Machine Learning-driven Predictive Modeling of Magnetohydrodynamic Double Diffusion of Non-Newtonian Hybrid Ferrofluids with Variable Thermophysical Properties within Corrugated Cylinders

✅ Published in Engineering Applications of Artificial Intelligence 2025 (Elsevier, Q1, Impact Factor: 8.0)
📄 Paper: ScienceDirect · DOI: 10.1016/j.engappai.2024.109455

Abstract

Introduction

Hybrid ferrofluids — colloidal suspensions of two or more magnetic nanoparticle types in a base liquid — offer superior thermal conductivity compared to mono-nanofluids. Double-diffusive natural convection, where simultaneous temperature and concentration gradients drive buoyancy, arises in chemical processing, alloy solidification, and ocean circulation. Combining MHD, non-Newtonian rheology, and double diffusion in corrugated geometries creates a highly nonlinear problem that challenges traditional CFD.

Key motivation: Corrugated cylinders are used in industrial heat exchangers to enhance mixing. Machine learning surrogates can replace costly parametric CFD sweeps with fast, accurate predictions.

🎯 Key Contributions

• MHD Double Diffusion in Hybrid Ferrofluid — Fe₃O₄ + CoFe₂O₄/water hybrid ferrofluid modeled with power-law viscosity and temperature-dependent thermal conductivity inside concentric corrugated cylinders.
• Four ML Surrogate Models — DT, RF, FFNN, 1D-CNN trained on 3,696 CFD points; 1D-CNN achieves near-perfect prediction accuracy (R² > 0.999).

• Full Parametric Study — Effects of Rayleigh number (Ra), Hartmann number (Ha), power-law index (n), volume fraction (φ), and wave number (N) on Nu̅, Sh̅,

  Ψ

  , and viscosity distribution.

• Viscosity Distribution Prediction — 1D-CNN predicts full spatial viscosity field (R² = 0.9652) using a dataset of 26,454,000 data points.

• Polynomial Regression Correlations — 2nd-degree polynomial regression equations derived for Nu̅, Sh̅, and

  Ψ

  (R² ≥ 0.953).

Methodology

Physical Setup

Figure 1: Schematic of the wavy corrugated cylinder. Inner cylinder (heated): T=T_H, C=C_H. Outer cylinder (cooled): T=T_C, C=C_C. Both walls: no-slip (u=v=0). Corrugation shape: x(η) = (r + A cos(2πηN))·sin(2πη), y(η) = (r + A cos(2πηN))·cos(2πη). Outer cylinder: A=0.15, N=5; Inner cylinder: A=0.1, N=5.

Thermophysical Properties of Nanoparticles

Non-Newtonian Viscosity Model (Power-law)

μ̄ₙf = μf / [1 − 34.87(dn/df)^{−0.3} φ^{1.03}] × [2(ūₓ)² + 2(v̄y)² + (ūy + v̄ₓ)²]^{(n−1)/2}

• n < 1: shear-thinning (lower viscosity at higher shear rate)
• n = 1: Newtonian
• n > 1: shear-thickening

Dimensionless Governing Parameters

ML Model Input Parameters

Dataset: 3,696 CFD simulation results. Train/Validation/Test split: 70%/10%/20%.

📊 Grid Independence & Validation

Grid Independence (n=0.8, 1, 1.4; φ=0.04; Ra=2×10⁶; Ha=15; Le=2; Br=1)

Selected: Mesh 3 (29,182 elements) for all simulations.

Code Validation vs Literature (Ra=10⁵, n=1, Pr=1, Le=2.0, φ=0, H/L=2)

Excellent agreement confirms code reliability for double-diffusive simulation.

📊 Results — Parametric Study

Effect of Rayleigh Number (Ra) on Flow Flux (|Ψ|_max)

Higher Ra → stronger buoyancy → more chaotic vortex structures, enhanced convective heat transfer, wavy isotherms replacing smooth conduction-dominated profiles.

Effect of Hartmann Number (Ha) on Flow Flux (n=0.8, Ra=2×10⁶)

Higher Ha → stronger Lorentz force opposing buoyancy → suppressed convection, smoother isotherms, thicker boundary layers. Nu̅ drops 55.39% and Sh̅ drops 50.45% (Ra=2×10⁶) as Ha increases from 5 to 25.

Effect of Power-Law Index (n) on Flow Flux (Ha=15, Ra=2×10⁶)

Higher n → more viscous → slower flow → conduction-dominated heat transfer replacing convection.

Effect of Volume Fraction (φ) on Flow Flux (Ha=15, n=0.8, Ra=2×10⁶)

Increased φ → slightly higher effective viscosity (reduces flow velocity) but enhanced thermal conductivity (improves heat transfer). Net effect: small increase in Nu̅, slight decrease in Sh̅.

Effect of Wave Number (N) on Flow Flux (Ha=15, n=0.8, Ra=2×10⁶)

📊 Table 4 — Effect of Wave Number (N) on Average Nu̅ and Sh̅

Increasing N from 3→7: Nu̅ increases by up to 47.55% and Sh̅ by up to 45.32% (total values), as longer arc length creates more vortices enhancing convective transport. Average values may decrease locally due to flow separation effects.

📊 Table 9 — ML Model Performance Comparison (Test Set)

1D-CNN and FFNN consistently outperform tree-based models. **1D-CNN is best for Nu̅ and Sh̅; FFNN is best for

Ψ

** in terms of RMSE.

📊 Table 11 — CFD vs FFNN Prediction Comparison (Nu̅ and Sh̅)

FFNN predictions closely match CFD results across all parameter combinations, confirming surrogate model reliability.

📊 ML Hyperparameters (Optimized)

Training/Prediction Times

1D-CNN achieves highest accuracy but requires ~153 seconds to train vs 10 ms for DT.

📊 Feature Importance (SHAP & Permutation Analysis)

Both FFNN and 1D-CNN agree: Ra and n are the dominant features for predicting Nu̅ and Sh̅.

📊 Polynomial Regression Correlations (2nd-degree, R²≥0.953)

Nu̅ = 12.2035 − 0.1470N + 1.08×10⁻⁵Ra − 0.4553Ha − 9.2873n
      + 6.1264φ_p1 − 6.1264φ_p2 + 0.0037Ha² + 0.2574Han + 2.0833n²
      + 8.3373nφ_p1 + 8.3373nφ_p2 + ...  (R² = 0.953)

Sh̅ = 19.7753 − 0.1865N + 2.032×10⁻⁵Ra − 0.7619Ha − 15.9467n
      − 5.1611φ_p1 − 5.1611φ_p2 + 0.0046Ha² + 0.4264Han
      + 0.6107Haφ_p1 + 0.6107Haφ_p2 + ...  (R² = 0.957)

|Ψ| = 244.5815 + 7.7899N + 0.0004Ra − 17.3095Ha − 187.4751n
      + 0.1877Ha² + 9.1834Han + 16.1567n² + ...  (R² = 0.960)

Key physics captured: Ra has positive effect on all targets; Ha and n have strong negative effects; φ increases Nu̅ but decreases Sh̅.

Key Conclusions

• Increasing n (0.8→1.4) reduces flow flux by 49.04% and significantly lowers Nu̅ and Sh̅; shear-thinning (n<1) promotes convection, shear-thickening (n>1) promotes conduction
• Increasing Ra (2×10⁴→2×10⁶) boosts flow flux by 769.22% + 354.21% in two steps
• Increasing Ha (5→25) reduces flow flux by 43.58% then 43.29%; magnetic Lorentz force suppresses buoyancy-driven convection
• Adding φ=4% nanoparticles increases flow flux by only 4.18% but improves thermal conductivity
• Wave count N enhancement: Nu̅ increases up to 47.55% and Sh̅ up to 45.32% (N=3→7)

• 1D-CNN best overall: R²=0.9996/0.9994/0.9995 for Nu̅/Sh̅/

  Ψ

  ; MAPE < 1% for Nu̅ and Sh̅

📚 Citation

@article{ahad2025mhd,
  title={Machine learning-driven predictive modeling of magnetohydrodynamic double diffusion of non-Newtonian hybrid ferrofluids with variable thermophysical properties within corrugated cylinders},
  author={Ahad, Jawad Ibn and Molla, Md Mamun and Siddiqa, Sadia and Naqvi, Sahrish Batool},
  journal={Engineering Applications of Artificial Intelligence},
  volume={141},
  pages={109455},
  year={2025},
  publisher={Elsevier},
  doi={10.1016/j.engappai.2024.109455}
}