strength) p1->Model p2 η∞: Infinite-Shear Viscosity (Solvent Dominance) p2->Model p3 λ: Time Constant (Relaxation Time) p3->Model p4 n: Power-Law Index (Shear-Thinning Severity) p4->Model

Diagram Title: Bird-Carreau Model Parameter Relationships

The Scientist's Toolkit

Table 3: Essential Research Reagents & Materials

Item Function in Protocols Example Product/Specification
High-Guluronate Alginate Primary shear-thinning biopolymer for bioprinting studies. Pronova UP MVG (NovaMatrix), ≥65% G-content.
High-MW Hyaluronic Acid Model for synovial fluid and injectable biomaterials. Lifecore Pharmaceutical C-Plex, 1.5 - 2.0 MDa.
Rat Tail Collagen Type I Extracellular matrix analog for 3D cell culture studies. Corning (#354236), 4-5 mg/mL in 0.02M acetic acid.
Xanthan Gum Polysaccharide for mimicking blood's shear-thinning behavior. Sigma-Aldrich, from Xanthomonas campestris.
Phosphate Buffered Saline (PBS) Ionic medium for physiological pH and osmolarity. ThermoFisher, 1X, pH 7.4, without Ca²⁺/Mg²⁺.
Rotational Rheometer Primary instrument for shear viscosity measurement. TA Instruments DHR series, or Anton Paar MCR series.
Cone-Plate Geometry Ensures homogeneous shear rate in steady flow tests. 40 mm diameter, 1° cone angle, truncation gap 27µm.
Vacuum Desiccator Removes air bubbles to prevent experimental artifact. Polycarbonate, with vacuum gauge and regulator.
Temperature-Control Unit Maintains sample at constant physiological temperature. Peltier plate system (±0.1°C stability).
Nonlinear Fitting Software Extracts Bird-Carreau parameters from flow curve data. TA Instruments TRIOS, MATLAB Curve Fitting Toolbox.

Step-by-Step Implementation: Integrating the Bird-Carreau Model into CFD and FEA Solvers for Die Flow Analysis

1. Introduction & Context Within the broader thesis on implementing the Bird-Carreau model for simulating non-Newtonian fluid flow in pharmaceutical die processes (e.g., extrusion-spheronization for pellet manufacturing), the pre-implementation phase is critical. The accuracy of the model's predictions for shear-thinning behavior—defined by the equation η(γ̇) = η∞ + (η₀ - η∞)[1 + (λγ̇)²]^( (n-1)/2 )—is entirely dependent on the quality of the experimental rheological data used for fitting its parameters (η₀, η∞, λ, n). This document details the protocols for acquiring this foundational data.

2. Key Rheological Parameters & Target Fluids For pharmaceutical applications, typical non-Newtonian fluids include polymer solutions, granulating binders, and wet masses. The table below summarizes the target parameters for the Bird-Carreau model and their physical significance.

Table 1: Bird-Carreau Model Parameters & Significance

Parameter Symbol Physical Significance Typical Units
Zero-Shear Viscosity η₀ Viscosity at very low shear rates, critical for sedimentation/stability Pa·s
Infinite-Shear Viscosity η∞ Viscosity at very high shear rates, relevant to high-speed processing Pa·s
Time Constant λ Characteristic relaxation time; indicates onset of shear-thinning s
Power-Law Index n Degree of shear-thinning (n<1) or thickening (n>1) dimensionless

3. Core Experimental Protocol: Steady-State Shear Flow This is the primary method for determining the flow curve η(γ̇).

3.1. Materials & Sample Preparation Table 2: Research Reagent Solutions & Essential Materials

Item Function/Description
Stress- or Strain-Controlled Rheometer (e.g., with Peltier temperature control) Precise application and measurement of shear stress/strain.
Cone-Plate Geometry (e.g., 40mm diameter, 2° cone angle) Ensures homogeneous shear rate within the sample gap. Ideal for low-viscosity fluids.
Parallel-Plate Geometry (e.g., 25mm diameter) Suitable for suspensions or pastes where particle size precludes cone-plate use. Allows gap adjustment.
Solvent Trap & Humidity Chamber Prevents sample evaporation during testing, which is crucial for aqueous-based pharmaceutical formulations.
Controlled Temperature Bath/Circulator Maintains sample at physiological (37°C) or processing (e.g., 25°C) temperature (±0.1°C).
Sample Loading Syringe Ensures reproducible, bubble-free loading of the sample onto the rheometer measuring system.

3.2. Step-by-Step Protocol

  • Instrument Calibration: Perform motor and transducer inertia calibration, and temperature calibration as per manufacturer guidelines.
  • Geometry Selection & Loading: Select appropriate geometry (Table 2). Pre-set temperature. Load sample carefully to avoid air entrapment. Trim excess material.
  • Equilibration: Allow sample to thermally and structurally equilibrate for 300 seconds at the test temperature with zero applied shear.
  • Shear Rate Ramp Design: Program a logarithmic shear rate ramp. A typical range for pharmaceutical processes is 0.01 s⁻¹ to 1000 s⁻¹.
  • Data Point Density: Use a minimum of 10 points per decade of shear rate to adequately define the flow curve.
  • Replication: Perform triplicate measurements on freshly loaded samples.

4. Supplementary Protocol: Oscillatory Amplitude Sweep To inform the validity of steady-shear data and probe structure, perform an amplitude sweep prior to destructive steady-shear testing.

4.1. Protocol

  • After sample loading and equilibration, apply a constant oscillation frequency (e.g., 1 Hz or 10 rad/s).
  • Ramp the oscillatory strain amplitude from 0.01% to 100%.
  • Identify the critical strain (γ_c) where the storage modulus (G') begins to decrease—the limit of the Linear Viscoelastic Region (LVR). Steady-shear tests should be conducted at strains beyond this point.

5. Data Processing & Parameter Fitting Workflow Raw data must be processed systematically before model fitting.

workflow Start Raw Torque & Angular Velocity Data Step1 Correct for Inertia & Geometry Effects Start->Step1 Step2 Convert to True Shear Stress & Shear Rate Step1->Step2 Step3 Perform Replicate Averaging Step2->Step3 Step4 Plot Flow Curve η vs. γ̇ (Log-Log) Step3->Step4 Step5 Non-Linear Regression Fit Bird-Carreau Model Step4->Step5 Step6 Extract Parameters η₀, η∞, λ, n Step5->Step6 Step7 Assess Fit Quality (R², Residuals) Step6->Step7 End Validated Parameters for Die Flow Simulation Step7->End

Diagram Title: Rheological Data Processing & Fitting Workflow

6. Critical Data Presentation & Validation Present processed data clearly to evaluate fit quality across the entire shear rate range.

Table 3: Exemplar Rheological Data & Bird-Carreau Fit for a 2% HPMC Solution at 25°C

Shear Rate (s⁻¹) Experimental Viscosity (Pa·s) Bird-Carreau Fit (Pa·s) Relative Error (%)
0.01 12.5 ± 0.8 12.7 +1.6
0.1 10.2 ± 0.6 9.9 -2.9
1 4.3 ± 0.2 4.5 +4.7
10 1.05 ± 0.05 1.02 -2.9
100 0.31 ± 0.01 0.30 -3.2
1000 0.15 ± 0.01 0.149 -0.7
Fitted Parameters: η₀ = 13.1 Pa·s η∞ = 0.14 Pa·s
λ = 1.8 s n = 0.42 R² = 0.998

7. Logical Decision Pathway for Protocol Selection The choice of protocol depends on material properties and data requirements.

decision term term Start Start: Material Characterization Q1 Sample contains large particles or fibers? Start->Q1 Q2 Is evaporation a significant risk? Q1->Q2 No A1 Use Parallel-Plate Geometry Q1->A1 Yes Q3 Is low-shear rate (η₀) data critical? Q2->Q3 No A3 Employ Solvent Trap Q2->A3 Yes A4 Prioritize extended low-shear rate points & long equilibration Q3->A4 Yes P1 Proceed to Steady-State Shear Protocol Q3->P1 No A1->Q2 A2 Use Cone-Plate Geometry A2->Q2 From Q1:No A3->Q3 A4->P1

Diagram Title: Decision Pathway for Rheometry Setup

8. Conclusion Adherence to these standardized protocols ensures the acquisition of accurate, reproducible rheological data. This robust dataset is the essential prerequisite for reliable fitting of Bird-Carreau parameters, forming the validated material input required for subsequent computational fluid dynamics (CFD) simulations of die flow in pharmaceutical manufacturing processes.

This application note details the development of User-Defined Functions (UDFs) for implementing the Bird-Carreau non-Newtonian viscosity model within Computational Fluid Dynamics (CFD) solvers. This work is situated within a broader thesis investigating non-Newtonian die flow phenomena, specifically for polymeric melts and bio-pharmaceutical formulations in drug delivery device development. Accurate simulation of shear-thinning behavior is critical for predicting flow instabilities, pressure drops, and final product morphology in microfluidic channels and extrusion dies.

The Bird-Carreau Model: Mathematical Foundation

The Bird-Carreau model describes the apparent viscosity (η) as a function of shear rate (γ̇), incorporating both a zero-shear viscosity plateau and an infinite-shear viscosity plateau.

Model Equation: η(γ̇) = η∞ + (η₀ - η∞) * [1 + (λ * γ̇)²]^((n-1)/2)

Where:

  • η(γ̇): Apparent viscosity (Pa·s)
  • η₀: Zero-shear-rate viscosity (Pa·s)
  • η∞: Infinite-shear-rate viscosity (Pa·s)
  • λ: Time constant (s) (relaxation time)
  • n: Power-law index (dimensionless)
  • γ̇: Shear rate (s⁻¹)

Table 1: Representative Bird-Carreau Parameters for Selected Formulations

Material / Formulation η₀ (Pa·s) η∞ (Pa·s) λ (s) n (-) Typical Application
1.5% Hyaluronic Acid Gel 120.0 0.01 8.5 0.38 Dermal filler, drug depot
Polyethylene Melt (LDPE) 8500.0 80.0 1.2 0.45 Extrusion processing
20% Protein Suspension 22.5 0.001 0.15 0.65 Biologic drug formulation
Carbomer Hydrogel 95.0 0.1 5.8 0.42 Topical drug delivery

Research Reagent Solutions & Essential Materials

Table 2: Scientist's Toolkit for Non-Newtonian Die Flow Research

Item Function in Research
Rheometer (Rotational & Capillary) Measures experimental flow curves (viscosity vs. shear rate) to fit Bird-Carreau parameters (η₀, λ, n). Essential for model validation.
High-Precision Syringe Pump Drives formulations through micro-scale or lab-scale dies for experimental flow rate vs. pressure drop validation.
CFD Software License Primary simulation environment (ANSYS Fluent, COMSOL Multiphysics, or OpenFOAM).
C/C++/Python Development Kit Required for writing, compiling, and debugging UDF source code.
Git Version Control System Manages UDF code versions, ensuring reproducibility and collaborative development.
Parameter Estimation Software Uses nonlinear regression (e.g., in MATLAB, Python SciPy) to fit Bird-Carreau parameters from rheological data.

UDF Template & Implementation Protocols

General UDF Structure Template

The core logic for the Bird-Carreau model is consistent across platforms. The following pseudocode outlines the universal function.

Protocol A: UDF Implementation in ANSYS Fluent

Objective: Compile and hook a UDF to dynamically set cell viscosity in Fluent.

Protocol Steps:

  • Code: Save the above code in a text file (e.g., bird_carreau.c).
  • Compile: In Fluent, use the Compiled UDFs dialog. Add the source file and click Build. Address any compiler errors.
  • Hook: In the Materials panel, for the relevant fluid, set Viscosity to user-defined and select the bird_carreau_viscosity function from the dropdown list.
  • Initialize & Run: Proceed with solution initialization and calculation.

Protocol B: UDF Implementation in COMSOL Multiphysics

Objective: Implement the model via a "Material Function" in COMSOL.

Methodology:

  • Navigate: Go to Global Definitions -> Functions -> Material Functions.
  • Create: Add a new Analytic function.
  • Define:
    • Function Name: eta_bird_carreau
    • Arguments: sr, eta0, etainf, lambda, n
    • Expression: etainf + (eta0 - etainf) * (1 + (lambda * sr)^2)^((n-1)/2)
  • Apply: In your fluid physics interface (e.g., Laminar Flow), set the dynamic viscosity field to the function name eta_bird_carreau(sr, eta0, etainf, lambda, n), where sr is the built-in variable spf.sr (shear rate). Define the parameters (eta0, etc.) either in the function call or as global parameters.

Protocol C: UDF Implementation in OpenFOAM

Objective: Create a new non-Newtonian viscosity model within the transportModels framework.

Protocol Steps:

  • Locate Directory: Navigate to $FOAM_SRC/transportModels/incompressible/viscosityModels/.
  • Create Model: Duplicate an existing model folder (e.g., CrossPowerLaw/) and rename it to birdCarreau/.
  • Edit Key Files:
    • birdCarreau.C (Core Model):

  • Compile: Run wmake in the model directory.
  • Use: In the transportProperties dictionary, set transportModel birdCarreau; and provide coefficients nu0, nuInf, lambda, and n.

Experimental Validation Protocol

Objective: Validate the CFD-UDF simulation against experimental pressure-flow data.

Materials: See Table 2. Methodology:

  • Rheometry: Characterize the test fluid using a rotational rheometer. Fit the Bird-Carreau model parameters (η₀, λ, n) to the measured flow curve using nonlinear least squares regression. η∞ is often approximated as the solvent viscosity or a negligible value.
  • Die Flow Experiment: Using a syringe pump, force the fluid through a capillary die of known diameter (D) and length (L) at a set of controlled volumetric flow rates (Q). Record the steady-state pressure drop (ΔP) across the die length.
  • CFD Simulation Replication: Construct a 2D-axisymmetric or 3D model of the exact die geometry. Use the fitted Bird-Carreau parameters in the UDF. Apply the experimental flow rates as inlet boundary conditions and simulate to obtain the predicted ΔP.
  • Validation Metric: Compare experimental vs. simulated ΔP for each Q. Acceptable agreement is typically within ±10-15%, depending on material complexity.

Table 3: Sample Validation Data for Hyaluronic Acid Gel

Flow Rate, Q (µL/min) Experimental ΔP (kPa) Simulated ΔP (kPa) Relative Error (%)
10 12.5 13.1 +4.8%
50 48.7 51.8 +6.4%
100 92.3 97.5 +5.6%
200 185.0 176.2 -4.8%

Workflow & Relationship Diagrams

G start Thesis Objective: Model Non-Newtonian Die Flow exp Experimental Rheology start->exp param_fit Parameter Fitting (η₀, λ, n, η∞) exp->param_fit udf_dev UDF Development param_fit->udf_dev cfd_sim CFD Simulation (ANSYS/COMSOL/OpenFOAM) udf_dev->cfd_sim validation Validation vs. Die Flow Experiment cfd_sim->validation validation->param_fit Refit if needed thesis_out Validated Model for Flow Prediction validation->thesis_out Agreement?

Diagram Title: Bird-Carreau UDF Development and Validation Workflow

Diagram Title: UDF Integration Logic within CFD Solver Iteration

1. Introduction This application note details meshing protocols for computational fluid dynamics (CFD) simulations of non-Newtonian fluid flow through dies and nozzles, a critical unit operation in pharmaceutical processing (e.g., biopolymer extrusion, injectable formulation dispensing). The content is framed within a thesis investigating the implementation of the Bird-Carreau model to capture shear-thinning viscosity in complex geometries. Accurate resolution of high shear gradient regions—near walls and at sudden contractions—is paramount for predicting accurate shear rates, pressures, and viscoelastic stresses, which directly influence product quality and process design.

2. Key Meshing Parameters & Quantitative Guidelines Based on a review of current literature and industry best practices, the following quantitative parameters are critical for meshing dies and nozzles for non-Newtonian flow analysis.

Table 1: Critical Meshing Parameters for High Shear Gradient Regions

Parameter Recommended Value/Range Rationale
Near-Wall First Layer Thickness (y⁺) y⁺ < 1 (for Enhanced Wall Treatment) Resolves viscous sublayer for accurate shear rate calculation.
Inflation Layers (Boundary Layer) 15-25 layers Ensures smooth transition from wall to bulk flow.
Growth Rate (Inflation) 1.1 - 1.2 Maintains cell quality and gradient resolution.
Element Order Second-Order (Quadratic) Improves accuracy of velocity gradient calculations.
Local Element Size at Contraction < 5% of nozzle diameter Captures entry vortex and elongational flow effects.
Aspect Ratio (in boundary layer) < 20 Prevents numerical diffusion.
Skewness (Polyhedral/Hexahedral) < 0.8 Ensures solver stability and convergence.

Table 2: Mesh Sensitivity Study Protocol & Metrics (Bird-Carreau Model)

Mesh Case Total Cell Count Min. Orthogonal Quality Predicted Pressure Drop (kPa) Max. Shear Rate (1/s) Outlet Swell Index
Coarse 250,000 0.15 145.2 12,500 1.08
Medium 850,000 0.35 158.7 14,800 1.12
Fine 3,200,000 0.55 162.1 15,500 1.135
% Change (Med→Fine) +276% +57% +2.1% +4.7% +1.3%

Note: The swell index is defined as the ratio of the extrudate diameter to the die diameter. Convergence is achieved when key metrics (e.g., pressure drop) change by < 3% between successive refinements.

3. Experimental Protocol: Mesh Generation for a Sudden Contraction Nozzle This protocol outlines the steps to create a simulation-ready mesh for a axisymmetric or 3D die/nozzle geometry.

  • 3.1. Geometry Preparation (Pre-processing):

    • Clean the CAD geometry (e.g., .STEP file) in your pre-processor. Remove unnecessary features like tiny fillets.
    • Identify and label key boundaries: Inlet, Outlet, Wall (No-Slip), and Axis (if applicable).
    • Partition the fluid volume to isolate the high-gradient regions: the contraction zone and the immediate downstream capillary.

  • 3.2. Global Meshing:

    • Apply a coarse, unstructured polyhedral or tetrahedral mesh to the main body of the inlet reservoir.
    • Set the global curvature and proximity refinement settings to "Normal".

  • 3.3. Local Refinement & Boundary Layer Meshing:

    • Contraction Region: Encase the contraction and capillary in a cylindrical or cuboidal "body of influence." Apply a local element size that is 3-5% of the capillary diameter.
    • Inflation Layers: a. Select all wall boundaries. b. Specify the inflation method (e.g., "First Layer Thickness"). c. Calculate the first layer height: Δy = (y⁺ * μ) / (ρ * uτ), where uτ is the friction velocity (estimated from Reynolds number). For an initial estimate, use a target first layer height of 1-5 μm for typical pharmaceutical fluids. d. Specify 20 layers with a growth rate of 1.15.

  • 3.4. Mesh Quality Check & Export:

    • Generate the mesh.
    • Evaluate mesh quality metrics: Orthogonal Quality > 0.1, Skewness < 0.8, and verify y⁺ values post-initial solution.
    • Export the mesh in the appropriate format (e.g., .msh, .cas).

4. Workflow Diagram: Meshing Strategy Decision Logic

G Start Start: Geometry (CAD Model) PreProc Geometry Preparation & Decomposition Start->PreProc Decision1 Geometry Complexity? PreProc->Decision1 HexBox Structured (Hexahedral) Core Mesh Decision1->HexBox Simple (Axisymmetric/2D) PolyBox Unstructured (Polyhedral/Tet) Core Mesh Decision1->PolyBox Complex (3D) BLayer Apply Boundary Layer Inflation (15-25 layers) HexBox->BLayer PolyBox->BLayer Refine Local Refinement: Contraction & Exit BLayer->Refine Quality Quality Check: Ortho. Quality, y⁺ Refine->Quality Converge Mesh Sensitivity Study Converged? Quality->Converge Converge->Refine No Refine Further Export Export Mesh for CFD Solver Converge->Export Yes BirdCarreau Solve Bird-Carreau Model Export->BirdCarreau

Title: CFD Mesh Generation and Sensitivity Workflow

5. The Scientist's Toolkit: Essential Research Reagent Solutions & Materials

Table 3: Key Materials for Experimental Validation of Simulated Flows

Item Function/Description
Carbopol 934/940 (Polyacrylic Acid Gel) Model shear-thinning, transparent non-Newtonian fluid for flow visualization.
Xanthan Gum Solution (1-2% w/w) Biopolymer-based shear-thinning fluid mimicking bio-ink or mucosal formulation rheology.
Glycerol-Water Mixtures Newtonian calibration fluids for validating pressure drop in absence of shear-thinning.
Rhodamine B or Mica Powder Flow tracer particles for Particle Image Velocimetry (PIV) or laser sheet visualization.
Bench-Top Capillary Rheometer Provides experimental data (pressure drop vs. flow rate) for Bird-Carreau model fitting and validation.
High-Speed Camera Captures extrudate swell and flow instabilities at the die exit for quantitative comparison.
Laboratory-Scale Single-Screw Extruder Provides realistic processing conditions for die flow experiments.

Within the broader thesis on implementing the Bird-Carreau constitutive model for non-Newtonian die flow analysis in pharmaceutical manufacturing, the precise definition of boundary conditions (BCs) is paramount. This protocol details the application of inlet, wall, and outlet BCs for simulating complex fluid flow relevant to drug formulation processes, such as extrusion and hot-melt extrusion for amorphous solid dispersions.

Core Boundary Condition Definitions & Data

Table 1: Summary of Critical Boundary Conditions for Die Flow Simulation

Boundary Type Mathematical Formulation (Typical) Physical Interpretation Key Parameter in Bird-Carreau Context
Inlet: Pressure-Driven p = p₀ (specified) A fixed total pressure is applied at the flow entrance. Common for equipment where pressure is the controlled variable (e.g., extrusion). Inlet pressure (p₀) indirectly determines the shear rate profile, affecting the apparent viscosity (η) calculated by the Bird-Carreau model.
Inlet: Flow Rate-Driven Q = ∫v·dA = Q₀ (specified) A fixed volumetric flow rate is applied. Common for precision pumping systems. Directly controls the average velocity, influencing the shear rate and thus the non-Newtonian viscosity field from the inlet.
Wall: No-Slip v = 0 Fluid velocity relative to the stationary wall is zero. Fundamental for viscous flow. Critical for generating the high shear rate gradient near the wall, where shear-thinning is most pronounced for Bird-Carreau fluids.
Outlet: Pressure p = p_out (specified), often p_out = 0 (gauge) A static pressure is fixed at the flow exit. Represents discharge to atmospheric or back pressure. Outlet pressure (p_out) sets the baseline for the pressure gradient driving the flow. Must be lower than inlet pressure.
Outlet: Outflow/Zero Gradient ∂v/∂n = 0 (approx.) Velocity and pressure are extrapolated from the interior. Used where flow is fully developed at exit. Assumes flow is fully developed, which may not be valid for highly shear-thinning fluids in short dies; requires domain length validation.

Experimental Protocol: Characterizing Inlet Condition for a Non-Newtonian Fluid

Objective: To determine the appropriate inlet boundary condition (pressure or flow rate) and necessary fluid parameters for a Bird-Carreau model simulation of a polymeric drug carrier (e.g., HPMCAS) in a capillary die.

Materials & Equipment:

  • Twin-screw extruder or precision piston-driven capillary rheometer.
  • Die assembly with known geometry (length L, diameter D).
  • Pressure transducers (at die entrance and along length if possible).
  • Temperature control unit.
  • Data acquisition system.

Procedure:

  • Fluid Preparation & Loading: Prepare the polymer/drug blend. Preheat the rheometer barrel and die to the target processing temperature (e.g., 150°C). Allow thermal equilibrium.
  • Flow Rate-Controlled Experiment: a. Set the piston to a constant velocity, corresponding to a specific volumetric flow rate (Q₁). b. Allow flow to stabilize (steady-state pressure readings). c. Record the steady-state pressure drop (ΔP) across the die length. d. Repeat steps a-c for a range of flow rates (Q₂...Qₙ) covering the expected shear rate range.
  • Data Processing for Bird-Carreau Parameters: a. For each (Q, ΔP) pair, calculate the apparent shear rate (γ̇app = 32Q/πD³) and apparent shear stress (τapp = ΔP D / 4L). b. Apply the Weissenberg-Rabinowitsch correction to obtain true wall shear rate (γ̇w). c. Fit the true (τw, γ̇_w) data to the Bird-Carreau model: η(γ̇) = η∞ + (η₀ - η∞) * [1 + (λγ̇)²]^((n-1)/2) where η₀ is zero-shear viscosity, η∞ is infinite-shear viscosity, λ is the time constant, and n is the power-law index. d. The fitted parameters (η₀, η∞, λ, n) are the input for the constitutive model in the simulation.
  • Inlet BC Selection:
    • If the experimental setup mimics a pump-driven system (constant Q), use a flow rate inlet in the simulation, with the value(s) from Step 2.
    • If the setup mimics an extrusion process where pressure is the driving force, use a pressure inlet, with the value(s) of P₀ derived from the recorded ΔP plus any known entrance pressure losses.

Protocol: Implementing and Validating No-Slip and Outlet Conditions in Simulation

Objective: To set up and verify the wall and outlet boundary conditions in a Computational Fluid Dynamics (CFD) model of die flow using the experimentally derived Bird-Carreau parameters.

Pre-Simulation Setup:

  • Geometry & Mesh: Create a 2D-axisymmetric or 3D model of the capillary die. Generate a mesh with significant refinement near the wall to capture the high shear rate gradient.
  • Solver Settings: Select a pressure-based, steady-state solver. Enable gravity if significant.

Procedure:

  • Material Definition: Input the fitted Bird-Carreau model parameters (from Table 2) into the non-Newtonian fluid property definition in the CFD software.
  • Boundary Condition Assignment: a. Inlet: Select either "Pressure Inlet" (set Gauge Pressure to experimental P₀) OR "Mass Flow Rate/Volume Flow Rate Inlet" (set to experimental Q). Specify flow direction. b. Walls: Apply the "No-Slip" condition. Set wall temperature as per experiment. For adiabatic flow, use thermally insulated condition. c. Outlet: Apply "Pressure Outlet" with a gauge pressure of 0 Pa (atmospheric) or a specified back pressure. This is generally more physically representative for die exit than outflow conditions.
  • Solution Initialization & Calculation: Hybrid initialize and run the calculation until residuals plateau and monitored parameters (e.g., outlet mass flow, average outlet pressure) stabilize.
  • Validation against Experiment: a. Extract the simulated pressure drop (ΔPsim) across the die length for the given inlet condition. b. Compare ΔPsim to the experimentally measured ΔP_exp for the same flow rate. c. A successful validation typically requires agreement within ±10%.

Table 2: Example Bird-Carreau Parameters for 20% Drug in HPMCAS at 150°C

Parameter Symbol Value Unit Determination Method
Zero-Shear Viscosity η₀ 1.2 x 10⁵ Pa·s Extrapolation from low shear rate rheometry
Infinite-Shear Viscosity η∞ 1.0 x 10⁻¹ Pa·s Estimated from high-shear capillary data
Time Constant λ 15.8 s Curve-fitting of shear stress vs. rate data
Power-Law Index n 0.45 - Slope of log-log plot at intermediate shear rates

Visualization of the Boundary Condition Implementation Workflow

bc_workflow Start Start: Define Simulation Goal ExpData Acquire Experimental Rheological Data Start->ExpData ModelFit Fit Data to Bird-Carreau Model ExpData->ModelFit BC_Select Select Inlet BC Type (Pressure or Flow Rate) GeoMesh Create Geometry & Mesh BC_Select->GeoMesh InputParams Input Parameters (η₀, η∞, λ, n) ModelFit->InputParams InputParams->BC_Select SetInlet Apply Inlet BC (p₀ or Q₀) GeoMesh->SetInlet SetWall Apply Wall BC (No-Slip, v=0) SetInlet->SetWall SetOutlet Apply Outlet BC (p_out=0) SetWall->SetOutlet Solve Run CFD Solver SetOutlet->Solve Validate Validate ΔP_sim vs ΔP_exp Solve->Validate Validate->ExpData Poor Agreement End Validated Simulation Model Validate->End ±10% Agreement

Diagram Title: Workflow for Implementing BCs in Non-Newtonian Die Flow CFD

The Scientist's Toolkit: Essential Research Reagents & Materials

Table 3: Key Materials for Bird-Carreau Die Flow Experimentation

Item Function/Justification Example/Specification
Polymeric Excipient Forms the shear-thinning matrix for drug delivery. Its molecular weight dictates η₀ and λ. Hydroxypropyl methylcellulose acetate succinate (HPMCAS), PVP-VA.
Model Drug Compound Active Pharmaceutical Ingredient (API) used to study the effect of loading on rheology. A BCS Class II drug (e.g., itraconazole, fenofibrate).
Plasticizer Modifies the glass transition temperature and processing viscosity of the blend. Triethyl citrate, polyethylene glycol.
Capillary Rheometer Instrument to apply controlled shear rates/pressures and measure the pressure drop for parameter fitting. Piston-driven with heated barrel, L/D ≥ 20 dies.
High-Pressure Differential Transducer Precisely measures the small pressure drop across the die for shear stress calculation. ±0.1% FS accuracy, temperature rated.
CFD Software with Non-Newtonian Solver Platform to implement the Bird-Carreau model and boundary conditions for flow field prediction. ANSYS Fluent, COMSOL, OpenFOAM.
Mesh Generation Software Creates the computational domain with necessary refinement at walls and inlets. ANSYS Mesher, Gmsh.

This application note details the protocols for solving coupled flow field variables within an extrusion or injection die, framed within a broader thesis implementing the Bird-Carreau constitutive model for non-Newtonian fluids. Accurate prediction of pressure (P), velocity (v), shear rate (\dot{γ}), and spatially variable viscosity (η) is critical in pharmaceutical manufacturing for ensuring uniform drug-polymer matrix distribution in hot-melt extrusion, achieving consistent coating thickness in film casting, and controlling API dispersion in injectable depots. The Bird-Carreau model effectively captures the shear-thinning behavior prevalent in polymeric melts and semi-solid formulations, making its implementation essential for precise die design and process optimization.

Governing Equations & Numerical Implementation

The flow of an incompressible, generalized Newtonian fluid is governed by the continuity and momentum conservation equations: ∇·v = 0 ρ(∂v/∂t + v·∇v) = -∇P + ∇·τ where ρ is density and τ is the stress tensor. For the Bird-Carreau model, the stress tensor is defined as τ = η(˙γ)˙γ, with the viscosity function given by: η(˙γ) = η_∞ + (η_0 - η_∞)[1 + (λ˙γ)^a]^((n-1)/a) where:

  • η_0 is the zero-shear viscosity,
  • η_∞ is the infinite-shear viscosity,
  • λ is the relaxation time constant,
  • n is the power-law index,
  • a is a dimensionless parameter governing the transition region,
  • ˙γ is the scalar shear rate, ˙γ = √(2D:D), with D as the rate-of-deformation tensor.

These equations form a tightly coupled system, as viscosity depends on shear rate, which is derived from the velocity field solution.

Research Reagent & Computational Toolkit

Table 1: Essential Research Reagents & Computational Tools

Item Function in Die Flow Research
Polymer/Drug Melt (e.g., PLGA, HPMC, PEO with API) The non-Newtonian test fluid. Its Bird-Carreau parameters (η_0, η_∞, λ, n, a) must be characterized via rheometry.
High-Pressure Capillary Rheometer Provides experimental validation data for pressure drop versus flow rate under high shear conditions relevant to die flow.
Computational Fluid Dynamics (CFD) Software (e.g., ANSYS Polyflow, COMSOL, OpenFOAM) Platform for implementing the User-Defined Function (UDF) for the Bird-Carreau model and solving the coupled momentum-mass conservation system.
Laser Doppler Velocimetry (LDV) or Particle Image Velocimetry (PIV) Enables non-invasive measurement of velocity profiles within transparent die analogs for model validation.
Pressure Transducers (Multiple, flush-mounted along die length) Measures axial pressure gradient, a key benchmark for simulation accuracy.

Experimental & Numerical Protocols

Protocol 4.1: Determination of Bird-Carreau Model Parameters

Objective: To empirically determine the parameters (η_0, η_∞, λ, n, a) for the test formulation.

  • Equipment: Controlled-stress rotational rheometer with parallel plate or cone-and-plate geometry.
  • Procedure: a. Perform a steady-state flow sweep across a relevant shear rate range (e.g., 0.01 to 1000 s⁻¹) at constant process temperature. b. Record the viscosity (η) versus shear rate (˙γ) data. c. Using nonlinear regression software (e.g., via the Levenberg-Marquardt algorithm), fit the Bird-Carreau equation to the collected data. d. Validate the fit quality by comparing predicted and measured viscosity across the shear rate range.
  • Output: A set of five fitted parameters for use in the CFD simulation.

Protocol 4.2: CFD Simulation of Planar Slit Die Flow

Objective: To solve for P, v, ˙γ, η distributions in a 2D planar slit die geometry.

  • Pre-processing (Geometry & Mesh): a. Create a 2D geometry of the die flow channel (e.g., length L=50 mm, height H=1 mm). b. Generate a structured, boundary-layer-refined mesh. Ensure mesh independence via grid convergence study.
  • Physics Setup: a. Select a steady-state, incompressible flow solver. b. Implement the Bird-Carreau viscosity model via a UDF, inputting the parameters from Protocol 4.1. c. Set boundary conditions: Inlet - volumetric flow rate (Q); Walls - no-slip (v=0); Outlet - atmospheric pressure (P=0 gauge).
  • Solver Settings: a. Use a coupled pressure-velocity scheme. b. Employ a second-order upwind discretization for momentum. c. Solve until scaled residuals fall below 1e-6 and key monitors (e.g., outlet pressure) stabilize.
  • Post-processing: Extract and visualize contours of pressure, velocity magnitude, shear rate, and viscosity. Plot velocity profile at die outlet and pressure drop along the centerline.

Protocol 4.3: Experimental Validation Using a Lab-Scale Extruder

Objective: To validate the CFD-predicted pressure drop and flow field.

  • Equipment: Twin-screw extruder with a instrumented slit die (multiple pressure transducers), LDV/PIV system for transparent die.
  • Procedure: a. Process the characterized material through the extruder at a set temperature and screw speed. b. Record the steady-state pressure readings from each transducer along the die. c. (If using a transparent die) Introduce tracer particles and use PIV to capture the velocity field at a plane of interest. d. Compare the experimental pressure gradient and velocity profile to the CFD predictions for the same operating conditions.

Representative Data & Results

Table 2: Simulation Output for a Model PLGA Melt (T=180°C, Q=5 cm³/s) in a Slit Die

Flow Field Variable Symbol Maximum Value Minimum Value Key Location/Note
Pressure P 4.72 MPa 0 MPa (Gauge) Inlet / Outlet
Velocity Magnitude v 75.4 mm/s 0 mm/s Channel Centerline / Wall
Shear Rate ˙γ 1508 s⁻¹ ~0 s⁻¹ Die Wall / Centerline
Viscosity η 1250 Pa·s 12.8 Pa·s Centerline (Low Shear) / Wall (High Shear)

Parameters: η₀=1250 Pa·s, η∞=10 Pa·s, λ=1.2 s, n=0.35, a=2.

Visualized Workflows & Relationships

G Start Start: Formulation (PLGA + API) Rheology Protocol 4.1: Rheological Characterization Start->Rheology Params Extract Bird-Carreau Parameters (η₀, λ, n, etc.) Rheology->Params CFD Protocol 4.2: CFD Simulation Setup (Geometry, Mesh, BCs) Params->CFD Solve Solve Coupled System: Continuity + Momentum with Bird-Carreau UDF CFD->Solve Output Obtain Flow Field: P, v, γ̇, η Distributions Solve->Output Val Protocol 4.3: Experimental Validation (Pressure, PIV) Output->Val Compare Compare Simulation & Experimental Data Val->Compare Valid Model Validated Compare->Valid Agreement Adjust Adjust Model Parameters/BCs Compare->Adjust Discrepancy Adjust->CFD

Title: Workflow for Solving Die Flow with Bird-Carreau Model

G BC_Params Bird-Carreau Parameters (η₀, η∞, λ, n, a) Viscosity Update Viscosity Field η(γ̇) = η∞ + (η₀-η∞)[1+(λγ̇)^a]^(ⁿ⁻¹⁄ₐ) BC_Params->Viscosity Velocity_Field Velocity Field (v) Solution Guess ShearRate Calculate Shear Rate Field (γ̇ = √(2D:D)) Velocity_Field->ShearRate Converged Solution Converged? Velocity_Field->Converged ShearRate->Viscosity Momentum Solve Momentum Equation ∇P = ∇·(η(γ̇)∇v) Viscosity->Momentum Continuity Solve Continuity Equation ∇·v = 0 (Correct Pressure) Momentum->Continuity Continuity->Velocity_Field Update v, P Converged->ShearRate No Results Final Coupled Fields: P, v, γ̇, η Converged->Results Yes

Title: Coupled Numerical Solution Algorithm Logic

This application note details critical post-processing protocols for capillary die flow analysis, framed within a doctoral thesis investigating the implementation of the Bird-Carreau constitutive model for simulating non-Newtonian fluid flow in pharmaceutical extrusion processes. Accurate prediction of extrudate swell, pressure drop, and wall shear stress is paramount for die design, ensuring uniform drug product quality, and mitigating processing issues such as degradation or uneven mixing. These results directly inform scale-up and validation in drug development.

Theoretical Foundation: The Bird-Carreau Model

The Bird-Carreau model describes the shear-thinning behavior of polymeric melts and solutions, common in pharmaceutical formulations. It relates the apparent viscosity (η) to the shear rate (γ̇):

η(γ̇) = η∞ + (η₀ - η∞) * [1 + (λγ̇)²]^((n-1)/2)

Where:

  • η₀: Zero-shear viscosity (Pa·s)
  • η∞: Infinite-shear viscosity (Pa·s)
  • λ: Relaxation time (s)
  • n: Power-law index (dimensionless)

This model is integral to the finite element simulations from which the key results (swell, pressure, stress) are derived.

Core Post-Processing Protocols

Protocol: Extrudate Swell (Die Swell) Prediction from Simulation Data

Objective: To predict the final diameter of the extrudate after it exits the die, a critical factor for pellet or filament size control.

Methodology:

  • Simulation Completion: Run a transient, free-surface flow simulation using the Bird-Carreau parameters for your specific formulation.
  • Data Extraction: At the die exit plane, extract the velocity vector field and the stress tensor components at the final steady-state simulation time.
  • Swelling Calculation: The swell ratio (B) is computed as the ratio of the extrudate diameter to the die diameter. A common predictive method uses the first normal stress difference (N₁) at the die wall:
    • N₁ = τ₁₁ - τ₂₂ (where τ are normal stress components).
    • An empirical relation: B ≈ 0.1 + [1 + (1/2 * (N₁/2τw)^2 )]^(1/6), where τw is the wall shear stress.
  • Validation: Compare predicted swell against experimental measurements from a capillary rheometer equipped with a laser die swell detector.

Table 1: Sample Extrudate Swell Predictions for Model Formulations

Formulation Code Bird-Carreau Parameters (η₀, λ, n) Shear Rate (s⁻¹) Predicted Swell Ratio Experimental Swell Ratio (± SD)
Frm-A (HPMC) 8500 Pa·s, 0.5 s, 0.45 100 1.32 1.28 ± 0.03
Frm-B (PEO) 12000 Pa·s, 1.2 s, 0.38 50 1.41 1.45 ± 0.05
Frm-C (API Loaded) 15000 Pa·s, 0.8 s, 0.42 75 1.38 1.35 ± 0.04

Protocol: Pressure Drop Calculation and Analysis

Objective: To determine the total and component pressure losses through a complex die, essential for equipment sizing and ensuring homogeneous pressure history.

Methodology:

  • Domain Segmentation: Divide the die geometry into logical sections (e.g., inlet reservoir, conical convergence, land region).
  • Probe Placement: Define pressure probe points at the entrance and exit of each section in the simulation model.
  • Data Collection: Extract the area-averaged pressure at each probe point under steady-state flow conditions.
  • Drop Calculation: Calculate the pressure drop (ΔP) for each section: ΔPsection = Pin - P_out.
  • Total Drop: The total pressure drop is the sum of all section drops or directly Pinlet - Poutlet (atmospheric).
  • Analysis: Compare the percentage contribution of each section, particularly the entry (elongational) and land (shear) regions.

Table 2: Pressure Drop Analysis for a Conical Die (Inlet Dia: 5mm, Land Dia: 1mm, L=10mm)

Die Section Pressure (MPa) Section ΔP (MPa) % of Total ΔP Dominant Flow Regime
Inlet Reservoir 12.5 0.5 4% Stagnant/Very Low Shear
Conical Convergence 12.0 8.2 66% Elongational Flow
Capillary Land 3.8 3.8 30% Shear Flow
Outlet (Atmospheric) 0.0 - - -
TOTAL - 12.5 100% -

Protocol: Wall Shear Stress Analysis

Objective: To quantify the shear stress at the die wall, a key parameter for predicting material degradation, interface slip, and ensuring reproducible flow.

Methodology:

  • Shear Stress Tensor: From the simulation, extract the full stress tensor (σ) at the elements adjacent to the die wall.
  • Wall Shear Stress (τw) Calculation: Compute the magnitude of the shear stress vector acting tangentially to the wall surface. In a cylindrical die land, this simplifies to: τw = (ΔP * R) / (2 * L), where R is radius and L is length.
  • Spatial Mapping: Create a contour plot of τ_w along the die wall surface to identify "hot spots" of high stress.
  • Critical Comparison: Compare the maximum τ_w value against the known critical shear stress of the formulation, beyond which surface defects or API degradation may occur.

Table 3: Calculated Shear Stress vs. Critical Stress for API Stability

Formulation Max τ_w at Wall (kPa) Critical Shear Stress (kPa) from Rheology τw / τcrit Ratio Risk of Degradation
Frm-A 85 120 0.71 Low
Frm-B 110 105 1.05 High (Marginal)
Frm-C 95 90 1.06 High (Marginal)

The Scientist's Toolkit: Essential Research Reagents & Materials

Table 4: Key Reagent Solutions and Materials for Die Flow Experimentation

Item Function/Explanation
Model Polymer Solutions (e.g., HPMC, PEO in Glycerol/Water) Well-characterized non-Newtonian fluids used to validate simulations before testing expensive API-loaded formulations.
Capillary Rheometer Primary experimental device for applying controlled shear rates, measuring pressure drop, and extruding material.
Laser-Based Die Swell Sensor Precisely measures the diameter of the extrudate as it emerges from the die to validate swell predictions.
Pressure Transducers (High-Temp) Installed along the barrel/die to provide experimental pressure drop data for model calibration.
Data Acquisition System Synchronizes data collection from pressure sensors, load cells, and laser sensors for correlated analysis.
Finite Element Software (e.g., COMSOL, ANSYS Polyflow) Platform for implementing the Bird-Carreau model and performing the 3D flow simulations.
High-Performance Computing (HPC) Cluster Enables running complex, transient free-surface simulations with realistic material models in feasible timeframes.

Visualized Workflows and Relationships

post_process thesis Thesis Core: Bird-Carreau Model Implementation sim 3D Non-Newtonian Flow Simulation thesis->sim data Raw Simulation Data (Velocity, Pressure, Stress Fields) sim->data proc1 Protocol 1: Extrudate Swell Prediction data->proc1 proc2 Protocol 2: Pressure Drop Calculation data->proc2 proc3 Protocol 3: Shear Stress Analysis data->proc3 res1 Predicted Swell Ratio & Diameter proc1->res1 res2 Sectional & Total ΔP (% Contribution) proc2->res2 res3 τ_w Map & Max Stress vs. Critical Stress proc3->res3 app Application: Die Design Optimization, Scale-up, API Degradation Risk Assessment res1->app res2->app res3->app

Diagram 1: Post-Processing Workflow for Die Flow Research

swell_logic visc Shear-Thinning Viscosity (η) n1 First Normal Stress Difference (N₁) visc->n1 Affects recoil Elastic Energy Storage & Recovery recoil->n1 Drives velocity Velocity Profile Reformation at Exit ratio Swell Prediction (B = D_final / D_die) velocity->ratio Determines n1->ratio Primary Predictor tau_w Wall Shear Stress (τ_w) tau_w->ratio Secondary Input

Diagram 2: Key Factors in Extrudate Swell

Solving Convergence Challenges: Troubleshooting Numerical Instabilities and Optimizing Bird-Carreau Simulations

Application Notes

Within the context of implementing the Bird-Carreau model for non-Newtonian die flow research in pharmaceutical development, a critical numerical challenge arises when simulating fluids with a low power-law index (n) under high shear rate conditions. This combination, common in polymeric solutions and some biological fluids, can lead to divergence in computational fluid dynamics (CFD) simulations, compromising the accuracy of die flow predictions essential for drug delivery system manufacturing.

The Bird-Carreau model describes the apparent viscosity (η) as: η(γ̇) = η∞ + (η₀ - η∞)[1 + (λγ̇)²]^((n-1)/2) where η₀ is the zero-shear viscosity, η∞ is the infinite-shear viscosity, λ is the time constant, γ̇ is the shear rate, and n is the power-law index.

The pitfall occurs because as n decreases (e.g., n < 0.3) and γ̇ increases, the term (λγ̇)² becomes large, and the exponent ((n-1)/2) becomes a negative number with a large magnitude. This causes the viscosity function to approach zero very rapidly, creating an extremely steep gradient. From a numerical standpoint, this leads to a poorly conditioned system matrix, causing iterative solvers (e.g., pressure-velocity coupling in SIMPLE/PISO algorithms) to fail to converge, resulting in divergence.

The impact on die flow research is significant: inaccurate prediction of pressure drops, flow front advancement, and shear stress distribution within the die, which are critical for controlling product uniformity (e.g., of transdermal patches or biodegradable implants) and preventing defects.

Table 1: Impact of Low n and High γ̇ on Simulation Stability

Power-Law Index (n) Shear Rate Range (γ̇) [1/s] Typical Fluid Type Relative Condition Number of System Matrix Observed Convergence Rate (%)
0.2 10⁴ - 10⁶ Highly shear-thinning polymer melt 10¹⁰ 15%
0.4 10⁴ - 10⁶ Pharmaceutical gel (e.g., Carbopol) 10⁷ 65%
0.6 10⁴ - 10⁶ Semidilute suspension 10⁵ 95%
0.8 10⁴ - 10⁶ Blood analog 10³ 99%

Table 2: Recommended Numerical Parameters for Stable Bird-Carreau Implementation

Parameter Standard Value Range Recommended Value for Low n (<0.3) Purpose
Under-Relaxation Factor (Momentum) 0.3 - 0.7 0.1 - 0.3 Slows solution update to prevent oscillation from steep gradients.
Solver Type (Viscosity Update) Explicit Semi-Implicit Couples viscosity update with momentum equation.
Tolerance (Pressure-Velocity) 10⁻³ - 10⁻⁴ 10⁻⁶ Tighter convergence criteria for difficult cases.
Maximum Iterations per Time Step 20 - 50 100 - 200 Allows more iterations for hard-to-converge steps.
Time Step Size (Transient) Adaptive 10⁻⁵ - 10⁻⁶ s Smaller steps capture rapid viscosity changes.

Experimental Protocols

Protocol 1: Determining Critical Numerical Stability Limits for Bird-Carreau Fluids

Objective: To empirically establish the combination of power-law index (n) and shear rate (γ̇) at which a standard finite-volume CFD solver diverges for a Bird-Carreau fluid in a simple die geometry.

Materials & Setup:

  • CFD Software: ANSYS Fluent 2023 R2 or OpenFOAM v10.
  • Geometry: 2D axisymmetric capillary die (Length: 10mm, Diameter: 1mm).
  • Mesh: Structured quadrilateral mesh with boundary layer refinement at wall (y+ < 1).
  • Boundary Conditions: Inlet: fully-developed velocity profile (alternatively, pressure inlet). Outlet: pressure outlet (atmospheric). Wall: no-slip.

Procedure:

  • Parameter Space Definition: Define a matrix of test parameters: n = [0.1, 0.2, 0.3, 0.4, 0.5] and inlet pressure leading to approximate wall shear rates of γ̇_wall = [10³, 10⁴, 10⁵, 10⁶] s⁻¹. Keep η₀ = 100 Pa·s, η∞ = 0.001 Pa·s, λ = 1.0 s constant.
  • Solver Configuration: Use a pressure-based coupled solver. Set discretization schemes to second-order upwind for momentum and PRESTO! for pressure.
  • Stability Enhancement: Activate the "High-Shear Viscosity Cutoff" option, if available, to limit the minimum allowable η to 0.1*η∞.
  • Iterative Solving: For each {n, γ̇} pair, run the simulation from zero initial conditions.
  • Divergence Detection: Monitor the scaled residuals for continuity and momentum equations. Define divergence as a continuous rise in residuals for >50 iterations or residuals exceeding 10¹⁰.
  • Data Logging: Record the number of iterations to convergence or divergence, final residual values, and computed pressure drop for each successful run.
  • Post-Processing: Calculate the apparent viscosity at the wall for each case and plot stability region maps.

Protocol 2: Validation Using a Controlled Rheometric Flow Cell

Objective: To validate the corrected CFD model predictions against experimental data for a low-n test fluid under high shear.

Materials:

  • Test Fluid: 0.4% Polyacrylamide in 60/40 water-glycerol solution (approximates n ≈ 0.25).
  • Equipment: High-pressure capillary rheometer with die L/D = 20.
  • Instrumentation: Precision pressure transducers, mass flow meter, temperature control.

Procedure:

  • Fluid Characterization: Using cone-and-plate rheometry, obtain full flow curve. Fit data to the Bird-Carreau model to extract precise η₀, η∞, λ, and n.
  • High-Shear Rheometry: Load fluid into capillary rheometer. Perform steady-state flow experiments at controlled piston speeds corresponding to wall shear rates from 10³ to 10⁶ s⁻¹.
  • Data Collection: Record the steady-state pressure drop (ΔP) across the die and the volumetric flow rate (Q) at each speed. Ensure isothermal conditions.
  • CFD Simulation: Construct a 3D model of the exact capillary rheometer die. Use the fitted Bird-Carreau parameters and the implemented stability mitigations (from Protocol 1).
  • Comparison: For each experimental Q, run the corresponding CFD simulation. Compare the predicted ΔP with the measured ΔP. Calculate the relative error.
  • Sensitivity Analysis: Vary the numerical parameters (under-relaxation, solver type) in the CFD model to quantify their impact on prediction error for the lowest-n, highest-γ̇ condition.

Visualizations

stability_mitigation Low-n High-Shear Rate Divergence Mitigation start Simulation Divergence (Residuals Rise) check_n Check Bird-Carreau Parameters: n < 0.4? start->check_n check_shear Check Max Shear Rate γ̇ > 10⁴ s⁻¹? check_n->check_shear Yes converge Stable Convergence Achieved check_n->converge No step1 Step 1: Apply Viscosity Floor η_min = 0.1*η∞ check_shear->step1 Yes check_shear->converge No step2 Step 2: Reduce Relaxation Factors Momentum < 0.3 step1->step2 step3 Step 3: Switch to Coupled Solver & Use Smaller Time Step step2->step3 step3->converge

bc_workflow Bird-Carreau Stability Validation Workflow A Rheological Characterization B Extract Model Parameters (η₀, η∞, λ, n) A->B C Define Critical Case (Low n, High γ̇) B->C D Baseline CFD Simulation C->D E Divergence? D->E F Apply Numerical Mitigations E->F Yes G High-Shear Capillary Experiment E->G No F->D H Compare ΔP, Q Validate Model G->H I Model Ready for Die Flow Research H->I

The Scientist's Toolkit: Research Reagent & Numerical Solutions

Table 3: Essential Materials & Numerical Tools for Low-n Bird-Carreau Research

Item Name/Software Category Function in Research Critical Note
Polyacrylamide (MW 5-10M Da) Research Reagent Model shear-thinning fluid for preparing low-n (<0.3) test solutions. Prepare in water-glycerol mixes to control zero-shear viscosity.
High-Pressure Capillary Rheometer (e.g., Rosand RH7) Equipment Generates controlled, high shear rate (up to 10⁷ s⁻¹) flow for experimental validation. Essential for collecting pressure-drop data under relevant die flow conditions.
"Viscosity Floor" or "Cutoff" UDF Numerical Tool User-Defined Function to impose a lower limit on calculated viscosity, preventing numerical underflow. Prevents η → 0, which causes matrix singularity.
Semi-Implicit Viscosity Update Algorithm Numerical Method Couples the viscosity calculation with the velocity field update within the solver iteration. Increases stability vs. explicit update but adds computational cost.
Adaptive Time-Stepping (Transient) Numerical Scheme Automatically reduces time-step size when residuals rise sharply during high shear events. Crucial for transient die-filling simulations.
Double-Precision Solver Computational Setting Uses 64-bit floating-point arithmetic, reducing round-off error in steep gradient calculations. Non-negotiable for low-n cases; single precision will almost certainly diverge.

This application note, situated within a broader thesis on Bird-Carreau model implementation for non-Newtonian die flow in pharmaceutical extrusion processes, details critical methodologies for optimizing Computational Fluid Dynamics (CFD) solver settings. Accurate simulation of non-Newtonian flow, essential for drug product development (e.g., hot-melt extrusion, film coating), hinges on the appropriate selection of discretization schemes, under-relaxation factors (URFs), and linear solvers to ensure convergence, accuracy, and computational efficiency.

Core Solver Settings: Quantitative Comparison & Protocols

Discretization Schemes for the Bird-Carreau Model

Discretization schemes approximate the solution of governing equations (momentum, continuity, energy) across the computational domain.

Table 1: Common Discretization Schemes for Advection Terms

Scheme Order of Accuracy Stability / Numerical Diffusion Recommended Use for Bird-Carreau Flow
First-Order Upwind 1st Highly stable, significant false diffusion Initialization, highly shear-thinning regions with steep gradients.
Power Law 1st-2nd Conditionally stable, less diffusive than upwind General purpose for moderate shear rates.
Second-Order Upwind 2nd More accurate, potentially unstable Refined simulations where accuracy is critical; requires good mesh quality.
QUICK 3rd for structured meshes High accuracy for structured hex meshes, can oscillate Accurate resolution of velocity profiles in die channels with structured grids.
MUSCL 2nd-3rd High resolution with flux limiters to prevent oscillations Preferred for capturing sharp gradients (e.g., near die walls) with complex geometries.

Protocol 1: Selection and Implementation of Discretization Schemes

  • Initialization Run: Begin simulations with a first-order upwind scheme for all transport equations to obtain a stable initial flow field.
  • Refinement Phase: Switch to higher-order schemes (e.g., Second-Order Upwind for momentum, QUICK for species/energy if applicable) for the final computation.
  • Monitoring: Closely monitor residuals during the switch. A sudden divergence indicates instability, requiring reduced under-relaxation factors or a hybrid scheme approach.
  • Validation: Compare the fully developed velocity and shear rate profile in a simple duct section against an analytical or semi-analytical solution for the Bird-Carreau model to gauge scheme accuracy.

Under-Relaxation Factor (URF) Optimization

URFs control the update of solution variables between iterations, stabilizing the convergence process for strongly non-linear equations like the Bird-Carreau model.

Table 2: Typical Under-Relaxation Factor Ranges for Non-Newtonian Flow

Equation / Variable Recommended URF Range (Steady-State) Adjustment Guidance
Pressure 0.1 - 0.3 Use lower values (0.1-0.2) for high apparent viscosity ratios.
Density 1.0 Typically kept at 1.0 for incompressible flow.
Body Forces 1.0 Typically kept at 1.0.
Momentum 0.5 - 0.7 Start at 0.5 for strongly shear-thinning fluids; increase as solution stabilizes.
Non-Newtonian Viscosity (Bird-Carreau) 0.7 - 0.9 Use high factors but monitor viscosity field oscillation.
Turbulence Quantities (if k-ε used) 0.5 - 0.8 Required for turbulent non-Newtonian flow regimes.

Protocol 2: Systematic Tuning of Under-Relaxation Factors

  • Baseline Configuration: Start with conservative URFs (lower end of ranges in Table 2) for pressure and momentum.
  • Convergence Monitoring: Run for 50-100 iterations. Observe the residual plot. Smooth, monotonic reduction indicates good stability.
  • Incremental Increase: If residuals decrease smoothly, incrementally increase URFs (in steps of 0.1 for momentum, 0.05 for pressure) to accelerate convergence.
  • Oscillation Response: If residuals oscillate or diverge, reduce the URF for the offending variable by 30-50%.
  • Final "Aggressive" Settings: Once near convergence (residuals reduced by ~3 orders of magnitude), slightly increase URFs to achieve final tolerance faster.

Solver Types and Algorithms

The choice of pressure-velocity coupling algorithm and linear equation solvers is paramount.

Table 3: Solver and Algorithm Selection for Pressure-Velocity Coupling

Algorithm Description Suitability for Non-Newtonian Die Flow
SIMPLE Semi-Implicit Method for Pressure Linked Equations. Robust, slower convergence. Default choice for most steady-state, laminar non-Newtonian flows. Highly robust.
SIMPLEC SIMPLE-Consistent. Often allows larger pressure URFs, faster convergence. Preferred over SIMPLE for faster convergence if stability is maintained.
PISO Pressure-Implicit with Splitting of Operators. Non-iterative, transient-focused. Recommended for transient simulations or highly skewed meshes.
Coupled Solves momentum and pressure equations simultaneously. Can be faster for steady-state, high-Rayleigh number flows; requires significant memory.

Protocol 3: Configuring the Linear Solvers and Controls

  • Pressure Equation: Use a geometric multigrid (GAMG) solver with a Gauss-Seidel smoother for complex geometries. For simple ducts, the Preconditioned Conjugate Gradient (PCG) solver is efficient. Set tolerance to 1e-6.
  • Momentum Equation: Typically use a smoother-based solver like Gauss-Seidel or an Incomplete Cholesky Preconditioned Conjugate Gradient (ICCG). Set tolerance to 1e-5.
  • Solution Limits: Apply strict solution limits for viscosity (e.g., between 1e-5 and 1e5 Pa·s) to prevent numerical overflow from the Bird-Carreau equation in low-shear regions.

Visualization of Optimization Workflow

G Start Start: Define Bird-Carreau Parameters & Geometry Mesh Generate Mesh (Quality Check) Start->Mesh BaseSettings Apply Base Solver Settings: - 1st Order Schemes - Low URFs - SIMPLE Algorithm Mesh->BaseSettings SolveInit Run Initial 100 Iterations BaseSettings->SolveInit CheckResid Check Residual Behavior SolveInit->CheckResid Stable Stable & Decreasing? CheckResid->Stable  Yes Increase Gradually Increase Scheme Order & URFs Stable->Increase  Yes NotStable Not Stable/Oscillating Stable->NotStable  No FinalSolve Run with Refined Settings to Convergence Increase->FinalSolve Decrease Reduce URFs Improve Mesh NotStable->Decrease Decrease->SolveInit Restart Validate Validate vs. Analytical/Experimental FinalSolve->Validate End Solution Ready for Analysis Validate->End

Title: Non-Newtonian Solver Optimization Workflow

The Scientist's Toolkit: Research Reagent Solutions & Essential Materials

Table 4: Key Computational & Material Reagents for Non-Newtonian Die Flow Research

Item / Reagent Function / Purpose in Research
CFD Software (e.g., ANSYS Fluent, OpenFOAM, COMSOL) Platform for implementing Bird-Carreau model, meshing, solving, and post-processing flow fields.
High-Performance Computing (HPC) Cluster Enables parallel processing for high-fidelity 3D transient simulations with complex geometries.
Rheometer (e.g., Capillary, Rotational) Essential for experimentally determining Bird-Carreau model parameters (η₀, η∞, λ, n) for the polymer melt or solution.
Model Pharmaceutical Polymer (e.g., HPMC, PVA, Eudragit) Non-Newtonian fluid substrate whose flow behavior is being studied for die design and process optimization.
Planar/Laser Doppler Velocimetry Setup Experimental apparatus for validating simulated velocity profiles in a transparent die or channel.
Differential Pressure Transducer Measures pressure drop across the die for comparison with CFD predictions, a key validation metric.
Structured/Unstructured Grid Generation Tool Creates the computational mesh; mesh quality near walls is critical for resolving high shear gradients.
Python/MATLAB Scripts For pre-processing rheological data, automating parametric studies, and custom post-processing of results.

Within the broader thesis on implementing the Bird-Carreau model for simulating non-Newtonian polymer flow in pharmaceutical die design, mesh dependency presents a critical challenge. Numerical solutions for shear-thinning fluids are highly sensitive to spatial discretization, particularly near wall boundaries where velocity gradients are steepest. This application note details protocols for achieving grid-independent solutions, a prerequisite for reliable predictions of drug-loaded polymer extrusion, mixing efficiency, and final product quality.

Core Concepts & Quantitative Benchmarks

Mesh Resolution Parameters

The table below summarizes key quantitative parameters for mesh sensitivity analysis in non-Newtonian die flow simulations using the Bird-Carreau model.

Table 1: Mesh Resolution Metrics and Target Values for Grid Independence

Parameter Symbol Coarse Mesh Medium Mesh Fine Mesh Target for Independence Function
Base Cell Size (mm) Δx 0.20 0.10 0.05 <0.025 Overall spatial resolution.
Near-Wall Layer Thickness (mm) δ₁ 0.10 0.05 0.025 ≤0.0125 Height of first cell adjacent to wall.
Number of Prism Layers Nₚ 5 10 20 ≥15 Cells to resolve boundary layer.
Wall y⁺ (Non-Dim.) y⁺ >5 ~1 <<1 <<1 (Low-Re req.) Dimensionless wall distance.
Global Element Count N 500k 2M 8M ≥4M* Total mesh cells.
Max. Aspect Ratio AR 50 25 15 <20 Cell width/height ratio.
Skewness - 0.8 0.5 0.3 <0.4 Measure of cell distortion.

*Target element count is problem-dependent; this is for a typical cylindrical die.

Solution Monitoring Variables

Table 2: Key Solution Variables for Grid Convergence Monitoring

Variable Location Measured Value (Example: Fine Mesh) Acceptable % Change (Final Mesh) Physical Significance
Wall Shear Stress (Pa) Die Wall 1.25e4 < 2% Determines viscous heating, stress on API.
Pressure Drop (kPa) Inlet-Outlet 850 < 1% Key for extrusion power and pump sizing.
Centerline Velocity (m/s) Die Center 0.15 < 0.5% Influences residence time.
Viscosity (Pa·s) High Shear Region 120 < 3% Critical non-Newtonian property.
Shear Rate (s⁻¹) Near Wall 5000 < 5% Directly impacts apparent viscosity.

Experimental & Numerical Protocols

Protocol: Systematic Mesh Refinement Study

Objective: To establish a mesh configuration that yields a grid-independent solution for Bird-Carreau die flow simulation. Materials: CAD model of die geometry, CFD software (e.g., ANSYS Fluent, OpenFOAM), High-Performance Computing (HPC) cluster. Procedure:

  • Geometry Preparation: Clean and defeature the die CAD model. Extract the fluid volume.
  • Baseline Mesh (Coarse): Generate an unstructured tetrahedral mesh with a global size per Table 1 (Coarse). Apply 5 prism layers on all walls with a first layer thickness calculated for y⁺ ~5.
  • Initial Solution: Run a steady-state simulation with the Bird-Carreau model until convergence (residuals < 1e-4). Record monitoring variables from Table 2.
  • Progressive Refinement: Sequentially generate Medium and Fine meshes (Table 1). For each: a. Global Refinement: Reduce base cell size by a factor of ~√2. b. Boundary Layer Refinement: Double the number of prism layers and halve the first layer thickness. c. Region-Based Refinement: Add cylindrical refinement zones in high shear regions (e.g., near the die lip).
  • Solution & Data Extraction: Run simulation for each mesh. Log all monitoring variables.
  • Grid Convergence Index (GCI) Calculation: Apply the Richardson Extrapolation method to calculate the GCI for key variables (e.g., pressure drop). A GCI < 3% between the two finest meshes indicates acceptable grid independence.
  • Final Validation Mesh: Create a mesh with specifications meeting or exceeding the "Target for Independence" in Table 1 based on GCI results.

Protocol: Near-Wall Treatment for Bird-Carreau Fluids

Objective: To accurately resolve the viscous sublayer where shear-thinning behavior is most pronounced. Materials: CFD software with low-Reynolds number k-ε or k-ω turbulence model capability, viscosity profile data. Procedure:

  • Mesh Generation for Low y⁺: Generate a mesh ensuring the centroid of the wall-adjacent cell lies within the viscous sublayer (y⁺ < 1). First layer thickness δ₁ = y⁺ * μ / (ρ * uτ), where uτ is the friction velocity (initially estimated).
  • Solver Settings: Select a low-Reynolds number turbulence model (e.g., SST k-ω). Disable wall functions. Ensure the numerical scheme is at least second-order accurate.
  • Bird-Carreau Parameters: Input the zero-shear viscosity (η₀), infinite-shear viscosity (η∞), time constant (λ), and power-law index (n) obtained from rheometry.
  • Iterative Adjustment: After an initial run, compute the actual y⁺ field. If y⁺ > 1 in critical areas, further refine the near-wall mesh and reiterate.
  • Verification: Plot the viscosity profile normal to the wall. Ensure it is smoothly resolved without abrupt jumps indicative of poor mesh resolution.

Visualizations

mesh_refinement_workflow Start Start: Geometry Definition M1 Generate Coarse Mesh (Global & Prism Layers) Start->M1 S1 Solve Bird-Carreau Model M1->S1 M2 Generate Medium Mesh (Refine Globally & Near-Wall) S1->M2 S2 Solve Bird-Carreau Model M2->S2 M3 Generate Fine Mesh (Target y+ << 1) S2->M3 S3 Solve Bird-Carreau Model M3->S3 Calc Calculate GCI & Monitor Variables S3->Calc Decision GCI < 3% ? Calc->Decision End Grid-Independent Solution Achieved Decision->End Yes Loop Further Refinement Required Decision->Loop No Loop->M3

Title: Mesh Independence Study Workflow

near_wall_modeling Wall Die Wall (No-Slip) Viscous Viscous Sublayer (y+ < 1) Wall->Viscous LogLayer Log-Law Region Viscous->LogLayer Core Turbulent Core Flow LogLayer->Core MeshReq Mesh Requirement: First Cell Centroid in Viscous Sublayer MeshReq->Viscous Physics Physics Resolved: Shear-Thinning Viscosity Steep Velocity Gradient Physics->Viscous Model Solver Model: Low-Re k-ω (SST) No Wall Functions Model->Viscous

Title: Near-Wall Mesh and Physics Modeling

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Materials & Reagents for Non-Newtonian Die Flow Study

Item Function/Benefit Example/Specification
Bird-Carreau Fluid Analog A well-characterized, shear-thinning test fluid to validate simulations. Aqueous solution of Carbopol or Xanthan Gum at specified concentration.
Capillary Rheometer Provides experimental data for viscosity vs. shear rate to fit Bird-Carreau model parameters (η₀, λ, n). Rosand RH7 or similar, with precision pressure transducers.
Laser Doppler Anemometry (LDA) / PIV System Non-invasively measures velocity profiles inside a transparent die model for direct CFD validation. 2D-PIV system with high-speed camera and Nd:YAG laser.
High-Performance Computing (HPC) Resources Enables the execution of multiple high-resolution (8M+ cells) CFD cases for mesh studies. Cluster with >= 64 cores, 256 GB RAM, and fast interconnects.
Structured/Unstructured Mesh Generator Creates the computational grid with advanced boundary layer control. ANSYS Mesher, Pointwise, or snappyHexMesh (OpenFOAM).
Convergence Monitoring Script Automates extraction and comparison of key variables (Table 2) across multiple mesh cases. Python/Matlab script interfacing with CFD solver output files.

1. Introduction and Thesis Context Within the broader thesis investigating the implementation of the Bird-Carreau constitutive model for simulating non-Newtonian polymer melt flow in pharmaceutical die design (e.g., hot-melt extrusion), managing numerical instabilities at extreme shear rates is paramount. The Bird-Carreau model describes shear-thinning viscosity (η) as: η(γ̇) = η∞ + (η₀ - η∞)[1 + (λγ̇)²]^( (n-1)/2 ) where η₀ is the zero-shear viscosity, η∞ is the infinite-shear viscosity, λ is the relaxation time, n is the power-law index, and γ̇ is the shear rate. At numerically extreme γ̇ → 0 or γ̇ → ∞, finite-precision arithmetic leads to overflow, underflow, or division-by-zero errors, corrupting computational fluid dynamics (CFD) solutions for die flow.

2. Numerical Limits and Stability Analysis Current analysis identifies critical computational limits in double-precision arithmetic:

Table 1: Numerical Limits for Double-Precision (64-bit) Implementation

Parameter Lower Safe Limit Upper Safe Limit Risk Beyond Limit
Shear Rate (γ̇) [s⁻¹] 1e-12 1e+12 Underflow/Overflow of (λγ̇)² term.
Viscosity (η) [Pa·s] 1e-6 1e+12 Invalid physical values disrupting solver matrix.
Term [1 + (λγ̇)²] --- --- Overflow at high γ̇, renders power law inactive at low γ̇ if <1.

3. Parameter Scaling Techniques (Protocol) Protocol 3.1: Non-Dimensionalization for Enhanced Stability Objective: Transform the Bird-Carreau equation into a scaled form to minimize the magnitude range of variables. Materials:

  • Reference Viscosity (η_ref): Typically η₀.
  • Reference Shear Rate (γ̇_ref): Characteristic process shear rate (e.g., at die wall).
  • CFD Solver with User-Defined Function (UDF) capability. Method:
  • Define scaled variables:
    • γ̇* = γ̇ / γ̇ref
    • η* = η / ηref
    • λ* = λ * γ̇_ref
  • Reformulate the model: η* = (η∞/η₀) + [1 - (η∞/η₀)] * [1 + (λ* γ̇*)²]^((n-1)/2)
  • Implement η* in the solver UDF. The computed dimensional viscosity is η = η* * η_ref. Validation: Run simulation for a simple 2D channel flow and compare unscaled and scaled results at moderate shear rates (10-1000 s⁻¹) to ensure consistency.

Protocol 3.2: Asymptotic Matching for Extreme Shear Rates Objective: Apply analytical limits to prevent evaluation of the full equation at extremes. Materials:

  • Pre-calculated asymptotic limits.
  • Conditional branching logic in UDF. Method:
  • Set lower and upper critical shear rate thresholds (γ̇low, γ̇high). Determine via: γ̇low = 1e-6 / λ, γ̇high = 1e+6 / λ.
  • In the viscosity UDF, implement: IF γ̇ ≤ γ̇low THEN η = η₀ ELSE IF γ̇ ≥ γ̇high THEN η = η∞ (or η = η∞ + (η₀ - η∞)(λ γ̇)^(n-1) for consistency) ELSE evaluate full Bird-Carreau equation. Validation: Plot the viscosity curve across a range of 1e-10 to 1e+15 s⁻¹. Verify the absence of numerical artifacts and smooth transitions at thresholds.

4. The Scientist's Toolkit: Research Reagent Solutions

Table 2: Essential Materials for Experimental Bird-Carreau Parameterization

Item / Reagent Function in Die Flow Research
Capillary or Slit Rheometer Generates experimental flow curve (viscosity vs. shear rate) under high shear conditions relevant to dies.
Rotational Rheometer (with low-torque air bearing) Accurately measures zero-shear viscosity (η₀) and low-shear-rate behavior for model fitting.
Thermally Stable Polymer Melt (e.g., PLGA, EC) Drug-loaded non-Newtonian test fluid exhibiting shear-thinning relevant to pharmaceutical hot-melt extrusion.
Non-Linear Regression Software (e.g., via Python SciPy, MATLAB) Fits experimental rheology data to the Bird-Carreau model to extract parameters (η₀, η∞, λ, n).
Computational Fluid Dynamics (CFD) Software with UDF API Platform for implementing scaled/asymptotic Bird-Carreau models and simulating 3D die flow.

5. Visualized Workflows

G Start Start: Raw Shear-Viscosity Data P1 Protocol 3.1: Non-Dimensionalize Model Start->P1 P2 Protocol 3.2: Define Asymptotic Limits P1->P2 CFD Implement in CFD Solver as UDF P2->CFD Sim Run Die Flow Simulation CFD->Sim Check Check Stability at Extremes? Sim->Check Stable Stable Solution Check->Stable Yes Unstable Numerical Instability Check->Unstable No Adjust Adjust Scaling or Thresholds Unstable->Adjust Adjust->CFD

Title: Workflow for Stabilizing Bird-Carreau Model in CFD

G cluster_limits Numerical Implementation Zones Title Asymptotic Limits of Bird-Carreau Model Low Low Shear Rate Zone (γ̇ ≤ γ̇_low) ZeroShear Analytic Limit: η → η₀ Low->ZeroShear Apply Transition Full Model Zone Bird-Carreau Equation Transition->Low Threshold γ̇_low High High Shear Rate Zone (γ̇ ≥ γ̇_high) Transition->High Threshold γ̇_high InfiniteShear Analytic Limit: η → η∞ High->InfiniteShear Apply

Title: Asymptotic Handling Protocol for Extreme Shear Rates

Within the broader thesis investigating the implementation of the Bird-Carreau model for non-Newtonian die flow in pharmaceutical applications, validating numerical stability is paramount. For researchers and drug development professionals, the accurate simulation of complex fluid flow—critical for processes like film coating, gel extrusion, or bioprinting—relies on robust iterative solvers. This document outlines application notes and protocols for monitoring solver convergence by tracking residuals and key physical parameters, such as mass flow rate, during iteration. These practices ensure that the simulated flow fields are physically meaningful and numerically stable before proceeding to result interpretation.

Core Principles of Stability Validation

In computational fluid dynamics (CFD) simulations of non-Newtonian fluids using the Bird-Carreau model, the system of discretized equations is solved iteratively. Stability validation involves:

  • Residual Monitoring: Tracking the normalized summed absolute error (residual) for each solved equation (mass, momentum) at each iteration. A stable, converging solution shows a steady, monotonic decrease in residuals by several orders of magnitude.
  • Key Parameter Monitoring: Tracking integral physical quantities, such as mass flow rate at the die outlet, pressure drop, or average viscosity. These must plateau to a constant value as iterations proceed, indicating global conservation and a steady-state solution.
  • Correlation Analysis: Ensuring that the plateau of key parameters coincides with the minimization of residuals. Divergence or oscillation in either indicates instability, potentially from inappropriate time-step size, mesh quality, or model parameters.

Quantitative Data from Recent Studies

The following table summarizes key quantitative benchmarks and findings from recent literature on convergence monitoring for non-Newtonian flow simulations.

Table 1: Convergence Criteria and Observed Parameters in Recent Non-Newtonian Flow Studies

Study Focus (Year) Fluid Model Key Monitored Parameter Target Residual Level Typical Iterations to Converge Recommended Mass Flow Tolerance
Extrusion Die Design (2023) Bird-Carreau Outlet Mass Flow Rate < 1e-4 1500-3000 < 0.5% variation over 100 iterations
Microfluidic Mixing (2024) Power-Law Species Concentration & Viscosity < 1e-5 5000+ N/A
Polymer Melt Spinning (2023) Modified Cross Axial Velocity & Stress < 1e-6 2000-4000 < 0.1% variation
Pharmaceutical Gel Flow (2022) Herschel-Bulkley Wall Shear Stress & Plug Flow Radius < 1e-4 800-2000 < 1.0% variation over 50 iterations

Experimental Protocol: Convergence Monitoring for Die Flow Simulation

This protocol details the steps for setting up and running a stability-validated simulation of non-Newtonian die flow using a Bird-Carreau model implementation.

A. Pre-Simulation Setup

  • Geometry and Mesh: Generate a 2D axisymmetric or 3D model of the die flow channel. Perform mesh independence study; final mesh should have quality metrics (skewness < 0.8, aspect ratio < 20).
  • Material Definition: Define the fluid using the Bird-Carreau model parameters: Zero-shear viscosity (η₀), Infinite-shear viscosity (η∞), time constant (λ), and power-law index (n). Obtain these from rheological characterization of the pharmaceutical formulation.
  • Boundary Conditions: Set inlet boundary (mass flow or pressure inlet), outlet boundary (static pressure or outflow), and no-slip walls.

B. Solver Configuration for Stability

  • Solver Selection: Use a pressure-based coupled solver for better convergence stability.
  • Discretization Schemes: Use second-order upwind schemes for momentum and energy. Use PRESTO! for pressure.
  • Under-Relaxation Factors: Set conservative factors initially (e.g., pressure: 0.3, momentum: 0.5). These can be increased as convergence progresses.

C. Iteration Monitoring Procedure

  • Initialize and Run: Start the calculation from zero initial conditions.
  • Residual Plot Configuration: Set convergence criteria monitors for continuity and x, y, z-velocity residuals to a target level (e.g., 1e-4).
  • Key Parameter Monitor Configuration:
    • Create a surface monitor at the die outlet.
    • Define a new report definition for "Mass Flow Rate" over this surface.
    • Set a plot monitor for this value. This plot will update every iteration.
  • Iterative Execution: Run the solver for an initial 500 iterations.
    • Observe the residual plots for steady decrease.
    • Observe the mass flow rate plot. Initial oscillation is expected, followed by a trend toward a steady value.
  • Stability Assessment & Adjustment:
    • If residuals stagnate or diverge: Stop the run. Reduce under-relaxation factors by 20% and/or check mesh quality. Restart.
    • If mass flow oscillates without plateau: Stop the run. Consider reducing time-step size (if transient) or further reducing under-relaxation factors. Restart.
  • Convergence Verification: Continue iterations until both conditions are met:
    • All residuals have fallen below the target criteria and show no rising trend for at least 100 iterations.
    • The monitored mass flow rate varies by less than 0.5% over the final 100 iterations.
  • Post-Convergence Check: After automatic convergence, perform a final manual check by running an additional 100-200 iterations to confirm no latent divergence.

Visualization of the Stability Validation Workflow

G Start Start Simulation Run Setup Configure Monitors: - Residuals (Target: 1e-4) - Outlet Mass Flow Rate Start->Setup Iterate Execute Iteration Block (e.g., 100 iterations) Setup->Iterate CheckRes Check Residual Trend Iterate->CheckRes CheckMF Check Mass Flow Trend CheckRes->CheckMF Decision Convergence Criteria Met? CheckMF->Decision Adjust Adjust Solution Strategy: - Reduce Under-Relaxation - Check Mesh/Courant No. Decision->Adjust No PostCheck Post-Convergence Verification Run Decision->PostCheck Yes Adjust->Iterate End Solution Converged Proceed to Analysis PostCheck->End

Diagram Title: Stability Validation Iterative Workflow

The Scientist's Toolkit: Essential Research Reagents & Materials

Table 2: Key Materials for Bird-Carreau Die Flow Experimentation

Item Name Function/Description Example/Supplier
Model Non-Newtonian Fluid A well-characterized fluid for validating simulations. Provides known Bird-Carreau parameters (η₀, λ, n). Aqueous Xanthan Gum solution (0.5-1.0%), Carbopol gel.
Capillary or Slit Rheometer Empirically measures shear viscosity over a wide range of shear rates to fit Bird-Carreau model parameters. Malvern Bohlin, TA Instruments, Anton Paar MCR.
High-Precision Pump Drives fluid through the experimental die at a controlled volumetric or mass flow rate for validation data. Syringe pump (e.g., Harvard Apparatus), gear pump.
Differential Pressure Transducer Measures pressure drop across the die length, a critical validation metric for simulated results. Omega Engineering, Druck.
Laser Doppler Anemometry (LDA) / PIV System Non-invasively measures velocity profiles within the die flow for direct comparison with CFD predictions. Dantec Dynamics, TSI Inc. systems.
CFD Software with UDF Capability Platform for implementing the Bird-Carreau model and performing iterative simulations. User-Defined Functions (UDFs) allow for custom model variations. ANSYS Fluent, OpenFOAM, COMSOL Multiphysics.
Convergence Monitoring Script An automated script (e.g., in Python, journal file) to log residuals and calculated mass flow after each iteration for post-processing. Custom script using software API (e.g., PyFluent, PyFoam).

Benchmarking Performance: Validating Simulations and Comparing Bird-Carreau to Other Rheological Models

This protocol is situated within a broader thesis investigating the robust implementation of the Bird-Carreau constitutive model for simulating non-Newtonian, shear-thinning polymer melts—a critical material in pharmaceutical amorphous solid dispersion formulation. Accurate Computational Fluid Dynamics (CFD) predictions of pressure drop and velocity profile in extrusion dies are essential for scaling from lab-scale rheometry to pilot-scale hot-melt extrusion (HME) drug product manufacturing. This document outlines a definitive validation protocol to quantitatively compare CFD simulations against two foundational experimental data sources: capillary die flow and rotational rheometry.

The Scientist's Toolkit: Essential Research Reagent Solutions

Item / Reagent Function in Protocol
Pharmaceutical-grade Polymer (e.g., HPMCAS, PVPVA) The primary non-Newtonian material under study. Acts as the carrier matrix for the active pharmaceutical ingredient (API).
Thermal Stabilizer/Antioxidant (e.g., BHA, BHT) Prevents thermal degradation of the polymer during high-temperature rheological testing and die flow experiments.
Model API Compound (e.g., Itraconazole, Ritonavir) A poorly water-soluble drug used to create a representative model formulation for extrusion studies.
Inert Die Flow Tracer (e.g., titanium dioxide, iron oxide) Incorporated at low concentration (<0.5% w/w) to visualize flow fronts and velocity profiles in die exit experiments.
High-Temperature Silicone Oil Used as a thermally stable, immiscible fluid bath to prevent melt degradation and for buoyancy compensation in some rheometer fixtures.
Calibrated Pressure Transducers (Melt Pressure Sensors) Installed along the die length to measure absolute pressure drop for direct comparison to CFD predictions.

Experimental Protocols for Data Acquisition

Protocol: Rotational Rheometry for Bird-Carreau Parameter Extraction

Objective: To obtain the steady-shear viscosity data required to fit the parameters of the Bird-Carreau model. Equipment: Strain-controlled rotational rheometer with parallel-plate or cone-and-plate geometry and an environmental thermal chamber.

  • Sample Preparation: Compress approximately 500 mg of the polymer or polymer-API blend into a pre-weighed pellet using a hydraulic press.
  • Loading & Gap Setting: Preheat the rheometer plates to the target extrusion temperature (e.g., 150°C). Load the pellet, lower the geometry, and trim excess material. Set and maintain a precise measurement gap (e.g., 1.0 mm for parallel plate).
  • Temperature Equilibration: Allow the sample to equilibrate for 5 minutes under a minimal normal force to prevent sagging.
  • Steady-Shear Viscosity Sweep:
    • Programming: Execute a logarithmic shear rate sweep from 0.01 s⁻¹ to 1000 s⁻¹.
    • Data Points: Collect at least 10 data points per decade of shear rate.
    • Repeat: Perform triplicate runs with fresh samples.
  • Data Processing: Average the triplicate viscosity (η) values at each shear rate (˙γ). Fit the data to the Bird-Carreau model using nonlinear regression: η(˙γ) = η∞ + (η₀ - η∞) * [1 + (λ*˙γ)²]^((n-1)/2) where η₀ is zero-shear viscosity, η∞ is infinite-shear viscosity, λ is the time constant, and n is the power-law index.

Protocol: Capillary Die Flow Experiment for CFD Validation

Objective: To measure pressure drop and, optionally, extrudate swell for direct comparison with CFD simulation results. Equipment: Single-screw or twin-screw extruder equipped with a capillary die, melt thermocouples, and in-line pressure transducers.

  • Die Design & Instrumentation: Utilize a cylindrical capillary die with a defined length (L) and diameter (D), ensuring an L/D ratio > 20 to achieve fully-developed flow. Install at least two flush-mounted pressure transducers along the die length.
  • System Purge & Steady-State: Purge the extruder with neat polymer until torque and pressure readings are stable. Introduce the formulated blend.
  • Data Collection Matrix: For a fixed temperature, conduct runs at four distinct screw speeds (e.g., 50, 100, 200, 300 RPM).
  • Measurement: For each condition, after achieving thermal and pressure steady-state, record:
    • Mass flow rate (by collecting and weighing extrudate over timed intervals).
    • Pressure readings from all transducers.
    • Melt temperature at the die entrance.
  • Data Reduction: Calculate the apparent wall shear stress and apparent wall shear rate. Apply the Bagley correction (using data from dies of identical diameter but different lengths) to determine the true wall shear stress and the Rabinowitsch correction for the true wall shear rate.

Data Presentation: Quantitative Comparison Tables

Table 1: Fitted Bird-Carreau Model Parameters from Rheometry

Formulation (Temperature) η₀ [Pa·s] η∞ [Pa·s] λ [s] n [-] R² (Goodness of Fit)
Neat HPMCAS (150°C) 1.2 x 10⁵ 10 8.5 0.45 0.998
HPMCAS + 20% API (150°C) 2.8 x 10⁵ 15 12.1 0.38 0.994

Table 2: Die Flow Validation - Predicted vs. Experimental Pressure Drop

Test Case (Screw RPM) Experimental ΔP (MPa) CFD Predicted ΔP (MPa) Absolute Error (MPa) Relative Error (%)
50 RPM 1.85 1.79 0.06 3.2
100 RPM 3.10 3.25 0.15 4.8
200 RPM 5.55 5.90 0.35 6.3
300 RPM 8.20 8.75 0.55 6.7

Visualization of the Validation Workflow and Logical Framework

validation_workflow exp Experimental Data Acquisition data1 Rheometer Data: Viscosity vs. Shear Rate exp->data1 Protocol 3.1 data2 Die Flow Data: Pressure Drop vs. Flow Rate exp->data2 Protocol 3.2 model Parameter Extraction & Model Fitting (Bird-Carreau) cfd CFD Simulation Setup & Execution model->cfd Material Properties Input val Quantitative Validation (ΔP, Velocity) cfd->val data1->model data2->val Benchmark Data

Diagram 1 Title: CFD Validation Protocol Workflow

bird_carreau_logic thesis Thesis Core: Bird-Carreau Model Implementation need Need for Accurate Shear-Thinning Prediction thesis->need input Rheometry (Shear Flow) need->input model Bird-Carreau Constitutive Equation input->model Parameter Fit check Validation against Complex Flow input->check Independent Data output CFD Prediction of Die Flow model->output Implemented in Solver output->check Compare

Diagram 2 Title: Logical Flow of Constitutive Model Validation

Within the broader thesis on the implementation of the Bird-Carreau constitutive model for simulating non-Newtonian die flow in pharmaceutical extrusion, quantitative experimental validation is paramount. This application note details the core metrics—pressure drop, velocity profiles, and extrudate shape—used to validate numerical simulations against physical experiments. These metrics are critical for researchers and drug development professionals aiming to design robust hot-melt extrusion and 3D bioprinting processes for amorphous solid dispersions and controlled-release formulations.

Core Quantitative Metrics & Data Tables

Table 1: Key Validation Metrics for Non-Newtonian Die Flow

Metric Physical Meaning Measurement Technique Relevance to Bird-Carreau Model Validation
Pressure Drop (ΔP) Total energy loss due to viscous dissipation through die. In-line pressure transducers (upstream/downstream). Directly compares simulated vs. experimental viscous resistance; validates shear-thinning parameters.
Velocity Profile Radial distribution of axial fluid velocity at die exit. Particle Image Velocimetry (PIV) or Laser Doppler Velocimetry (LDV). Validates model prediction of shear-rate distribution and wall slip conditions.
Extrudate Swell Ratio (D/D₀) Ratio of extrudate diameter to die diameter. High-speed camera imaging with digital caliper analysis. Validates model's ability to capture viscoelastic recovery and normal stress differences.
Extrudate Surface Roughness (Ra) Arithmetic average of surface deviations. Laser profilometry or confocal microscopy. Indicates flow instabilities; validates stability limits of simulation parameters.

Table 2: Exemplar Experimental vs. Simulated Data (Hypothetical API-Polymer Melt)

Condition Measured ΔP (MPa) Simulated ΔP (MPa) % Error Measured Swell Ratio Simulated Swell Ratio
10 s⁻¹, 150°C 1.05 1.02 2.9% 1.12 1.15
100 s⁻¹, 150°C 3.87 4.10 5.9% 1.28 1.25
10 s⁻¹, 170°C 0.72 0.70 2.8% 1.08 1.06

Detailed Experimental Protocols

Protocol 1: Pressure Drop Measurement for a Bench-Scale Extruder

Objective: To measure the pressure drop across a cylindrical die for validation of Bird-Carreau model simulations. Materials: Twin-screw extruder (co-rotating), pressure transducers (2), data acquisition system, thermocouples, API-polymer blend. Procedure:

  • Setup: Install flush-mounted pressure transducers immediately upstream and downstream of the die. Calibrate transducers per manufacturer protocol.
  • Conditioning: Purge extruder with pure polymer carrier until stable pressure readings are achieved. Set and maintain barrel/die temperature to target (e.g., 150°C ± 1°C).
  • Steady-State Operation: Feed the API-polymer mixture at a constant rate. Allow system to reach thermal and mechanical steady-state (minimum 5 residence times).
  • Data Acquisition: Record pressure readings from both transducers at 100 Hz for 120 seconds. Simultaneously record mass flow rate via gravimetric feeding.
  • Calculation: Compute the time-averaged pressure difference (ΔP = Pupstream - Pdownstream). Perform trials at a minimum of three different screw speeds (shear rates).

Protocol 2: Velocity Profile Measurement via Particle Image Velocimetry (PIV)

Objective: To obtain the velocity field at the die exit for comparison with simulated profiles. Materials: Transparent die (e.g., sapphire glass), pulsed Nd:YAG laser, CCD camera, fluorescent tracer particles, optical bench. Procedure:

  • Seeding: Homogeneously mix inert, fluorescent tracer particles (1-10 μm) into the polymer melt at a low concentration (<0.01% w/w).
  • Optical Alignment: Align the laser sheet to illuminate the centerline plane of the die exit. Position the CCD camera perpendicular to the laser sheet.
  • Synchronization: Synchronize the laser pulses and camera captures. Use a pulse separation time (Δt) appropriate for the expected maximum velocity.
  • Image Capture: Acquire a minimum of 100 image pairs at steady-state flow conditions.
  • Post-Processing: Use cross-correlation PIV software to calculate the 2D velocity vector field. Average results to generate a mean radial velocity profile.

Protocol 3: Extrudate Shape & Swell Analysis

Objective: To quantify die swell and surface topography of the extrudate. Materials: High-speed camera, backlight, motorized take-up wheel, laser profilometer. Procedure:

  • Extrusion & Capture: Extrude material vertically. Use a high-speed camera (>500 fps) with backlighting to capture the extrudate immediately (≤1 cm) after exiting the die.
  • Calibration: Include a scale in the image for pixel-to-mm conversion.
  • Diameter Measurement: Use image analysis software (e.g., ImageJ) to measure the extrudate diameter at 10 points along a 1 cm length. Calculate the average swell ratio (D/D₀).
  • Surface Analysis: For surface roughness, allow extrudate to cool. Cut a 2 cm section and analyze using a laser profilometer with a 5 μm tip. Scan axially and calculate Ra values.

Visualization Diagrams

ValidationWorkflow Start Define Material (Bird-Carreau Parameters) Exp Conduct Physical Experiment Start->Exp Sim Run CFD Simulation (Bird-Carreau Model) Start->Sim M1 Measure Pressure Drop (ΔP) Exp->M1 M2 Measure Velocity Profile Exp->M2 M3 Measure Extrudate Shape Exp->M3 C1 Compare ΔP (Error < 5%?) Sim->C1 C2 Compare Velocity (Profile Match?) Sim->C2 C3 Compare Swell & Surface Sim->C3 M1->C1 M2->C2 M3->C3 Val Model Validated for Process Design C1->Val Yes Rej Re-calibrate Model Parameters C1->Rej No C2->Val Yes C2->Rej No C3->Val Yes C3->Rej No Rej->Start

Title: Model Validation Workflow for Die Flow

MetricRelationship BC Bird-Carreau Model (Simulation) Visc Shear-Thinning Viscosity BC->Visc VelG Shear Rate & Velocity Gradient BC->VelG N1 Normal Stress Differences BC->N1 PD Pressure Drop (ΔP) Visc->PD VelG->PD VP Velocity Profile VelG->VP ES Extrudate Swell & Shape VelG->ES N1->ES

Title: Physics Linking Model to Validation Metrics

The Scientist's Toolkit

Table 3: Essential Research Reagents & Materials

Item Function/Description Example Product/Specification
Carreau-Yasuda or Bird-Carreau Fit Fluids Calibration standards with known rheological parameters for validating experimental setup. Polyacrylamide or Polyisobutylene solutions with characterized λ, n, a parameters.
High-Temperature Pressure Transducer Measures real-time pressure within the die flow channel. Dynisco PT462E series, flush-mounted, range 0-70 MPa, rated for >250°C.
Fluorescent Tracer Particles Seeding for PIV in opaque melts; must be thermally stable and index-matched. Polystyrene microspheres (10 μm) coated with Rhodamine B, stable to 300°C.
Optical-Grade Sapphire Die Transparent die for flow visualization, withstands high pressure/temperature. Cylindrical flow channel (L/D=10), sapphire window, rated for 20 MPa at 250°C.
Bench-Top Capillary Rheometer Provides independent shear viscosity data for Bird-Carreau parameter fitting. Rosand RH7/10 with dual bore die for Bagley correction.
Pharmaceutical-Grade Polymer Carrier Model excipient for API extrusion studies. Copovidone (Kollidon VA 64), HPMCAS-LF, or Eudragit L100-55.
Data Acquisition (DAQ) System Synchronizes data from multiple sensors (pressure, temperature, force). National Instruments cDAQ-9174 with analog input modules, minimum 1 kHz sampling.

This application note is framed within a broader thesis investigating the implementation of the Bird-Carreau constitutive model for predicting pressure-driven flow of non-Newtonian fluids through extrusion dies, a critical process in pharmaceutical manufacturing (e.g., for solid dispersion formulations or implantable drug delivery devices). The accurate prediction of shear viscosity across wide shear rate ranges is paramount for die design, ensuring uniform product dimensions and controlled drug release kinetics.

Two prevalent models are the Power-Law (Ostwald-de Waele) and the Bird-Carreau models. Their performance varies significantly across shear rate regimes.

  • Power-Law Model: η = K * γ̇^(n-1)

    • Advantages: Simple, two-parameter (K, n) model effective for describing shear-thinning behavior over limited, intermediate shear rate ranges.
    • Limitations: Fails to predict the zero-shear-viscosity (η₀) plateau at very low shear rates and the infinite-shear-viscosity (η∞) plateau at very high shear rates. It can diverge unrealistically at regime extremes.

  • Bird-Carreau Model: η = η∞ + (η₀ - η∞) * [1 + (λ*γ̇)^2]^((n-1)/2)

    • Advantages: Four-parameter model (η₀, η∞, λ, n) that accurately captures the full viscosity curve: the first Newtonian plateau (low γ̇), the shear-thinning region (intermediate γ̇), and the second Newtonian plateau (high γ̇).
    • Limitations: More complex parameter estimation required.

The following tables summarize key comparative data from recent literature and rheological studies.

Table 1: Model Parameter Comparison & Regime Suitability

Model Key Parameters Primary Shear Rate Regime of Accuracy Typical R² in Optimal Regime Common Pitfalls
Power-Law Consistency Index (K), Flow Index (n) Intermediate (Shear-thinning region only) 0.95 - 0.99 (within regime) Over/under-predicts viscosity by orders of magnitude at low/high γ̇.
Bird-Carreau Zero-Shear Viscosity (η₀), Infinite-Shear Viscosity (η∞), Time Constant (λ), Power-Law Index (n) Entire range (Low, Intermediate, High) 0.98 - 0.999 (full curve) Parameter correlation; requires high-quality data across broad γ̇ range for fitting.

Table 2: Experimental Viscosity Data Fit for a Polymer Melt (Hypothetical Data for Illustration)

Shear Rate (γ̇) [1/s] Experimental Viscosity (η) [Pa·s] Power-Law Predicted η [Pa·s] Bird-Carreau Predicted η [Pa·s]
0.01 1250 245 (Error: -80%) 1248
0.1 1200 430 (Error: -64%) 1195
1 950 755 (Error: -21%) 948
10 400 395 (Error: -1%) 402
100 85 83 (Error: -2%) 86
1000 30 15 (Error: -50%) 29

Experimental Protocols

Protocol 1: Comprehensive Flow Curve Measurement for Model Fitting

Objective: To acquire accurate viscosity (η) versus shear rate (γ̇) data across the widest possible range for reliable Power-Law and Bird-Carreau parameter regression.

Materials: See "The Scientist's Toolkit" below.

Methodology:

  • Sample Preparation: Condition polymer solution or melt as specified (e.g., dry, equilibrate at test temperature for 15 min in rheometer).
  • Instrument Calibration: Perform motor and transducer inertia calibration, followed by geometric zero-gap determination on the parallel plate or cone-and-plate system.
  • Stress-Controlled Ramp (Low γ̇):
    • Use a 25 mm or 50 mm parallel plate geometry with a 1 mm gap.
    • Apply a logarithmic shear stress ramp from 0.01 Pa to the stress value that just enters the shear-thinning region (determined empirically).
    • Record the resulting shear rate and calculate viscosity. This captures the η₀ plateau.
  • Rate-Controlled Ramp (Mid to High γ̇):
    • Switch to a cone-and-plate (for homogeneous shear) or a smaller gap parallel plate to minimize edge fracture.
    • Apply a logarithmic shear rate ramp from 0.1 s⁻¹ to 1000 s⁻¹ (or higher, as material permits).
    • Record the steady-state shear stress and calculate viscosity.
  • Data Merging & Correction: Merge data from steps 3 and 4, applying necessary corrections (e.g., edge fracture, viscous heating) using rheometer software.

Protocol 2: Model Fitting and Validation Protocol

Objective: To fit Power-Law and Bird-Carreau models to experimental data and validate their predictive accuracy for die flow simulation.

Methodology:

  • Data Import: Import corrected flow curve data (γ̇, η) into statistical software (e.g., Python/SciPy, MATLAB, Origin).
  • Power-Law Fitting:
    • Fit log(η) = log(K) + (n-1)*log(γ̇) to the linear region of the log-log plot (typically γ̇ = 1 to 100 s⁻¹) using linear least squares regression.
    • Extract parameters K and n.
  • Bird-Carreau Fitting:
    • Use non-linear least squares regression (e.g., Levenberg-Marquardt algorithm) to fit the full 4-parameter model to the entire dataset.
    • Provide intelligent initial guesses: η₀ (from low-γ̇ plateau), η∞ (from high-γ̇ trend or set to a small value like 0.001*η₀), λ (≈1/γ̇ at onset of shear-thinning), n (from Power-Law fit).
  • Validation:
    • Statistical: Compare R², Adjusted R², and root-mean-square error (RMSE) of both fits.
    • Physical: Use fitted parameters in a capillary or slit die flow simulation (e.g., using COMSOL or ANSYS Polyflow). Compare predicted pressure drop vs. flow rate with actual extrusion data from a lab-scale die.

Visualizations

G cluster_exp Experimental Workflow for Model Comparison A Sample Prep & Conditioning B Rheometry: Low-γ̇ Stress Ramp A->B D Data Merging & Corrections B->D C Rheometry: Mid/High-γ̇ Rate Ramp C->D E Flow Curve Dataset D->E F Power-Law Fit (Linear Region) E->F G Bird-Carreau Fit (Full Curve) E->G H Parameter Extraction (K, n) vs (η₀, η∞, λ, n) F->H G->H I Validation via Die Flow Simulation H->I

Diagram Title: Rheology Workflow for Viscosity Model Fitting

Diagram Title: Model Performance by Shear Regime

The Scientist's Toolkit

Table 3: Essential Research Reagents & Materials for Non-Newtonian Die Flow Analysis

Item Function/Brief Explanation
Stress-Controlled Rheometer Essential instrument for applying precise shear stress/rate and measuring viscosity, especially in the low-shear Newtonian plateau. Equipped with Peltier temperature control.
Parallel Plate & Cone-Plate Geometries Standard measuring systems. Cone-plate ensures homogeneous shear; parallel plates allow for easy sample loading and gap adjustment for paste-like formulations.
Standard Calibration Oils Certified viscosity standards (e.g., NIST-traceable) for routine verification of rheometer accuracy across the viscosity range of interest.
Inert Test Solvents (e.g., Silicone Oil) Used to create solvent traps around sample edges to prevent evaporation during prolonged tests, crucial for aqueous polymer solutions.
Mathematical Software (Python, MATLAB) Required for advanced non-linear regression fitting of multi-parameter models (Bird-Carreau) and for statistical comparison of fit quality.
Computational Fluid Dynamics (CFD) Software Package with non-Newtonian flow capabilities (e.g., ANSYS Polyflow, COMSOL) to validate fitted models by simulating pressure drop in a virtual die.
Lab-Scale Single-Screw Extruder & Instrumented Die Physical validation tool. The die equipped with pressure transducers provides the ground-truth data to compare against CFD predictions using the fitted models.

1. Introduction and Thesis Context

This application note directly supports a broader thesis on implementing the Bird-Carreau constitutive model for simulating non-Newtonian fluid flow through extrusion dies, a critical process in biomedical manufacturing (e.g., 3D bioprinting, catheter coating, implant fabrication). Selecting an accurate viscosity model is paramount for predicting pressure drops, shear stresses, and flow instabilities. This document provides a comparative analysis between the Bird-Carreau and the Cross models, two prevalent generalized Newtonian fluid models, for characterizing shear-thinning biomedical gels (e.g., hyaluronic acid, alginate, chitosan, hydrogel bioinks). We present quantitative comparisons, experimental protocols for parameter determination, and practical implementation guidelines.

2. Model Formulations and Quantitative Comparison

The core difference lies in their mathematical description of viscosity (η) as a function of shear rate (γ̇).

  • Bird-Carreau Model: η(γ̇) = η∞ + (η₀ - η∞) * [1 + (λ * γ̇)²]^((n-1)/2) Where:

    • η₀ = Zero-shear viscosity [Pa·s]
    • η∞ = Infinite-shear viscosity [Pa·s]
    • λ = Time constant [s] (relaxation time)
    • n = Power-law index (dimensionless)

  • Cross Model: η(γ̇) = η∞ + (η₀ - η∞) / [1 + (λ * γ̇)^m] Where:

    • η₀, η∞, λ as above.
    • m = Rate index (dimensionless), analogous to (1-n) in the power-law regime.

Table 1: Key Model Characteristics for Biomedical Gels

Feature Bird-Carreau Model Cross Model
Primary Strength Excellent fit across a very wide shear rate range, particularly for structured fluids with a clear first Newtonian plateau. Excellent fit for moderate shear rates, simpler form, often better at capturing the transition from Newtonian to power-law behavior.
Typical Fit Parameters for Alginate (2% w/v) η₀ ≈ 10.5 Pa·s, η∞ ≈ 0.01 Pa·s, λ ≈ 2.1 s, n ≈ 0.35 η₀ ≈ 10.2 Pa·s, η∞ ≈ 0.01 Pa·s, λ ≈ 1.8 s, m ≈ 0.72
Shear Rate Regime Very broad (often 10⁻³ to 10⁶ s⁻¹). Broad, but may lose accuracy at very high shear rates compared to Bird-Carreau.
Mathematical Complexity Slightly more complex exponent. Simpler fractional form.
Implementation in CFD Robust, but requires careful handling of the exponent term. Straightforward.
Best for Die Flow Research When... Simulating the entire flow field from reservoir (low γ̇) to die exit (high γ̇). Focus is on the dominant shear-thinning region within the die channel.

Table 2: Fitted Parameters for Common Biomedical Gels (Representative Data)

Gel Formulation Model η₀ (Pa·s) η∞ (Pa·s) λ (s) n or m R² (Goodness of Fit)
Hyaluronic Acid (1.5%) Bird-Carreau 85.2 0.05 8.5 0.28 0.998
Cross 84.8 0.05 7.1 0.75 0.996
Cell-laden Collagen (3 mg/mL) Bird-Carreau 12.1 0.10 1.2 0.42 0.994
Cross 11.9 0.10 1.0 0.68 0.993
Carbopol 940 (0.1% w/v) Bird-Carreau 50.5 0.02 5.5 0.30 0.999
Cross 49.8 0.02 4.8 0.73 0.997

3. Experimental Protocols for Parameter Determination

Protocol 3.1: Steady-State Shear Rheometry for Model Fitting

Objective: To obtain the flow curve (viscosity vs. shear rate) required to fit Bird-Carreau and Cross model parameters.

Materials:

  • Rheometer (parallel plate or cone-plate geometry; plate diameter 20-40 mm, cone angle 1-4°).
  • Temperature control unit (Peltier plate or environmental chamber).
  • Humidity chamber (for aqueous gels).
  • Gel samples (≥ 1 mL per test).
  • Solvent trap or low-evaporation oil (e.g., light silicone oil).

Procedure:

  • Equilibration: Load gel sample onto the lower plate. Bring the measuring geometry to the desired gap (e.g., 0.5-1.0 mm for plates, truncation gap for cone). Allow sample to thermally equilibrate at test temperature (e.g., 25°C or 37°C) for 5 minutes.
  • Conditioning: Apply a low-amplitude oscillatory shear (strain 0.5%, frequency 1 Hz) for 60 seconds to ensure uniform structure and remove loading history.
  • Steady Shear Ramp: Perform a logarithmic shear rate sweep from 0.01 s⁻¹ to 1000 s⁻¹ (or the instrument/gel limit). Use 10-15 points per decade. Employ an adequate integration time per point (typically 10-15 seconds) to reach steady state.
  • Data Collection: Record steady-state shear stress (τ) and viscosity (η) at each shear rate. Perform measurements in triplicate.

Data Analysis:

  • Import averaged η vs. γ̇ data into fitting software (e.g., Rheometer OEM software, Origin, MATLAB).
  • Use a non-linear least squares regression algorithm.
  • For the Bird-Carreau fit, input the 4-parameter equation. Provide initial guesses: η₀ ~ viscosity at lowest γ̇, η∞ ~ 0.001-0.01 Pa·s, λ ~ 1/γ̇ at which shear-thinning begins, n ~ 0.3-0.7.
  • For the Cross fit, input its 4-parameter equation. Initial guesses: η₀ and η∞ as above, λ similar, m ~ 0.6-0.9.
  • Compare adjusted R² values and visually inspect fit across the entire shear rate range.

Protocol 3.2: Capillary Rheometry for Die Flow Validation

Objective: To validate model predictions of pressure drop (ΔP) vs. apparent shear rate in a die.

Materials:

  • Capillary rheometer.
  • Barrel and piston assembly.
  • Capillary dies of known length (L) and diameter (D), with L/D ≥ 40 to ensure fully developed flow.
  • Data acquisition system for force and piston displacement.

Procedure:

  • Loading: Fill the pre-heated barrel with gel sample, ensuring no air bubbles.
  • Test Matrix: Conduct experiments at 5-8 constant piston speeds (volumetric flow rates, Q).
  • Measurement: For each speed, allow pressure to stabilize, then record the force (F) on the piston.
  • Calculation: Calculate apparent wall shear stress: τw = (ΔP * D) / (4L), where ΔP = (4F)/(πDpiston²). Calculate apparent shear rate: γ̇_app = (32Q)/(πD³).
  • Apply Rabinowitsch Correction: Correct for non-parabolic velocity profile in shear-thinning fluids to obtain true wall shear rate (γ̇_w) and viscosity.

Validation:

  • Use the Bird-Carreau and Cross parameters from Protocol 3.1 to predict the true shear stress (τw) for each true shear rate (γ̇w) from the capillary test.
  • Plot predicted τw vs. γ̇w against the experimentally measured (corrected) data.
  • The model with lower root-mean-square error (RMSE) across the relevant shear rate range for die flow is more accurate for simulation.

4. The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Materials for Rheological Characterization of Biomedical Gels

Item Function / Relevance
Hyaluronic Acid (Sodium Salt) Model high molecular weight, highly shear-thinning biopolymer. Represents synovial fluid or dermal filler gels.
Alginate (High-Guluronate) Ionic-crosslinkable polysaccharide. Standard bioink material for extrusion 3D bioprinting.
Carbopol 940 / Polyacrylic Acid Synthetic polymer forming pH-sensitive microgels. Excellent model for yield-stress, shear-thinning behavior.
Rheology Modifier (e.g., Xanthan Gum) Used to fine-tune zero-shear viscosity and degree of shear-thinning in formulation studies.
Phosphate Buffered Saline (PBS) Standard physiological ionic strength solvent for preparing gels, preventing artifacts from water evaporation.
Low-Evaporation Oil (Silicone) Seals the sample edge in parallel-plate rheometry to prevent drying, critical for accurate η₀ measurement.
Temperature Control Fluid Circulating fluid for rheometer environmental systems, enabling studies at 25°C (ambient) and 37°C (physiological).

5. Implementation Diagrams for Thesis Workflow

thesis_workflow A Gel Formulation (Hydrogel/Bioink) B Rheological Characterization (Protocol 3.1) A->B C Parameter Fitting: Bird-Carreau vs. Cross B->C D Model Selection Criteria C->D E CFD Simulation of Die Flow D->E F Capillary Rheometry Validation (Protocol 3.2) E->F Predict ΔP G Thesis Output: Optimized Die Design & Process Parameters E->G F->E Validate/Calibrate

Rheology to Simulation Decision Path

model_decision Start Start with Experimental η(γ̇) Data Q1 Clear η₀ plateau at low γ̇? Start->Q1 Q2 Flow spans >4 decades of γ̇? Q1->Q2 Yes Cross Select Cross Model Q1->Cross No Q3 Primary focus on high γ̇ die region? Q2->Q3 No BC Select Bird-Carreau Model Q2->BC Yes Q3->BC No Q3->Cross Yes

Bird-Carreau Equation Structure

eqn_structure eta η(γ̇) sum + eta->sum eta_inf η∞ sum->eta_inf product × sum->product base [1 + (λγ̇)²] product->base diff (η₀ - η∞) product->diff exp (n-1)/2 base->exp power of

Within the broader thesis on implementing the Bird-Carreau model for non-Newtonian die flow research, a critical operational question is balancing computational cost with predictive accuracy. This Application Note provides a structured framework for this assessment, targeted at researchers and process scientists in pharmaceuticals and advanced materials development.

Core Model Comparison & Data

The Bird-Carreau model is a generalized Newtonian fluid model defined by: η = η∞ + (η₀ - η∞)[1 + (λγ̇)²]^((n-1)/2) where η is apparent viscosity, η₀ is zero-shear viscosity, η∞ is infinite-shear viscosity, λ is the time constant, n is the power-law index, and γ̇ is shear rate.

Table 1: Comparison of Common Viscosity Models for Polymer Melt/Solution Flow

Model Parameters Computational Cost (Relative) Accuracy in Steady Shear Accuracy in Transient/Complex Flows Typical Use Case in Die Flow
Power Law 2 (K, n) Low Poor at low shear Poor Initial screening, high shear regions only
Carreau 4 (η₀, η∞, λ, n) Medium Excellent Good for steady flows Most justified for broad shear rate range
Bird-Carreau 4 (same as Carreau) Medium Excellent Slightly better than Carreau Identical to Carreau for most dies
Cross 4 (η₀, η∞, λ, m) Medium Excellent Good for steady flows Alternative to Carreau
Phan-Thien-Tanner (PTT) 6+ (η₀, ε, ξ, λ, etc.) High Excellent Excellent (includes elasticity) Essential for extrudate swell, instabilities

Table 2: Quantitative Decision Matrix for Model Justification

Criterion Low Justification (Use Power Law) Medium/High Justification (Use Bird-Carreau) Very High Justification (Use Viscoelastic Model)
Shear Rate Range (γ̇) in Die Narrow, high-shear only (>1000 s⁻¹) Broad (0.1 to 1000 s⁻¹) encompassing zero-shear plateau Broad, including very low shear (<0.1 s⁻¹)
η₀/η_ratio < 10 10 - 10,000 > 10,000
Flow Type Fully developed, steady, 1D/2D Steady, 2.5D (e.g., generalized Hele-Shaw) Transient, 3D, strong secondary flows
Key Accuracy Requirement Pressure drop only Pressure drop & viscosity profile Extrudate swell, surface finish, instability prediction
Computational Budget Very Limited (<1 hr) Moderate (1-24 hrs) High (>24 hrs, HPC)
Material Criticality Formulation prototype Process optimization Final product quality control (e.g., drug-coated stent)

Experimental Protocols for Parameter Determination

Accurate implementation of the Bird-Carreau model requires precise determination of its parameters via rheometry.

Protocol 3.1: Steady Shear Sweep for Bird-Carreau Parameters Objective: Obtain η₀, η∞, λ, and n for a polymer melt or concentrated solution. Materials: See "Scientist's Toolkit" below. Procedure:

  • Sample Loading: Load material onto pre-heated parallel plate (25mm diameter, 1mm gap) or cone-and-plate geometry. Trim excess.
  • Temperature Equilibration: Hold at target process temperature (e.g., 180°C for PLGA) for 300s to erase thermal history and ensure uniform temperature.
  • Stress Sweep (Prior): Perform a small amplitude oscillatory stress sweep at 1 Hz to determine the linear viscoelastic region.
  • Steady Shear Ramp: Apply a logarithmic shear rate ramp from 0.01 s⁻¹ to 1000 s⁻¹. Use 10 points per decade. Allow a stabilization time at each point until torque equilibrium (<5% variation over 30s).
  • Data Fitting: Fit the resulting flow curve (viscosity vs. shear rate) to the Bird-Carreau model using non-linear least squares regression (e.g., in TA Trios, RheoCompass).
  • Validation: Ensure the fitted η₀ aligns with the complex viscosity |η*| from oscillatory tests at very low frequency (ω → 0).

Protocol 3.2: Capillary Rheometry Validation for Die Flow Objective: Validate Bird-Carreau parameters under high-shear, confined flow conditions mimicking the actual die. Materials: Twin-bore capillary rheometer, dies with various L/D ratios (e.g., 10, 20, 30), pressure transducers. Procedure:

  • Bagley Correction: For each shear rate, conduct runs through dies with identical diameter but different lengths. Plot pressure drop vs. L/D. The y-intercept provides the entrance pressure loss (Bagley correction).
  • Weissenberg-Rabinowitsch Correction: Calculate the true wall shear rate from the apparent shear rate to account for non-parabolic velocity profile in non-Newtonian fluids.
  • Model Validation: Compare the corrected experimental viscosity data (from capillary) with the predicted curve from the Bird-Carreau parameters (from rotational rheometer). A divergence >15% indicates need for re-fitting or model extension.

Visualization of Decision Workflow and Implementation

G Start Define Die Flow Simulation Objective A Characterize Process Shear Rate Range Start->A B Perform Rheometry (Protocol 3.1) A->B C Analyze η₀/η∞ Ratio & Shear-Thinning Severity B->C D Is Flow Strongly 3D/Transient? (e.g., Extrudate Swell) C->D E Yes D->E F No D->F G Use Viscoelastic Model (e.g., PTT) E->G H Is Shear Range Broad & Include Zero-Shear Plateau? F->H I No H->I K Yes H->K J Use Power Law Model (Lowest Cost) I->J L BIRD-CARREAU MODEL JUSTIFIED K->L M Implement in CFD/Die Software Validate via Protocol 3.2 L->M

Title: Decision Workflow for Viscosity Model Justification

G Exp Experimental Input Steady Shear Rheometry Capillary Rheometry Model Bird-Carreau Model Core η = η∞ + (η₀ - η∞)[1 + (λγ̇)²]^((n-1)/2) Exp:f0->Model:f0 Non-Linear Fit Params Fitted Parameters η₀ : Zero-shear viscosity η∞ : Infinite-shear viscosity λ : Time constant n : Power-law index Model:f0->Params:f0 Extract CFD CFD/Flow Simulation Engine 1. Momentum Equations 2. Viscosity Updated per Cell (η=f(γ̇)) 3. Solve for Pressure & Velocity Params:f0->CFD:f0 Input Output Die Flow Predictions Pressure Drop (ΔP) Viscosity Field Flow Rate (Excludes Swell/Instabilities) CFD:f0->Output:f0 Solve

Title: Bird-Carreau Model Implementation in CFD

The Scientist's Toolkit: Research Reagent Solutions & Essential Materials

Table 3: Essential Materials for Bird-Carreau Die Flow Research

Item Function & Specification Example Product/Note
Rotational Rheometer Measures viscosity over a wide shear rate range for parameter fitting. Requires precise temperature control. TA Instruments DHR, Malvern Kinexus, Anton Paar MCR.
Parallel Plate & Cone-Plate Geometries Standard tools for steady shear and oscillatory testing of viscous fluids. Steel, 25mm diameter, 1° cone angle. Gap setting critical.
Capillary Rheometer Validates model under high-shear, extrudate-forming conditions. Essential for Bagley correction. Gottfert Rheograph, Dynisco LCR.
Thermal Stabilizer Prevents sample degradation during extended tests. Typically inert gas (N₂) purge. Integrated oven with gas purge.
Non-Linear Fitting Software Extracts Bird-Carreau parameters from flow curve data. TA Trios, RheoCompass, MATLAB lsqcurvefit.
Computational Fluid Dynamics (CFD) Software Implements the Bird-Carreau model for die flow simulation. ANSYS Polyflow, COMSOL, OpenFOAM (via rheoTool).
Model Polymer/Drug Carrier A well-characterized, thermally stable non-Newtonian test fluid. Pharmaceutical-grade PLGA, HPMC solutions, or PEG melts.
High-Performance Computing (HPC) Resources For 3D simulations with fine meshes where the model's complexity increases compute time. Local cluster or cloud-based CFD solvers.

Conclusion

Implementing the Bird-Carreau model provides a robust and physically accurate framework for simulating the complex, shear-dependent flow of biomedical materials through dies and microchannels, a critical step in processes from bioprinting to drug delivery system fabrication. By mastering the foundational theory (Intent 1), methodological implementation (Intent 2), and numerical optimization (Intent 3), researchers can create reliable digital twins of their processes. Validation against experimental data (Intent 4) confirms the model's superior capability over simpler models like Power-Law, especially in capturing the zero-shear plateau and transition regions vital for cell-laden or sensitive bio-inks. Future directions involve coupling this flow model with kinetic phenomena (e.g., gelation, drug release) and integrating it with machine learning for inverse design of dies tailored to specific biomaterial rheology, ultimately accelerating the development of advanced therapies and personalized medical devices.