ANN-based thermal analysis of 3D MHD hybrid nanofluid flow over a shrinking sheet via LMA

Abstract

The study accounts for the heat transfer by a 3D mixed convection magnetohydrodynamic (MHD) hybrid nanofluid (HNF) flow along a shrinking sheet using an engine oil-based fluid emulsified with $\,Al_{2} O_{3} ,and\,Fe_{3} O_{4} \,$ nanoparticles. The governing nonlinear equations are simplified through similarity transformations and solved numerically using MATLAB’s bvp4c method. An artificial neural network (ANN), trained using the Levenberg–Marquardt algorithm (LMA), is employed to learn from numerical data with high accuracy and speed, thereby predicting both flow and temperature fields. Compared to conventional numerical methods, the ANN-based approach offers improved computational efficiency. The effects of dominant parameters, Grashof number, thermal radiation, magnetic field strength, nanoparticle volume fraction, and internal heat generation, on velocity and temperature profiles are analyzed. Heat transfer improves with increased nanoparticle volume fraction and Grashof number, while magnetic fields and slip reduce fluid velocity. ANN predictions exhibit close agreement with numerical results, confirming the model’s accuracy. Specifically, the LMA–ANN model achieved a regression coefficient of $\,R = 1\,$ and mean squared errors as low as $\,1.2181 \times 10^{ - 9} \,$ for velocity and $\,7.9615 \times 10^{ - 9} \,$ for temperature. These results demonstrate the model’s effectiveness in capturing nonlinear thermal-fluid behavior with minimal error. The novelty lies in integrating ANN with traditional numerical techniques to simulate hybrid nanofluid flows, offering valuable insights for applications in electronic cooling, energy systems, and thermal process optimization.

Introduction

Background and literature

Nanofluids (NFs), developed by Choi and Eastman¹, are next-generation heat transfer fluids created through the dispersion of nanoparticles in a base fluid, which increases thermal conductivity and flow behavior appreciably. Their utilization is diverse and includes aeronautical engineering, automobile systems, and the energy domain. Hashemi et al.² showed enhanced heat transfer coefficients in NFs as a result of nanoparticle aggregation. Shamshuddin et al.³ considered ferromagnetic and ohmic influences in NF flows over rotating disks, whereas Mishra et al.^4,5 considered radiative heat transfer and sensitivity analysis for intricate boundary conditions. Research work by Rajput et al.⁶ and Sarfraz and Khan⁷ emphasized the influence of nanoparticle diameter and magnetic fields on regulating slip conditions and fluid speed. Further research by Dawar et al.⁸ and Pasha et al.⁹ dealt with thermophoresis phenomena and entropy generation in NF flows. Together, these works highlight the promise of nanofluids in thermal performance improvement. Hybrid nanofluids (HNFs), a more advanced subcategory, also extends thermal characteristics through mixing more than one kind of nanoparticle¹⁰.

HNFs are formed by amalgamating two or more metals in a homogenous phase, resulting in improved heat transmission and other characteristics relative to single metal NFs. They are extensively used in heat exchangers, electronic cooling, machining, vehicles, nuclear reactors, solar energy, and industrial cooling, among others. Figure 1 illustrates certain industrial applications of HNFs, including electronic cooling, heat exchangers, nuclear reactors, solar collectors, and biomedical devices. It also verifies the incentive for using HNF in engineering systems due to their enhanced thermal conductivity and heat transfer performance. It constitutes the practical background justifying the selection of a HNF ($\,Al_{2} O_{3} ,Fe_{3} O_{4}$/engine oil) in the present study. Chen, J. et al.¹¹ discuss closed-loop plasma flow control of a turbulent cylinder wake flow using machine learning at Reynolds number of 28 000.

Fig. 1

Several applications of hybrid nanofluids.

This work considers engine oil (EO) as base fluid, enhanced with gold (Au) and zinc oxide (ZnO) nanoparticles to form a hybrid nanofluid (HNF). Yadav^12,13 examined radiative heat transfer effects in HNFs over curved surfaces in porous media. Enamul and Ontela¹⁴ analyzed entropy generation in Hall-effect-driven TiO₂–CoFe₂O₄/EO-based HNF flows between rotating porous disks under convective thermal boundaries. Similarly, SK and Ontela¹⁵ studied entropy generation in Darcy–Forchheimer Reiner–Rivlin flows involving Cu–Al₂O₃/EO HNFs. Deng, Z. et al.¹⁶ developed he model segregation of polydisperse granular materials in developing and transient free-surface flows. Sachhin et al.¹⁷ investigated HNF flow with temperature jump and Navier slip effects, showing that blade-shaped ternary hybrid nanoparticles offered superior heat transfer enhancement compared to other geometries.

Sachhin et al.¹⁸ analyzed NF flow including heat and mass transfer, MHD, and mass transpiration both by computational and ANN-based methods. Figure 2 shows some of the most important engineering applications of MHD flow, such as electromagnetic pumps, magnetic drug targeting, plasma control, and MEMS devices, all involving electrically conducting fluids in magnetic fields, directly relevant to the subject of this study. Though Fig. 2 illustrate some general uses of HNFs and MHD flows respectively, the current model combines both, considering a 3D MHD-HNF (engine oil-based) flow over a shrinking sheet with slip conditions, a heat source, radiation, and viscous dissipation. Such a combined visualization validates the pragmatic applicability of the proposed model in several fields. MHD generates a resistive Lorentz force within conductive fluids, which has significant effects on temperature and concentration fields, promoting delay in boundary layer separation as well as turbulence transition. There have been various studies incorporating magnetic field influences: Wang et al.¹⁹ proposed a self-scheduled direct thrust control model for gas turbine engines based on an EME-based approach with parameter variability. Sun et al.²⁰ analyzed convection interaction in heating waxy crude oil storage. Rai and Mishra²¹ investigated NF boundary layer flow with slip and magnetic effects, reporting increased thermal and concentration fields and lower velocity profiles under magnetic influence.

Fig. 2

Diverse engineering applications of MHD.

Moreover, several studies Qureshi et al.^22,23,24 have underscored the importance of magnetic impacts on fluid dynamics research, further validating the critical function of MHD in this domain.

Mixed convection entails concurrent action of natural and forced convection where thermal energy is moved from a region of high temperature to one of low temperature. Forced convection occurs due to externally forced motion, and free convection due to buoyancy arising from variation in fluid density. This phenomenon finds significant application in cooling nuclear reactors, pipelining transportation, and thermal control of electronic devices. Sun, X. et al.²⁵ developed the heat transfer augmentation, endothermic pyrolysis and surface coking of hydrocarbon fuel in manifold microchannels at a supercritical pressure. Zhu, D. et al.²⁶ discuss the robust macroscale super lubricity in humid air via designing amorphous DLC/Crystalline TMDs friction pair.

The use of ANN in fluid dynamics has increased for enhancing computational techniques and addressing intricate challenges. ANN computing is used in fluid mechanics for flow management and optimization²⁷. Qureshi et al.²⁸ suggested a Machine learning application for dissipative ternary nanofluid thermal exchange over a stretchy, wavy cylinder with thermal slip. Qureshi et al.²⁹ used Research on machine learning for agrivoltaics using a tri-magnetized Sutterby nanofluidic model with Joule heating. Planned integrated methodology is assessed consuming synthetic datasets that depict analogous physical events, resulting in an average inaccuracy of 3.5%. Kwon et al.³⁰ conducted a computational analysis of cooling process in the excavator’s combustion chamber using ANN, demonstrating the capacity of ANN-generated models to accurately and swiftly predict and optimize an excavator’s regeneration scheme inexpensively. Li et al.³¹ studied on forecast of service consistency of wireless telecommunication arrangement via distribution regression. Zhang et al.³² analyzed anharmonic consequence of main responses of imperative intermediate species in NH3/DME mixed combustion. More References by Qureshi et al.^33,34 provide contemporary research on ANN.

The particular hybrid nanofluid (engine oil with Au and ZnO nanoparticles) employed in this work possesses enhanced thermal conductivity and magnetic sensitivity, rendering it highly efficient for various applications. Some of the applications include MHD-based cooling in MEMS, lubrication and thermal management in rotating machines, magnetic braking systems, and heat dissipation in aerospace. Also, such fluids find application in solar collectors, nuclear cooling, and drug delivery. The selected flow model 3D shrinking sheet in an absorbent medium with MHD and heat generation, replicates actual problems like heat exchangers, metal drawing, and thermal coating.

Motivation to carry out study

With increasing demands for efficient heat transfer in advanced applications such as electronic cooling, energy systems, and biomedical applications, the shortcomings of traditional heat transfer fluids and modeling techniques have become more evident. Computer simulations offer an economical and versatile means of simulating virtually thermal systems under a variety of conditions, minimizing the need for extensive physical experimentation. ANNs, when used in conjunction with conventional computation, enhance predictive modeling by the capability to model complex nonlinear relations of data enabling accurate optimization of heat transfer performance. Specifically, the cooperative action of $\,Al_{2} O_{3} ,and\,Fe_{3} O_{4}$ nanoparticles suspended in EO enhances thermal conductivity and fluid viscosity, resulting in enhanced thermal performance. Furthermore, understanding dissipative heat characteristics of HNFs is essential for attaining optimal efficiency in high-performance thermal systems. The use of appropriate models of viscosity and thermal conductivity also adds precision and accuracy in simulating fluid behavior. Finally, methods based on ANN are best suited to identification and prediction of singularity in heat transfer occurrences, making them a computational gem for modern fluid dynamics problems.

Objective of the proposed investigation

The current investigation, as highlighted in the literature survey, encompasses several objectives, which are as follows:

• Improve a model that precisely envisages heat transfer rate in a radiative NF comprising nanoparticles ($Al_{2} O_{3} ,and\,Fe_{3} O_{4}$) within an engine oil as a base fluid.

• Examine impact of viscous dissipation for heat transfer performance above a diminishing surface.

• Employ 3D mixed convection MHD flow of $\,Al_{2} O_{3} ,Fe_{3} O_{4}$/Engine oil HNF and thermal conductivity models to attain auxiliary consistent and realistic outcomes.

• Implement neuro-computing performances to confirm efficient and precise computational resolutions.

• The primary purpose of employing the ANN simulation is to provide a reasonable prediction model that learns from numerical data (obtained by the bvp4c method) and approximates the nonlinear effect of velocity and temperature profiles with good accuracy and effectiveness.

The novelty of the study

When it comes to solving fluid flow issues, ANNs provide clear benefits over numerical or analytical approaches. Compared to traditional techniques, ANNs improve forecast accuracy by effectively understanding intricate non-linear correlations inside data. This method is used to a variety of fluid flow issues without requiring presumptions about the physics or geometry at play. This increases their flexibility and versatility in a wide range of situations. An effective substitute for conventional numerical or analytical approaches in solving fluid flow issues, ANNs are taught to provide predictions at a much quicker pace. ANNs are a useful tool for resolving fluid flow difficulties because they can learn and advance their prediction accuracy over time^35,36,37. A sheet for increasing the rate of heat transmission was extended in light of the previous discussion on the 3D mixed convection MHD movement of an $\,Al_{2} O_{3} ,Fe_{3} O_{4}$/engine oil HNF. The present problem’s uniqueness is emphasized as:

• The outline of a HNF ($\,Al_{2} O_{3} ,Fe_{3} O_{4} \,in\,EO$) investigates its distinctive thermal properties and competent features.

• It examines the impression of viscosity and amended thermal conductivity on HNF heat transfer, particularly on shrinking surfaces.

• The novel concept of neuro-computing procedures for enhancing heat transfer rates signifies a substantial expansion in this field.

• Analyzing the projected viscosity and thermal conductivity enhances understanding of thermophysical properties.

• In addition, an ANN is utilized to validate and predict the nonlinear dynamics of the system, hence this study is a novel combination of physical modeling and intelligent data-driven analysis.

Research questions

To guide the current investigation, the following research questions are addressed:

• I. How does the addition of hybrid nanoparticles ($\,Al_{2} O_{3} ,Fe_{3} O_{4}$) to engine oil influence the heat transfer rate over a shrinking sheet under MHD?

• II. What are the effects of Grashof number, magnetic field strength, radiation, and heat source on the velocity and temperature profiles of the HNF flow?

• III. To what extent can the nonlinear fluid flow of heat be predicted by an ANN using the LMA, and how effective is it compared to conventional numerical methods?

• IV. What is the effect of nanoparticle shape factor and slip conditions on thermal conductivity and reduction of skin friction?

Mathematical formulation of the flow configuration

The current work investigates MHD flow due to 3D mixed convection in a spongy intermediate on a vertical expanding plate. The model takes into account elements like Lorentz force, form factor, solar radiation, and the impacts of hydrogen. Furthermore, a small range for the induced magnetized/electric square is indicated by the low Reynolds number. A fluid combination of $\,Al_{2} O_{3} ,Fe_{3} O_{4}$ hybrid nanoparticles floating in engine oil interacts with the porous plate. The Prandtl number of the base fluid employed in this research is 205, which is a lower thermal diffusivity.

The issue is illustrated in Fig. 3. To clearly define the modeling framework, the mathematical model is premised on a steady-state, laminar, and incompressible HNF flow subject to inspiration of a transverse magnetic field. The analysis is premised on a single-phase continuum model where the nanoparticles are uniformly dispersed and are in thermal equilibrium with base fluid. Viscous dissipation, Joule heating, and effects of the induced magnetic field are all considered negligible in the low Reynolds number condition. The fluid flow is regulated by mixed convection along with a porous medium and slip boundary conditions, thermal radiation, Brownian motion and thermophoresis effects of nanoparticles. Based on assumptions made, governing non-linear differential equations are derived as follows^38,39:

$$\frac{\partial u}{{\partial x}} + \frac{\partial v}{{\partial y}} + \frac{\partial w}{{\partial z}} = 0,$$

(1)

$$u\frac{\partial u}{{\partial x}} + v\frac{\partial u}{{\partial y}} + w\frac{\partial u}{{\partial z}} - 2\omega v - \upsilon_{hnf} \frac{{\partial^{2} u}}{{\partial z^{2} }} - \frac{g}{{\rho_{hnf} }}\left( {\rho \beta } \right)_{hnf} \left( {T - T_{\infty } } \right) + \left( {\frac{{\sigma_{hnf} B_{0}^{2} }}{{\rho_{nf} }} + \frac{{\upsilon_{hnf} }}{{k_{p}^{ * } }}} \right)u = 0,$$

(2)

$$u\frac{\partial v}{{\partial x}} + v\frac{\partial v}{{\partial y}} + w\frac{\partial v}{{\partial z}} - 2\omega \,u - \upsilon_{hnf} \frac{{\partial^{2} v}}{{\partial z^{2} }} - \frac{g}{{\rho_{hnf} }}\left( {\rho \beta } \right)_{hnf} \left( {T - T_{\infty } } \right) + \left( {\frac{{\sigma_{hnf} B_{0}^{2} }}{{\rho_{hnf} }} + \frac{{\upsilon_{hnf} }}{{k_{p}^{ * } }}} \right)v = 0,$$

(3)

$$\begin{gathered} \frac{{\left( {\rho c_{p} } \right)_{hnf} }}{{\left( {\rho c_{p} } \right)_{f} }}\left( {u\frac{\partial T}{{\partial x}} + v\frac{\partial T}{{\partial y}} + w\frac{\partial T}{{\partial z}}} \right) - \frac{{\mu_{hnf} }}{{\left( {\rho c_{p} } \right)_{f} }}\left( {\frac{\partial w}{{\partial z}}} \right)^{2} \hfill \\ - \frac{1}{{\left( {\rho c_{p} } \right)_{f} }}\frac{\partial }{\partial z}\left( {k_{hnf} \left( T \right)\frac{\partial T}{{\partial z}}} \right) - \frac{1}{{\left( {\rho c_{p} } \right)_{f} }}\frac{{\partial q_{r} }}{\partial z}\frac{{\partial^{2} T}}{{\partial z^{2} }} = 0, \hfill \\ \end{gathered}$$

(4)

Fig. 3

Flow diagram of the problem.

The radiative heat flux, as specified in Eq. (4), is represented by the Rosseland approximation as $q_{r} = - \frac{{4\sigma^{ * } \partial T^{4} }}{{4k^{ * } \partial y}},$ where $k^{ * }$ denotes attraction coefficient and $\sigma^{ * }$ represents the Stefan-Boltzmann constant. Assuming limited diversity in stream temperature, $T^{4}$ is approximated using the Taylor expansion as $\,\left( {4TT_{\infty }^{3} - 3T_{\infty }^{3} \approx T^{4} } \right)$.By incorporating the relationships defined for $\left( {q_{r} } \right)\,$ in Eq. (4), the energy equation is ultimately derived as follows:

$$\begin{gathered} \frac{{\left( {\rho c_{p} } \right)_{hnf} }}{{\left( {\rho c_{p} } \right)_{f} }}\left( {u\frac{\partial T}{{\partial x}} + v\frac{\partial T}{{\partial y}} + w\frac{\partial T}{{\partial z}}} \right) - \frac{{\mu_{hnf} }}{{\left( {\rho c_{p} } \right)_{f} }}\left( {\frac{\partial w}{{\partial z}}} \right)^{2} \hfill \\ - \frac{1}{{\left( {\rho c_{p} } \right)_{f} }}\frac{\partial }{\partial z}\left( {k_{hnf} \left( T \right)\frac{\partial T}{{\partial z}}} \right) - \frac{{16\sigma^{ * } T_{\infty }^{3} }}{{3k^{ * } \left( {\rho c_{p} } \right)_{f} }}\frac{{\partial^{2} T}}{{\partial z^{2} }} = 0, \hfill \\ \end{gathered}$$

(5)

Allowing for hypothesis outlined in the problem, initial boundary conditions for Eqs. (1–3) and (5) are as:

$$\left. \begin{gathered} u = ax + \frac{2 - \omega }{\omega }\Lambda \frac{\partial u}{{\partial z}},\,\,v = by + \frac{2 - \omega }{\omega }\Lambda \frac{\partial v}{{\partial z}},\,w = - W \hfill \\ - k_{f} \frac{\partial T}{{\partial z}} = h_{f} \left( {T_{f} - T} \right),\,\,at\,\,z = 0, \hfill \\ T \to T_{\infty } ,\,\,u = v \to 0,\,\,\,at\,\,\,z \to \infty . \hfill \\ \end{gathered} \right\}$$

(6)

In this context, $h_{f}$ represents convective heat transfer coefficient, $\Lambda$ denotes mean free path of molecules, and $W$ is the absorption velocity, with $W$ being greater than zero. The authors implemented a novel procedure to alter heat transfer mechanisms trendy recently studied liquids. The authors of this article assert that the application of hybrid nanoparticles, in conjunction with various formations of nanoparticles besides hybrid base liquids, is an effective process. The current approach is anticipated to effectively enhance heat transfer progression in liquids. The thermo-physical properties of the HNF and NF influenced by the shape factor are presented in Table 1. In given table, $m$ represents shape factor, which corresponds to various nanoparticle shapes illustrated in Table 2.$\,\phi = \phi_{1} + \phi_{2} \,$ Represents volume element of nanoparticles. The thermo-physical possessions of nanoparticles and base fluid are presented in Table 3. The subsequent descriptions are provided to facilitate simplification of partial PDEs to ODEs as follows³⁹:

$$\left. \begin{gathered} u = axf^{\prime}\left( \xi \right),\,\,w = - \sqrt {a\upsilon_{f} } \left( {f\left( \xi \right) + g\left( \xi \right)} \right),\,\,v = ayg^{\prime}\left( \xi \right), \hfill \\ \xi = z\sqrt {\frac{a}{{\upsilon_{f} }}} ,\,\,T = T_{\infty } \left( {\left( {\theta_{w} - 1} \right)\theta \left( \xi \right) + 1} \right). \hfill \\ \end{gathered} \right\}$$

(7)

Table 1 Thermo-physical attributes of HNF^39,40.

Table 2 Nanoparticles shape with their shape factor. Ref^39,40.

Table 3 Thermophysical properties.

The application of the upper assumptions in conjunction with Eqs. (1–3) and (5), (6) lead to the formulation of governing ODEs, accompanied by the pertinent boundary conditions, as follows:

$$\frac{{f^{\prime\prime\prime}}}{{\chi_{1} }} + f^{\prime\prime}\left( {f + g} \right) - f^{{\prime}{2}} + 2\gamma g^{\prime} + \frac{{\chi_{4} }}{{\chi_{2} }}Gr\theta - \left( {\frac{{k_{p} }}{{\chi_{1} }} + \frac{{\chi_{3} }}{{\chi_{2} }}M} \right)f^{\prime} = 0,$$

(8)

$$\frac{{g^{\prime\prime\prime}}}{{\chi_{1} }} + g^{\prime\prime}\left( {f + g} \right) - g^{{\prime}{2}} - 2\gamma f^{\prime} + \frac{{\chi_{4} }}{{\chi_{2} }}Gr\theta - \left( {\frac{{k_{p} }}{{\chi_{1} }} + \frac{{\chi_{3} }}{{\chi_{2} }}M} \right)g^{\prime} = 0,$$

(9)

Here $\,\xi \,$ is the similarity variable, and $\,f\left( \xi \right),\,g\left( \xi \right)\,and\,\theta \left( \xi \right)\,$ are the horizontal velocity, vertical velocity, and temperature, respectively.

$$\left( {\left( {\chi_{6} + Nr\left( {1 + \left( {\theta_{w} - 1} \right)\theta } \right)^{3} } \right)\theta^{\prime} + \chi_{6} \Gamma \theta \theta^{\prime}} \right)^{\prime } + \Pr \left( {\chi_{5} \left( {f + g} \right)\theta^{\prime} + \frac{{Ec\left( {f^{\prime} + g^{\prime}} \right)^{2} }}{{\chi_{1} }}} \right) = 0,$$

(10)

$$\left. \begin{gathered} f\left( \xi \right) + g\left( \xi \right) = f_{0} ,\,\,f^{\prime}\left( \xi \right) = 1 + \lambda f^{\prime\prime}\left( \xi \right),\,\,g\left( \xi \right) = 0, \hfill \\ g^{\prime}\left( \xi \right) = \alpha + \lambda g^{\prime\prime}\left( \xi \right), \hfill \\ \theta^{\prime}\left( \xi \right) = - Bi\left( {1 - \theta \left( \xi \right)} \right),\,\,at\,\,\xi = 0, \hfill \\ \theta \left( \xi \right) \to 0,\,\,f^{\prime}\left( \xi \right) \to 0,\,\,g^{\prime}\left( \xi \right) \to 0,\,\,as\,\,\xi \to \infty . \hfill \\ \end{gathered} \right\}$$

(11)

The convection boundary condition employed here simulates a real-world thermal interaction when the surface is having a heat exchange with an ambient fluid through convection. This type of condition frequently occurs in engineering problems such as electronic component cooling, heat exchangers, and thermal insulation problems.

In which;

$$\left. \begin{gathered} \chi_{1} = \left( {1 - \phi_{1} } \right)^{2.5} \left( {1 - \phi_{2} } \right)^{2.5} , \hfill \\ \chi_{2} = \left( {1 - \phi_{2} } \right)\left[ {\left( {1 - \phi_{1} } \right) + \phi_{1} \left( {\frac{{\rho_{s1} }}{{\rho_{f} }}} \right)} \right] + \phi_{2} \frac{{\rho_{s2} }}{{\rho_{f} }}, \hfill \\ \chi_{3} = \left( {1 + \frac{{\left( {3\left( {m - 1} \right)\left( {\frac{{\sigma_{2} }}{{\sigma_{f} }} - 1} \right)\phi_{2} } \right)}}{{\left( {\frac{{\sigma_{2} }}{{\sigma_{f} }} + 2} \right) - \left( {\frac{{\sigma_{2} }}{{\sigma_{f} }} - 1} \right)\left( {m - 1} \right)\phi_{2} }} + \frac{{\left( {3\left( {m - 1} \right)\left( {\frac{{\sigma_{1} }}{{\sigma_{f} }} - 1} \right)\phi_{1} } \right)}}{{\left( {\frac{{\sigma_{1} }}{{\sigma_{f} }} + 2} \right) - \left( {\frac{{\sigma_{1} }}{{\sigma_{f} }} - 1} \right)\left( {m - 1} \right)\phi_{1} }}} \right)\sigma_{f} \hfill \\ \chi_{4} = \left( {1 - \phi_{2} } \right)\left[ {\left( {1 - \phi_{1} } \right) + \phi_{1} \left( {\frac{{\left( {\rho \beta } \right)_{s1} }}{{\left( {\rho \beta } \right)_{f} }}} \right)} \right] + \phi_{2} \frac{{\left( {\rho \beta } \right)_{s2} }}{{\left( {\rho \beta } \right)_{f} }}, \hfill \\ \chi_{5} = \left( {1 - \phi_{2} } \right)\left[ {\left( {1 - \phi_{1} } \right) + \phi_{1} \left( {\frac{{\left( {\rho C_{p} } \right)_{s1} }}{{\left( {\rho C_{p} } \right)_{f} }}} \right)} \right] + \phi_{2} \frac{{\left( {\rho C_{p} } \right)_{s2} }}{{\left( {\rho C_{p} } \right)_{f} }}, \hfill \\ \chi_{6} = \frac{{k_{s2} + \left( {m - 1} \right)k_{bf} - \left( {m - 1} \right)\phi_{2} \left( {k_{bf} - k_{s2} } \right)}}{{k_{s2} + \left( {m - 1} \right)k_{bf} + \phi_{2} \left( {k_{bf} - k_{s2} } \right)}}.\frac{{k_{s1} + \left( {m - 1} \right)k_{f} - \left( {m - 1} \right)\phi_{1} \left( {k_{f} - k_{s1} } \right)}}{{k_{s1} + \left( {m - 1} \right)k_{f} + \phi_{1} \left( {k_{f} - k_{s1} } \right)}} \hfill \\ \end{gathered} \right\}$$

(12)

Succeeding are dimensionless number and parameters:

$Gr = \frac{{Gr_{T} }}{{{\text{Re}}^{2} }}$ Represents Grashof number, where $\gamma = \frac{\omega y}{{ax}}$ denotes the rotation parameter, with ${\text{Re}} = \frac{{ax^{2} }}{\upsilon }$ indicating local Reynolds number. $M = \frac{{\sigma_{f} B_{0}^{2} }}{{a\rho_{f} }}$ Defines magnetic square parameter, $k_{p} = \frac{{\upsilon_{f} }}{{ak_{p}^{ * } }}$ signifies porosity parameter, and $\theta_{w} = \frac{{T_{w} }}{{T_{\infty } }}$ indicates temperature ratio parameter. $Rd = \frac{{16\sigma^{ * } T_{\infty }^{3} }}{{3k^{ * } k_{f} }}$ Represents radiation parameter, while $Ec = \frac{{a\mu_{f} }}{{\left( {\theta_{w} - 1} \right)T_{\infty } \left( {\rho C_{p} } \right)_{f} }}$ denotes Eckert number. $\Pr = \frac{{\upsilon_{f} \left( {\rho C_{p} } \right)_{f} }}{{k_{f} }}$ Defines Prandtl number, and $\alpha = \frac{b}{a}$ indicates the stretching factor, where $a,\,b > 0$ are the stretching constants. Additionally $\,\lambda = \frac{2 - \omega }{\omega }\Lambda \sqrt {\frac{a}{{\upsilon_{f} }}} \,$ denotes the slip parameter, $f_{0} = \frac{W}{{\sqrt {a\upsilon_{f} } }}$ indicates the injection $\,\left( {f_{0} < 0} \right)/\left( {f_{0} > 0} \right)\,$ parameter, and $Bi = \frac{{h_{f} }}{{k_{f} }}\sqrt {\frac{{\upsilon_{f} }}{a}}$ represents Biot number. Surface drag forces and local Nusselt number are imperative physical quantities that influence flow and heat transfer.

The ODEs characterize the distribution of these quantities in this context as follows⁴²:

$$Nu_{x} = - \left[ {\frac{{xk_{nf} }}{{k_{f} \left( {\theta_{w} - 1} \right)T_{\infty } }} + \frac{x}{{k_{f} \left( {\theta_{w} - 1} \right)T_{\infty } }}\frac{{16\sigma^{ * } }}{{3k^{ * } }}T^{3} } \right]\left( {\frac{\partial T}{{\partial z}}} \right)_{z = 0} ,$$

(13)

$$\tau_{wx} = \mu_{nf} \left( {\frac{\partial u}{{\partial z}}} \right)_{z = 0}$$

(14)

$$\tau_{wx} = \mu_{nf} \left( {\frac{\partial v}{{\partial z}}} \right)_{z = 0} ,$$

(15)

By applying similarity transformations to Eqs. (13–15), Local Nusselt number and surface drag forces for the standard shape are determined as:

$$Nu_{x} {\text{Re}}_{x}^{{ - \frac{1}{2}}} = - \left( {\chi_{6} + Rd\left( {\left( {\theta_{w} - 1} \right)\theta \left( 0 \right) + 1} \right)^{3} } \right)\theta^{\prime}\left( 0 \right).$$

(16)

$$Cf_{y} {\text{Re}}_{y}^{\frac{1}{2}} = \sqrt[{ - 3}]{\alpha }\frac{{g^{\prime\prime}\left( 0 \right)}}{{\chi_{1} \chi_{2} }},$$

(17)

$$Cf_{x} {\text{Re}}_{x}^{\frac{1}{2}} = \frac{{f^{\prime\prime}\left( 0 \right)}}{{\chi_{1} \chi_{2} }},$$

(18)

The local Reynolds numbers in y and x directions are defined as ${\text{Re}}_{y} = y\sqrt {\frac{a}{\upsilon f}}$ and ${\text{Re}}_{x} = x\sqrt {\frac{a}{\upsilon f}}$ respectively, as indicated in Eqs. (17–18).

Thermophysical properties

The Table 3 represents thermophysical properties^43,44 of both base fluid and nanoparticles. The impact of nanoparticle shape factor $\,\psi \,$ verses dynamic viscosity of the HNF is implemented via a revised Brinkman-type model $\,\mu_{hnf} = \mu_{f} \left( {1 - \phi_{1} } \right)^{ - 2.5\psi } \,$ where $\,\mu_{hnf} \,$ is the effective viscosity of the HNF,$\,\mu_{f} \,$ is the base fluid viscosity,$\,\phi \,$ is the volume fraction of the nanoparticles, and $\,\psi \, = 1$ is the geometry-dependent dimensionless shape factor. Higher shape factors (spherical) increase effective viscosity because of greater hydrodynamic resistance. This equation conforms to recent studies on the impact of particle morphology on viscosity in NFs by Verma et al.⁴¹.

In this work, the thermal conductivity of the HNF is predicted based on a modified Maxwell model with a shape factor correction. The incorporation of the nanoparticle shape factor $\,\psi \,$ adjusts effective conductivity to incorporate realistic heat transfer mechanisms as a function of particle geometry. The adjustment allows for more accurate simulation of heat transfer in NFs with non-spherical nanoparticles such as platelets, cylinders, and bricks. Thermal conductivity from the geometry confirms experimental findings reported by Verma et al.⁴¹ and enhances physical realism of the simulations.

Graphical representations of thermophysical properties

The following figures show nanoparticles ($Al_{2} O_{3}$,$Fe_{3} O_{4}$) and base fluid Engine Oil (EO) thermophysical properties from the references^43,44. The following figures show nanoparticles ($Al_{2} O_{3}$,$Fe_{3} O_{4}$) and base fluid Engine Oil (EO) thermophysical properties from the references^43,44. The density values of the considered nanoparticles are illustrated in Fig. 4, while the heat capacity of the investigated nanoparticles is presented in Fig. 5. Furthermore, the thermal conductivity of the discussed nanoparticles is depicted in Fig. 6.

Fig. 4

Density of concerned nanoparticles.

Fig. 5

Heat capacity of deliberated nanoparticles.

Fig. 6

Thermal conductivity of discussed nanoparticles.

Dimensional consistency and continuity verification

To check for the physical consistency of the similarity transformations, dimensional analysis is carried out. The similarity variable $\,\xi$, which is given by $\,\xi = z\sqrt {\frac{a}{{\upsilon_{f} }}}$, is dimensionless because $\,z\,$ is of the order $\,\left[ m \right],\,$ $\,a\,$ is of the order $\,\left[ {1/s} \right],\,$ and $\,\upsilon_{f} \,$ is kinematic viscosity $\,\left[ {m^{2} /s} \right]$. Therefore:

$$\xi = \left[ m \right] \cdot \sqrt {\frac{{\left[ {{1 \mathord{\left/ {\vphantom {1 s}} \right. \kern-0pt} s}} \right]}}{{\left[ {{{m^{2} } \mathord{\left/ {\vphantom {{m^{2} } s}} \right. \kern-0pt} s}} \right]}}} = \frac{\left[ m \right]}{{\left[ m \right]}} = 1$$

(19)

Thus $\,\xi$ dimensionless, the transformed functions and $\,f\left( \xi \right),\,g\left( \xi \right)\,and\,\theta \left( \xi \right)\,$ are the horizontal velocity, vertical velocity, and temperature, respectively are formulated as non-dimensional versions of velocity, temperature, and concentration. Replacing similarity variables in 3D continuity equation:

$$\frac{\partial u}{{\partial x}} + \frac{\partial v}{{\partial y}} + \frac{\partial w}{{\partial z}} = 0,$$

(20)

Verifies that the equation is identically satisfied and supports the validity of the similarity transformation.

Methodology

This paper adopts a hybrid approach that combines a classical numerical technique and intelligent computing. The nonlinear governing equations are first resolved via using MATLAB’s bvp4c function, which solves boundary value problems by a collocation scheme. Concurrently, an ANN and LMA are employed to model the interaction between leading parameters (e.g., Grashof number, thermal radiation) and corresponding fluid behavior. The ANN structure is trained in a supervised learning model using data generated from the bvp4c algorithm. The two-method approach ensures accurate numerical solutions and effective predictive modeling with enhanced interpretability along with computational efficiency. The numerical approach used in this study, which is based on a modified framework of ODEs, is thoroughly evaluated in this part. The flowchart shown in Fig. 7 below provides a clear illustration of the procedure. The dynamics of heat transfer in a 3D mixed convection MHD flow of a HNF ($Al_{2} O_{3}$,$Fe_{3} O_{4}$) blended with engine oil as the base fluid next to a spongy vertical stretching plate is thoroughly investigated computationally in this paper. The suction/infusion process, thermal ray, and shape factor of the nanoparticles have all been examined using the shooting technique. ANNs are used in the bvp4c solver methodology’s computer programming for simulation. Both approaches will be explained in this section.

Fig. 7

Flow model of the methodology.

Numerical scheme

This investigation uses bvp4c method to analyze a nonlinear scheme of ODEs presented in Eqs. (8–10) and (11). For this method, it is essential to convert the higher-order ODEs into a system of 1^st-order nonlinear ODEs. In this case, the boundary amount concern needs to be converted into the main amount matter. For computational stability and accuracy, a convergence test was performed by varying the initial mesh size with the bvp4c solver. Numerical solutions were found to be mesh-independent for over 100 discretization points with negligible difference in the velocity and temperature profiles. The MATLAB bvp4c solver was programmed with relative and absolute tolerances of 10^–6, and convergence was achieved consistently. Such environments offered numerical stability and uniform output for all the values of the tested parameters. The complete procedure is delineated in detail as follows:

$$\begin{gathered} m_{1} = f,m_{2} = f^{\prime},m_{3} = f^{\prime\prime},m^{\prime}_{3} = f^{\prime\prime\prime}, \hfill \\ m_{4} = g,m_{5} = g^{\prime},m_{6} = g^{\prime\prime},\,m^{\prime}_{6} = g^{\prime\prime\prime}, \hfill \\ m_{7} = \theta ,m_{8} = \theta^{\prime},m^{\prime}_{8} = \theta^{\prime\prime},\, \hfill \\ \end{gathered}$$

(21)

$$m^{\prime}_{3} = \chi_{1} \left( {m_{2}^{2} - m_{3} \left( {m_{1} + m_{4} } \right) - 2\gamma m_{5} - \frac{{\chi_{4} }}{{\chi_{2} }}Grm_{7} + \left( {\frac{{k_{p} }}{{\chi_{1} }} + \frac{{\chi_{3} }}{{\chi_{2} }}M} \right)m_{2} } \right),$$

(22)

$$m^{\prime}_{6} = \chi_{1} \left( {m_{5}^{2} - m_{6} \left( {m_{1} + m_{4} } \right) + 2\gamma m_{2} - \frac{{\chi_{4} }}{{\chi_{2} }}Grm_{7} + \left( {\frac{{k_{p} }}{{\chi_{1} }} + \frac{{\chi_{3} }}{{\chi_{2} }}M} \right)m_{5} } \right),$$

(23)

$$m^{\prime}_{8} = - \frac{{\Pr \left( {\chi_{5} \left( {m_{1} + m_{4} } \right)m_{8} + \frac{{Ec\left( {m_{2} + m_{5} } \right)^{2} }}{{\chi_{1} }}} \right)}}{{\left( {\left( {\chi_{6} + Nr\left( {1 + \left( {\theta_{w} - 1} \right)m_{8} } \right)^{3} } \right) + \chi_{6} \Gamma m_{8} } \right)}},$$

(24)

$$\left. \begin{gathered} m_{1} \left( \xi \right) + m_{4} \left( \xi \right) = f_{0} ,\,\,m_{2} \left( \xi \right) = 1 + \lambda m_{3} \left( \xi \right),\,m_{4} \left( \xi \right) = 0, \hfill \\ m_{5} \left( \xi \right) = \alpha + \lambda m_{6} \left( \xi \right), \hfill \\ m_{8} \left( \xi \right) = - Bi\left( {1 - m_{7} \left( \xi \right)} \right),\,\,at\,\,\xi = 0, \hfill \\ m_{7} \left( \xi \right) \to 0,\,\,m_{2} \left( \xi \right) \to 0,\,\,m_{5} \left( \xi \right) \to 0,\,\,as\,\,\xi \to \infty . \hfill \\ \end{gathered} \right\}$$

(25)

ANN computational procedure

A single hidden layer feedforward-ANN is employed in this study, which is trained using LMA with fast convergence and improved precision in solving nonlinear regression problems based on training data produced by bvp4c. The ANN is fed with input features such as the magnetic parameter $\,M,\,$ thermal radiation parameter $\,Rd,\,$ Prandtl number $\,\Pr ,\,$ Brownian motion parameter $\,Nb,\,$ and thermophoresis parameter $\,Nt,\,$ among others. All input and output datasets were normalized to range [0, 1] by min–max scaling for improved convergence. The network was trained by the LMA with MATLAB’s default setting. The data were split into 80% training, 10% validation, and 10% testing. Early stopping on validation performance was activated to prevent overfitting. No dropout was applied, as the network complexity and the data size were moderate and did not require additional regularization. Figure 8 shows the internal neural architecture for weights, the 1-neuron structure of the output layer, the 10-neuron hidden layer, and the internal architecture for both the input and output layers. While biases enable the model to translate the activation function’s output, enabling it to learn and fit the data and identify fine-grained relationships, weights in neural networks are used to specify the strength of connections between neurons and assess the significance of input features. Weights and biases, which are learnt during training, play crucial roles in neural network testing and validation. They specify how input data is mapped, enabling the model to precisely predict unknown data.

Fig. 8

ANN-based architecture.

Graphical representation of data distribution

The data is split into three subsets to ensure the model works well when applied to fresh data: training 80%, validation 10% (for performance verification and monitoring to prevent over-fitting), and testing 10% (to evaluate the final performance). Figure 9 makes it evident.

Fig. 9

ANN-data distribution.

Validation of the numerical model

To ascertain accuracy of the present numerical model, result comparisons were undertaken with those of Ghadikolaei & Gholinia³⁹ for 3D mixed convection MHD flow of HNFs over a vertical surface. These validation results are summarized in Table 4, for several cases with the same parameters (i.e., magnetic parameter $\,M$, Grashof number $\,Gr$, and nanoparticle volume fraction $\,\phi$), the Nusselt numbers obtained in the present study were found to be satisfactory compared with those in the literature, with deviations less than 0.6%. This confirms the validity of the shooting method and the ANN-based method used here.

Table 4 Validation of the numerical model.

Results and discussion

This segment examines the influences of various parameters, considering the influence of nanoparticle shape factor versus temperature and velocity profiles, local Nusselt number, and skin friction coefficient for hybrid systems. The analysis of NF and its phases was conducted using graphical representations. The ranges of various parameters examined in existing study have been derived by Usman et al.⁴⁵. Simultaneously, once each parameter is evaluated, the constant values of the other dimensionless parameters are as follows^43,45.

$$\begin{gathered} 0.1 \le Gr \le 1.2,\,0.01 \le \phi_{1} = \phi_{2} \le 0.04,\,0.2 \le Rd \le 0.8,\, \hfill \\ 0.1 \le Q \le 0.4,\,0.1 \le \beta \le 0.4,\,0.1 \le \lambda \le 0.9,\,0.1 \le M \le 0.9. \hfill \\ \end{gathered}$$

Figures 10 and 11 illustrate the impact of variations in the incremental quantity of Grashof number ($Gr$) on the horizontal ($f^{\prime}$) and vertical ($g^{\prime}$) velocity functions, individually. An growth in $Gr\,$ causes the velocity profile to rise. As the Grashof number increases, the disparity between surface and environmental temperatures also increases. This temperature difference generates a temperature gradient, which temperature gradient results in an increase in velocity.

Fig. 10

Effect of Grashof number on horizontal velocity.

Fig. 11

Consequence of Grashof number on vertical velocity.

The influence of nanoparticle volume fractions on both velocities is seen in Figs. 12 and 13 for engine oil-based HNFs ($Al_{2} O_{3}$,$Fe_{3} O_{4}$) and conventional NFs. Figure 12 illustrates that the axial velocity components ($f^{\prime}$) and ($g^{\prime}$) enhance with an increased nanoparticle volume percentage. Conversely, an augmentation is shown in Fig. 13 with a rise in the values of nanoparticle volume fraction $\,\phi_{1} = \phi_{2}$. This is a consequence of an increase in collisions among the nanoparticles. Furthermore, hybrid suspension has a superior profile in comparison to conventional NF. The dynamics of fluids and particles within a system are significantly affected by two key factors. The nanoparticle fractional volume parameter significantly influences flow manners of fluid-particle structures. The 2^nd reason is temperature profile of arrangement, which delineates distribution of temperature between fluid and particles. The two influences are essential for comprehending the performance of fluid-particle arrangements. Concentration of nanoparticles in fluid-particle systems significantly influences temperature distribution. With an intensification in nanoparticle concentration, the temperature distribution exhibits greater variability, attributable to the differing thermal properties of fluid and the elements. A developed concentration of nanoparticles leads to an extra uniform temperature circulation due to enhanced heat transfer between fluid and particles. Figure 14 shows that an increase in nanoparticle volume fraction $\,\phi_{1} = \phi_{2} \,$ increases thermal conductivity, which promotes a higher and more homogeneous temperature distribution in the fluid.

Fig. 12

Effect of volume fraction parameter on horizontal velocity.

Fig. 13

Consequence of volume fraction parameter on vertical velocity.

Fig. 14

Significance of volume fraction parameter on temperature profile.

Figure 15 shows that an increase in the thermal radiation parameter increases the temperature distribution owing to augmented radiative heat flux. This phenomenon is particularly useful in high-temperature engineering devices like combustion chambers and spacecraft structures. In terms of physical properties, this radiation increases surface heat flow, which raises the boundary layer’s predicted temperature. The impact of a heat source $\,\left( Q \right)$ on temperature is seen in Fig. 16. It is observed that as the fluid’s temperature is raised the heat source is increased. The primary cause of this effect is that the heat source raises the fluid’s thermal and kinetic energy. In the case of heat source fluids, the thermal and momentum boundary layers are thin.

Fig. 15

Effect of radiation factor over temperature profile.

Fig. 16

Importance of heat source above temperature profile.

Figure 17 illustrates how the rotation factor $\,\left( \lambda \right)$ affects vertical velocity. A high rotation rate is linked to lesser longitudinal acceleration, which eventually surpasses the stretching rate, according to the rotation parameter, which is often represented by the symbol $\,\lambda$. A rise in the rotation parameter increases the centrifugal force, pushing the fluid radially outward. This action enhances convective transport from the surface, enhancing thermal dissipation, a process useful in rotating devices such as turbines and disk reactors. Additionally, Fig. 18 shows how horizontal profiles decrease as the quantity of the slip parameter $\,\left( \beta \right)$ increases. In terms of body, the thickness of the momentum barrier layer decreases as the slip parameter increases because the fluid flow area grows significantly.

Fig. 17

Consequence of rotation factor on vertical velocity.

Fig. 18

Effect of slip parameter on horizontal velocity.

The velocity field is diagrammatically represented for a range of magnetic values $\,\left( M \right)$ in Figs. 19, 20, and 21, respectively. The pattern of a reduction in fluid’s velocity as amount of $\,M$ increases is seen in Figs. 19 and 20. A heightened intensity of the magnetic component causes a resistive force, commonly known as the Lorentz force, to be produced. The fluid’s movement is impeded by this Lorentz force, which causes the fluid’s velocity to decrease. Consequently, anytime the amount of $\,M$ grows, a physical phenomenon happens where the velocity distribution diminishes. Temperature and longitudinal fields have both been shown to exhibit the reversal behavior. Figure 21 shows that the temperature profile increases with stronger magnetic fields. This is because magnetic field affects the decrease of convective currents, which results in heat energy building up close to the surface.

Fig. 19

Effect of magnetic value on horizontal velocity.

Fig. 20

Consequence of magnetic value on vertical velocity.

Fig. 21

Importance of magnetic value above temperature profile.

These findings have immediate applications in practical situations like cooling of microelectronic components, lubrication, energy-efficient heat exchangers, and biomedical devices under the control of a magnetic field. These results of parametric influences of Grashof number, magnetic field, and nanoparticle concentration are enlightening for the optimization of such thermal systems in real-world application scenarios.

Artificial neural network (ANN) training

The calibrated ANN model significantly reduces computational cost in parameter prediction and serves as a quick and reliable surrogate to real-time thermal analysis. The ANN-LMA technique is an effective method for addressing the complexities presented in our study (Huda et al.⁴⁶, Kumar et al.^47,48). ANNs represent an advanced computational approach for predicting various physical characteristics of everyday issues, grounded in the concept that human brain functions as an interrelated network of neural cells. Furthermore, the alignment of ANN training with the established numerical scheme, bvp4c, is demonstrated. The ANN-LMA adjustable learning rate and momentum are established features that enable effective performance across various deep learning tasks, including the current investigation (Darvesh et al.⁴⁹).

The ANN-LMA feed-forward neural network consists of two hidden layers, each containing eleven neurons. The dataset is separated into training (80%), validation (10%), and testing (10%) subsets to ensure effective model performance on new data, with the validation set used for monitoring to prevent overfitting. Additionally, a learning rate parameter, the combination of $Rd$(0.2, 0.4, 0.6, 0.8) and $Gr$(0.1, 0.4, 0.8, 1.2) were utilized. The network undergoes training for 1000 epochs. Furthermore, the effectiveness of ANN training in aligning with the bvp4c, a proposed numerical scheme, is demonstrated.

The progress of ANN parameter training is shown in Figs. 22(a-f) and 23(a-f). Graphical depictions of regression analysis, autocorrelation, training states, error histograms, mean squared errors, and function fitting are provided for the ANN training process. The mean square error (MSE) for each example declined throughout training and eventually stabilized, indicating diminishing returns with further training. The incorporation of testing and validation of the MSE signifies that model is not overfitting and has robust generalization abilities. The gradient plot depicts model’s merging, indicating stabilization at a low value across several situations as training advances. The models have attained an established situation with tiny overfitting, as shown by the error histograms reflecting good accuracy, with the greatest errors mostly clustered around zero. R-values of 1 signify perfect correlation, and regression graphs validate model’s exceptional routine across all datasets, with fit lines indicating ideal exactness and robust generalization, closely coinciding with desired output and displaying tiny bias. Moreover, during neural training, the representations of the ANN are documented for each effective parameter.

Fig. 22

ANN-LMA training for Grashof number ($Gr$) against velocity. (a) Mean squared error (MSE) during training; (b) Training state (gradient and validation checks); (c) Error histogram between targets and predictions; (d) Fitting plot of network output vs. target data; (e) Regression plot showing R-values for training, validation, and testing; (f) Autocorrelation of residuals.

Figure 22(a-f) shows an illustration of the ANN for the Grashof number ($Gr$) for velocity. The ANN achieved its highest optimal validation performance MSE is $1.2181\times {10}^{-09}$, $Mu=1\times {10}^{-08}$, with utilizes Histograms comprising 20 bins across 335 epochs and the maximum achieved Gradient is $9.9084\times {10}^{-08}$, where the function fit output element output is 1 and the best output target of Regression Analysis is $1.7\times {10}^{-08}$ with training value $R=1$ also its validation checks at $0$. While the autocorrelation of errors with confidence limits has a lower confidence limit -0.2 and the upper confidence limit is 0.2.

The ANN aimed at the thermal radiation ($Rd$) on behalf of temperature profile is demonstrated in Fig. 23 (a-f). Using histograms with 20 bins over 250 epochs, the ANN achieved its highest optimal validation performance $MSE=7.9615\times {10}^{-09}$ and $Mu=1\times {10}^{-08}$. The maximum achieved gradient was $9.8907\times {10}^{-08}$, where the function fit output element output was 1 and the best output target of Regression Analysis was $2.2\times {10}^{-08}$ with training value $R=1$ and its validation checks at $0$. The maximum confidence limit for the autocorrelation of errors with confidence limits is 0.2, whereas the lower confidence limit is -0.2.

Fig. 23

ANN-LMA training for thermal radiation ($Rd$) for temperature profile. (a) Mean squared error (MSE) during training; (b) Training state (gradient and validation checks); (c) Error histogram between targets and predictions; (d) Fitting plot of network output vs. target data; (e) Regression plot showing R-values for training, validation, and testing; (f) Autocorrelation of residuals.

Validation and performance evaluation of ANN

The ANN model was trained on 1,000 samples all drawn from the verified bvp4c numerical solver and no experimental data were used. A single-hidden-layer feedforward model with 12 neurons was used. Trial-and-error approach was used to select the learning rate (0.2–0.8) and momentum (0.1–1.2) hyper parameters for training error minimization and convergence. All the data were also normalized before training, and 80:10:10 split was used for training, validation, and testing. The training was conducted on a system with an Intel(R) Core(TM) i5-9400 CPU @2.90 GHz, × 64-based processor, 32 GB RAM, and MATLAB R2023a. The total computational time to produce the bvp4c dataset and train the ANN was approximately 4.2 min, which is a high level of efficiency of the model. The performance of the ANN was in very good accordance with bvp4c results, with a mean squared error (MSE) as low as $4.06 \times 10^{ - 10}$ and regression coefficients $\,R \approx 0.99999,\,$ as given in Table 5. Even though the ANN employed here has only a single output neuron for a prediction, several solution curves were produced by presenting the trained model with various sets of input parameters or $\,\eta \,$ values. Each of the curves represents one output prediction at a time, allowing the same ANN architecture to model different thermal and flow responses.

Table 5 Quantitative comparison between ANN predictions and bvp4c.

To provide insight into the configuration and performance of the trained ANN-LMA model, the key hyperparameters and training results are presented in Table 6. These comprise the number of layers, neurons, learning rate, training epochs, and MSE and R-values achieved for both velocity and temperature profiles.

Table 6 Summary of ANN-LMA model hyperparameters and performance values.

Conclusions

This study used a NF made of $\,Al_{2} O_{3} - Fe_{3} O_{4} \,$ particles suspended in engine oil to examine fluid flow, heat transfer rate, and thermal performance in a 3D vertical sheet. The study took into account the impacts of Grashof numbers, mixed convection, and MHD. We looked at different values of the volume friction of nanoparticles ($\phi_{1} = \phi_{2}$), Slip parameter ($\beta$), Grashof number ($Gr$), Thermal radiation ($Rd$), magnetic field ($M$), and Heat source ($Q$). To perform solution of the equations numerically bvp4c technique was used via MATLAB. Moreover, the LMA-ANN technique was used for accurate results. These findings are helpful for optimizing heat transfer in such systems as electronic cooling devices, high-performance heat exchangers, and magnetic field-controlled thermal systems, and thus the model has industrial and energy applications. The following are the investigation’s main findings:

• The heat transfer rate is significantly influenced by the concentration of HNF; better heat transfer results are obtained from higher concentrations of hybrid nanoparticles.

• Fluid flow in the vertical 3D sheet and heat transmission are negatively impacted by MHD. Lower slip parameter, less fluid movement, and fewer temperature gradients are all caused by a higher Grashof number.

• The vertical and horizontal velocities are growing with the augmentation of the Grashof number ($Gr$), and weakening with the improvement of the magnetic field ($M$) and slip parameter ($\beta$).

• Furthermore, the temperature profile and fluid velocity are enhanced due to a higher value of the volume slip parameter ($\phi_{1} = \phi_{2}$), temperature profile is also increasing with growing heat source ($\phi_{1} = \phi_{2}$), heat source ($Q$), and magnetic field ($M$).

• The findings can be applied directly to the thermal system optimization of such devices as heat exchangers, engine cooling systems, electronic component cooling systems, solar collectors, and magnetic biomedical devices, where precise thermal management and enhanced heat transfer are essential.

• The information produced by the bvp4c was successfully learned by the ANN model and produced accurate predictions, justifying its deployment in real-time prediction and control of engineering systems.

Limitations

The prevailing inquiries may have the succeeding limitations:

• Only particular mixes of engine oil (base fluid) and nanoparticles ($Al_{2} O_{3} - Fe_{3} O_{4}$) are included in the analysis. One can experiment with various nanoparticle combinations with various base fluids to develop the thermal performance.

• Non-Newtonian effects, turbulence, and temperature-dependent variations in nanoparticle concentrations were not taken into account. By adding these factors, Capabilities may make this investigation more applicable.

• Finite element analysis or computational fluid dynamics simulations might be used as an alternative to the numerical method (the bvp4c) used in this investigation.

• Although we scrutinized the ANN-LMA exercise technique in this study, alternate training methods such as adaptive moment estimation, the Bayesian Regularization algorithm (BRA), or scaled conjugate gradient may also be used to improve the model’s routine and convergence.

• This study assumes a steady, laminar, hybrid nanofluid with constant thermophysical properties. Nanoparticle aggregation, unsteady effects, and experimental validation are not considered here. These aspects will be addressed in future work

Future work

There are several methods to broaden the current inquiry. For instance,

• Different non-Newtonian fluid models may be implemented for thermal transportation analysis in a ternary NF with the same shape by including changing circumstances. Moreover, the physical model can incorporate elements like non-uniform heat sources and the summary of nanoparticle movements.

• It is possible to integrate sophisticated computational structures, like multi-layer managed neural network processes constructed on various neurons.

• Although no stability assessment was carried out, the computational method is confirmed in this study by comparing findings to publicly available data. Future studies might identify the stability criteria for the suggested method to demonstrate how it can be used to more complicated issues.

• Use stochastic numerical computation techniques like Monte Carlo simulations or stochastic differential equation solvers, as well as finite element and finite volume techniques, to handle complicated geometries and moving boundaries.