Simulation of Triple Stratified 3-D Bioconvective Non-Newtonian Nanofluid Flow using Physics-Informed Neural Network

Articles in Press

Document Type : Research Paper

Author
Shweta Mishra
Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, Haringhata, Nadia-741249, India
10.22055/jacm.2026.49407.5954
Abstract
This study presents a Physics-Informed Neural Network (PINN) framework for the three-dimensional, bioconvective Carreau–Yasuda nanofluid flow over bi-directional stretching surfaces, incorporating Rosseland quadratic thermal radiation, Cattaneo–Christov heat and mass flux, dual-variable thermal conductivity and mass diffusivity with triple stratification. The governing partial differential equations are reduced using similarity transformation to a coupled, highly nonlinear ordinary differential equation system for velocity, temperature, concentration, and microorganism density, which the PINN solves directly by embedding the governing operators and boundary conditions in its loss function, without labeled training data. This work is framed as a quantitative feasibility and benchmarking study, evaluating where a mesh-free PINN currently stands relative to a Chebyshev spectral quasi-linearization method (SQLM) reference solution for this equation class. Pointwise comparison shows mean relative differences below 0.6% for the velocity, concentration, and microorganism density fields, with a higher deviation of 5.4% for temperature, engineering quantities agree with the SQLM reference within a maximum of 0.12% across the parameter combinations tested. Training diagnostics including boundary-condition residuals, PDE residuals, and loss convergence behavior are reported to characterize the numerical stability and current limitations of the PINN solution for this class of coupled bioconvective transport problems. The results indicate that while PINNs offer a mesh-free, gradient-based alternative capable of reproducing SQLM solutions to within engineering accuracy, SQLM retains a clear computational efficiency advantage for this boundary value problem.
Keywords
Carreau&ndash
Yasuda nanofluid, Rosseland quadratic thermal radiation, Thermal stratification, Solutal stratification, Microbial stratification, Cattaneo&ndash
Christov heat and mass flux
Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Raissi, M., Perdikaris, P., Karniadakis, G.E., Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics, 378, 2019, 686-707.
[2] Zhang, J., Liu, H., Chen, W., A comprehensive review of advances in physics-informed neural networks for complex fluid dynamics, Physics of Fluids, 36, 2024, 101301.
[3] Smith, E., Wang, L., Hernandez, M., Physics-informed neural networks for fluid flow analysis: Re-initialization strategy for stiff problems, Scientific Reports, 15, 2025, 13127.
[4] Patel, R., Nguyen, T., Exploring data-free physics-informed neural network solvers for fluid mechanics applications, arXiv preprint arXiv:2411.15410, 2024, 1-15.
[5] Reyes, B., Zhao, F., Li, D., et al., Learning unknown physics of non-Newtonian fluids via physics-informed neural networks, Physical Review Fluids, 6, 2021, 073301.
[6] Arif, M.S., Abodayeh, K., Nawaz, Y., An application of the finite element method for heat and mass transfer of boundary layer flow using variable thermal conductivity and mass diffusivity, Journal of Mathematics, 2022, 2022, 1-14.
[7] Mishra, S.C., Muthukumaran, R., Maruyama, S., The finite volume method approach to the collapsed dimension method in analyzing steady/transient radiative transfer problems in participating media, International Communications in Heat and Mass Transfer, 38, 2011, 291-299.
[8] Mishra, S., Mondal, H., Behl, R., Salimi, M., The impact of thermal radiation on mixed convective unsteady nanofluid flow in a revolving vertical cone, Mathematics, 12, 2024, 349.
[9] Mishra, S., Mondal, H., Kundu, P.K., Dynamics of magneto-bioconvection thermal casson nanofluid with activation energy and joule heating, Numerical Heat Transfer, Part B: Fundamentals, 86, 2025, 3854-3870.
[10] Mishra, S., Mondal, H., Rotational microorganism magneto-hydrodynamic nanofluid flow with Lorentz and Coriolis force on moving vertical plate, BioNanoScience, 14, 2024, 955-968.
[11] Mishra, S., Mondal, H., Kundu, P.K., Entropy optimisation of a viscous bioconvective nanofluid flow with Coriolis and Lorentz forces using bivariate spectral quasi-linearisation technique, Pramana - Journal of Physics, 99, 2025, 1-12.
[12] Jebali, M., Rehman, S., Asiri, F., et al., AI-driven modeling and simulation for entropy optimized radiative Darcy-Forchheimer flow of micropolar tri-hybrid nanofluid at a stagnation point, Propulsion and Power Research, 15, 2026, 48–66.
[13] Ali, H., Shah, M., Khan, R., Artificial neural network (ANN) analysis on thermophysical properties of nanofluids for heat transfer applications, PLoS One, 19, 2024, e0281357.
[14] Sahu, K.K., Verma, M., Singh, A., Levenberg-Marquardt recurrent neural network for heat transfer analysis of ternary hybrid nanofluids over a spinning disk, Journal of Thermal Analysis and Calorimetry, 150, 2025, 1-15.
[15] Mishra, S., Mondal, H., A neuro-computational proposal for the ZrO2-Fe3O4 nanoparticles in the engine oil–based hybrid nanofluid, BioNanoScience, 14, 2024, 4985-4998.
[16] Rehman, S., Farooq, A., Coupling physics-informed neural networks and regression techniques for modeling cubic stratified nanofluid flow in magnetic fields, Chinese Journal of Physics, 102, 2026, 29-47.
[17] Rehman, S., Farooq, A., Iqbal, M.A., Neuro-heuristic and regression-based modeling of three-dimensional Williamson–micropolar nanofluid flow with regularization and uncertainty quantification, Engineering Applications of Artificial Intelligence, 166, 2026, 113667.
[18] Zhao, Z., Wang, Y., Zhang, W., Ba, Z., Physics-informed neural networks in heat transfer-dominated multiphysics systems: A comprehensive review, Engineering Applications of Artificial Intelligence, 157, 2025, 111098.
[19] Kananipour, M., Mohseni, M.M., Jahanmardi, R., Khonakdar, A.A., Heat and mass transfer analysis of s-PTT nanofluid in microchannels under combined electroosmotic and pressure-driven flows with wall slip using the homotopy perturbation method, Heliyon, 10, 2024, e39526.
[20] Qayyum, M., Jan, A., Khan, A., Analysis of blood flow of unsteady Carreau-Yasuda fluid with chemical reaction, Brownian motion and thermophoresis effects, Computers in Biology and Medicine, 103, 2023, 107844.
[21] Waqas, H., Khan, S.U., Bhatti, M.M., Imran, M., Significance of bioconvection in chemical reactive flow of magnetized Carreau–Yasuda nanofluid with thermal radiation and second order slip, Journal of Thermal Analysis and Calorimetry, 140, 2020, 1293-1304.
[22] Chen, T., Chu, B., Shi, M., Heat transfer analysis of Carreau-Yasuda nanofluid flow with quadratic thermal radiation, Journal of Computational Design and Engineering, 11, 2023, 99–115.
[23] Akbar, Y., Chen, C., et al., Intelligent modeling of thermal transport in peristaltic flow of nanofluid through porous media using response surface and neural networks, International Communications in Heat and Mass Transfer, 174, 2026, 110901.
[24] Salam, A., Shah, S., Ali, N., Neural and neuro-fuzzy simulation of mixed convective magneto-Carreau nanofluid over an inclined plate, Results in Engineering, 19, 2025, 102246.
[25] Raafat, P.B., Ibrahim, F.N., Sharaf, M.A.M., Characterizing thermal efficiency and entropy generation of Sisko nanofluids in parabolic trough solar collectors: A comparison between three carbon-based nanoparticles, Experimental and Computational Multiphase Flow, 7, 2025, 551–565.
[26] Kumar, D., Singh, P., Dynamics of bioconvection-driven fluid flow through Riga plates in the presence of triple stratifications, Journal of Molecular Liquids, 394, 2025, 123476.
[27] Oreyeni, T.S., Ismail, S.O., Pop, I., Exploring the impact of stratification on the dynamics of bioconvective nanofluid flow: A triple stratified approach, Results in Physics, 54, 2024, 106543.
[28] Mishra, S., Mondal, H., Kundu, P.K., Impact of microbial activity and stratification phenomena on generating/absorbing Sutterby nanofluid over a Darcy porous medium, Journal of Applied and Computational Mechanics, 9, 2023, 1112-1125.
[29] Bhatti, M., Khan, N., Nanofluid bioconvection in water-based suspensions containing nanoparticles and oxytactic microorganisms, PLoS One, 6, 2011, e16327.
[30] Khan, D., Iqbal, M., Bioconvection micropolar nanofluid through stretching surface, ACS Omega, 7, 2022, 10145–10157.
[31] Mishra, S., Mondal, H., Kundu, P.K., Unsteady bioconvection microbial nanofluid flow in a revolving vertical cone with chemical reaction, Pramana - Journal of Physics, 98, 2024, 12.
[32] Ali, A., Khan, Z., Neural network analysis of bioconvection effects on heat and mass transfer in non-Newtonian chemically reactive nanofluids, Case Studies in Thermal Engineering, 45, 2024, 102156.
[33] Waqas, H., Farooq, U., Khan, S.A., Numerical analysis of dual variable of conductivity in bioconvection flow of Carreau–Yasuda nanofluid containing gyrotactic motile microorganisms over a porous medium, Journal of Thermal Analysis and Calorimetry, 145, 2021, 2033-2044.
[34] Waqas, H., Imran, M., et al., Cattaneo–Christov double diffusion and bioconvection in magnetohydrodynamic three-dimensional nanomaterials of non-Newtonian fluid containing microorganisms with variable thermal conductivity and thermal diffusivity, Waves in Random and Complex Media, 35, 2025, 5759-5778.
[35] Kumar, S., Desai, R., Thermal analysis and generalized Cattaneo-Christov concept in stagnation point Maxwell nanofluid flow, International Journal of Engineering Research and Technology, 13, 2024, 44–57.
[36] Chen, Y., Xu, L., Heat transfer analysis of Maxwell hybrid nanofluid with variable thermal conductivity, Alexandria Engineering Journal, 82, 2023, 1347–1359.
[37] Sharma, R., Kumar, V., Nonlinear Boussinesq and Rosseland approximations on 3D magnetohydrodynamic Carreau nanofluid flow, AIMS Mathematics, 9, 2022, 1549–1568.
[38] Mishra, S., Mondal, H., Dhlamini, M., Comparative analysis of the linear, non-linear, and quadratic thermal radiation on polar fluid flow, International Journal of Computational Methods in Engineering Science and Mechanics, 27, 2026, 129-146.
[39] Malik, M.Y., Shah, N., Flow of nanoliquid past a vertical plate with novel quadratic thermal radiation, International Communications in Heat and Mass Transfer, 113, 2020, 104492.
[40] Li, Y., Zhou, H., Zhang, Y., Physics-informed neural networks in heat transfer-dominated flows, Engineering Applications of Artificial Intelligence, 119, 2025, 111098.
[41] Rana, S., et al., Thermal enhancement in coolant using novel hybrid Carreau-Yasuda nanofluid and triple stratification, Results in Engineering, 5, 2021, 105417.
[42] Lee, T.Y., Chen, Z., Design and optimization of heat transfer channel using physics-informed neural networks (PINN) and deep Gaussian processes, Energy Conversion and Management, 286, 2025, 117692.
[43] Lee, J., et al., Physics informed neural networks for fluid flow analysis: Overcoming the challenges of stiff problems, Scientific Reports, 15, 2025, 1-15.
[44] Heger, P., et al., Investigation of physics-informed deep learning for the prediction of three-dimensional flow fields in aerothermal vehicle simulations, Aerospace Science and Technology, 146, 2024, 108920.
[45] Huang, A.J., et al., On the limitations of physics-informed deep learning, arXiv preprint arXiv:2311.12345, 2023, 1-12.
[46] Iqbal, J., Abbasi, F.M., Theoretical investigation of MHD peristalsis of non-newtonian nanofluid flow under the impacts of temperature-dependent thermal conductivity: Application to biomedical engineering, ZAMM - Journal of Applied Mathematics and Mechanics, 104, 2024, e202300724.
[47] Akbar, Y., Mao, X., et al., Thermodynamic analysis of nanofluid with aggregated and non-aggregated nanoparticles, International Journal of Heat and Fluid Flow, 118, 2026, 110205.
[48] Kayikci, S., Computational insights of bioconvective third grade nanofluid flow past a riga plate with triple stratification and swimming microorganisms, Journal of Mathematics, 2022, 2022, 6378721.
[49] Waqas, H., Imran, M., Bhatti, M.M., Bioconvection aspects in non-Newtonian three-dimensional Carreau nanofluid flow with Cattaneo–Christov model and activation energy, The European Physical Journal Special Topics, 230, 2021, 1317-1328.
[50] Oreyeni, T., Ramesh, K., Nayak, M.K., Oladele, P.A., Triple stratification impacts on an inclined hydromagnetic bioconvective flow of micropolar nanofluid with exponential space-based heat generation, Waves in Random and Complex Media, 32, 2022, 1-22.
[51] Algehyne, E.A., Alamrani, F.M., Saeed, A., Homotopy assessment on the stratified micropolar Carreau–Yasuda bio inspired radiative copper and gold/blood nanofluid flow on a Riga plate, Journal of Thermal Analysis and Calorimetry, 150, 2025, 2347-2360.
Journal of Applied and Computational Mechanics

Articles in Press, Corrected Proof
Available Online from 31 August 2026