Artificial neural network prediction of nonlinear radiative heat transfer and entropy generation in tangent hyperbolic tetra hybrid nanofluid flow over a porous wedge

Abstract

An artificial neural network (ANN)-based surrogate modelling approach for forecasting entropy production and heat transfer properties in a tetra-hybrid tangent-hyperbolic nanofluid flow over expanding and contracting wedge surfaces under nonlinear thermal radiation is presented in this work. The generated numerical data are subsequently employed to develop an ANN surrogate model for the rapid prediction of the skin-friction coefficient and local Nusselt number. Numerical analysis demonstrates that increasing the wedge index reduces free-stream acceleration and, in turn, the velocity and temperature fields, while a larger Weissenberg number increases flow elasticity and heat transfer. Nonlinear thermal radiation significantly impacts heating, whereas magnetic field and porous-medium effects increase temperature and entropy generation by impeding convective heat transfer. The developed ANN model achieved excellent predictive accuracy, with test () values of 0.9815 for and 0.9983 for , demonstrating excellent agreement with the numerical solutions. The proposed numerical-ANN framework provides a computationally efficient approach for analyzing complex thermal transport in porous wedge flows, with potential applications in electronic cooling, and high-performance heat process.

Introduction

In comparison with standard fluids, nanofluids are considered to be one of the most effective heat-transfer agents due to their better thermal properties. They are obtained by dispersing very tiny metallic, non-metallic, or carbon-based nanoparticles in a base liquid, thereby resulting in significant enhancements in thermal conductivity and heat-transport capability, as first indicated by Choi [1] in 1995. The promise of such advantages has led to the wide-scale inquiry into their flow and thermal behavior under different physical conditions. Jan et al. [2] investigated the convective behavior of the Reiner–Philippoff nanofluid over a curved stretching surface and found that magnetic strength, surface curvature, and radiation have a dramatic effect on wall shear stress and heat transfer. Farooq et al. [3] studied the MHD nanofluid flow over an inclined stretching surface and added porosity, radiation, and internal heat generation or absorption in their model. They concluded that magnetic and porosity parameters restrain velocity, while Grashof, Eckert, and radiation parameters increase temperature. In a bid to further improve thermophysical performance, scientists have developed hybrid nanofluids, in which two different nanoparticles are dispersed in a base fluid. Compared with a single-nanoparticle system, these exhibit improved stability, thermal conductivity, and heat-transfer rates. Kodi et al. [4] analyzed how Hall currents, Joule heating, Brownian motion, and thermophoretic forces combined with chemical reactions affect hybrid nanofluids under no-slip conditions. On the other hand, Krishna and Chamkha [5] have informed that velocity and flow structure are changed drastically by the parameters of slip, Hall, and buoyancy in the case of hybrid nanofluids flowing over partially ionized surfaces illuminated by solar radiation. In fact, the latest developments have led to the invention of tri-hybrid nanofluids, in which three different types of nanoparticles are evenly dispersed in a base liquid to achieve higher thermal efficiency, improved radiative response, and enhanced porous-medium interactions. Also, their uses are not limited to biomedical cooling; they may also include pharmaceutical processes and bioelectrochemical systems. Alqudah et al. [6] researched the effect of ternary nanoparticles in Carreau fluid on thermal transport over a 3D wedge and found that the nanoparticles have a substantial impact on thermal transport. Ayub et al. [7] worked with fuzzy hybrid nanofluids under inclined magnetic fields, while Al-Kouz et al. [8] found that magnetic and radiative effects considerably raise the heat transfer rates in cross-ternary hybrid nanofluids flowing over a vertical cylinder. The next logical step in this evolution is tetra-hybrid nanofluids, in which four different nanoparticles are uniformly dispersed in a base fluid. These designed colloidal mixtures provide more control over heat, electrical, and mass transport properties and can be customized for very demanding heat-transfer conditions.

The tetra-hybrid nanoparticle combination used in this study was selected to benefit from the complementary thermophysical characteristics of its constituent parts. Gold (Au) has an exceptionally high thermal conductivity, enabling rapid heat transfer, whilst titanium dioxide () has excellent thermal and chemical stability at high temperatures. Aluminum oxide () enhances convective heat-transfer capability due to its favorable thermal properties and broad technical use, whilst silicon dioxide () improves solution stability and helps maintain uniform nanoparticle dispersion. The synergistic interaction of these four nanoparticles provides a balanced improvement of thermal conductivity, stability, and transport characteristics, making tetra-hybrid nanofluids attractive for advanced thermal management systems, electronic cooling devices, porous heat exchangers, renewable energy systems, and other high-performance heat-transfer applications. Sajid et al. [9], investigating tetra-hybrid nanofluid flow over a Riga wedge, observed that an increase in the Weissenberg and Casson parameters results in a decrease in velocity. In contrast, temperature decreases under heat generation, radiation, and variable conductivity. Mehmood et al. [10] were focusing on unsteady thin-film flow of a power-law tetra-hybrid nanofluid with slip effects, and they discovered that the temperature goes up as the solid-volume fraction is increased and at the same time the velocity is lowered.

Out of various non-Newtonian fluid models, the tangent-hyperbolic fluid is the most preferred one to describe shear-thinning materials that show both viscous and elastic responses. It is quite an accurate representation of the flow of emulsions, biological liquids, polymeric solutions, food slurries, and textile suspensions. Pop and Ingham [11] were very much convinced about the importance of tangent-hyperbolic rheology for accurately capturing fluid behavior. Abdal et al. [12], in their research on time-dependent tangent-hyperbolic nanofluid flow over a Riga wedge, argued that changes in activation energy, internal heat generation, and Hartmann number have a very strong effect on mobility. Nadeem et al. [13] focused on transient tangent-hyperbolic hybrid nanofluid flow over exponentially stretched surfaces and highlighted fuzzy parameters as very important. Choudhary et al. [14] explored tangent-hyperbolic flow in a diverging permeable channel with suction/blowing and heat generation. Aisha et al. [15] provided evidence that a Weissenberg number in the tangent-hyperbolic nanofluid flow over a permeable wedge lessens heat transfer and skin friction, whereas the wedge angle accelerates energy dissipation.

Processes for thermal management, such as metal-plate cooling, diffuser design, and petroleum recovery, require the most accurate modeling of heat and mass transport. Kumar et al. [16] used the Buongiorno model to study wedge flows with nanoparticles of different shapes under the influence of magnetic fields and a heat source, and thus demonstrated that non-spherical nanoparticles notably increase the Nusselt number. Mahariq et al. [17], on the other hand, demonstrated that ternary hybrid nanofluids improve the thermal performance of dusty blood flow. Nonlinear thermal radiation becomes dominant in high-temperature systems and cannot be accurately captured by linearized Stefan-Boltzmann formulations. Li et al. [18] were of the opinion that nonlinear radiation is the major reason for the changes in heat-transfer patterns, while Razaq et al. [19] demonstrated its significant impact on bioconvective wedge flows with convective boundary conditions. Alqudah et al. [6] went beyond these discoveries with 3D Carreau ternary-hybrid nanofluids under magnetic and radiative effects. Shakir et al. [20] compared linear and nonlinear radiation models for tetra-hybrid nanofluid flow over Riga wedges and found significant differences in temperature distribution, heat absorption, and thermal boundary layer thickness.

Entropy generation is one of the most important parameters for evaluating system irreversibility. Thabet et al. [21] pointed out that the main sources of entropy in magnetized nanofluids are frictional and thermal losses. Ali et al. [22] demonstrated that Carreau tri-hybrid nanofluids under radiation exhibit competing effects of enhanced heat transfer and increased irreversibility. Mahanta et al. [23] investigated electromagnetic Darcy–Forchheimer trihybrid nanofluid flow and coupled entropy generation with regression and sensitivity analyses. Their results demonstrated that statistical validation techniques effectively identified the dominant transport parameters governing heat transfer and entropy production, providing a reliable framework for optimizing thermodynamic performance. Zhang et al. [24] explored the Casson nanofluid flow over nonlinear wedge surfaces and found that buoyancy, radiation, and the Eckert number significantly contribute to the reduction of surface drag and the alteration of heat-transfer behavior. Sharma et al. [25] showed that nonlinear radiation increases both entropy generation and the Bejan number based on ANN-based entropy predictions. One of the most reliable recent advances in analyzing nonlinear transport phenomena is the rise of machine learning, especially artificial neural networks (ANNs). Raissi et al. [26] showed that neural networks can integrate physical laws directly into their structure, thus serving as highly efficient predictors of complex flow fields. Khan et al. [27] used bvp4c-generated datasets to train their ANN models to study ternary hybrid nanofluid flow over wedges and achieved high predictive accuracy. Parida et al. [28] predicted the skin friction and Nusselt number in electro-magnetohydrodynamic penta-hybrid nanofluid flow with entropy optimization by employing a multilayer artificial neural network and MATLAB bvp4c solver. Their results showed an excellent prediction accuracy and demonstrated the effectiveness of ANN as a computationally efficient surrogate for complex nonlinear thermal transport problems. Kumar et al. [29] investigated the thermal behavior of a non-Newtonian Maxwell ternary hybrid nanofluid over a cylindrical surface by combining the MATLAB bvp4c solver with artificial neural networks. Their study demonstrated that ANN provides a highly accurate prediction of heat-transfer characteristics while substantially reducing the computational effort required for repeated numerical simulations. Yaseen et al. [30] proposed a numerical and artificial intelligence approach for the flow of ternary hybrid nanofluid in an anisotropic porous channel. They successfully trained an artificial neural network to forecast the skin-friction coefficient and Nusselt number with high accuracy after generating numerical solutions using the MATLAB bvp4c solver, demonstrating the effectiveness of ANN as a computationally efficient surrogate for nonlinear transport problems in porous media. Further, the effectiveness of soft computing was demonstrated by Bartwal et al. [31], who correctly forecasted the Nusselt number for the ternary hybrid nanofluid flow over a rotating disk by combining the numerical data from MATLAB bvp4c with artificial neural networks and fuzzy particle swarm optimization. The results indicated that the ANN-based surrogate models were better suited to predicting complicated nonlinear heat transfer problems.

Motivation

• Advanced heat systems such as cooling channels, porous reactors, biomedical devices, and solar-assisted technologies require fluids with excellent heat-transfer performance, high stability, and controllable flow behavior.

• Traditional nanofluids and also hybrid nanofluids may not be able to deliver the necessary thermal enhancement under complicated scenarios that include magnetic fields, nonlinear radiation, porous resistance, and Non-Newtonian rheology.

• Tetra-hybrid nanofluids, when combined with tangent-hyperbolic rheology and nonlinear radiative heating, offer a powerful route to achieve enhanced thermal performance, reduced irreversibility, and improved flow control-yet their behavior over wedge geometries remains largely unexplored.

• There is a need for accurate and computationally efficient prediction of entropy-controlled heat transfer in complex tetra-hybrid non-Newtonian thermal systems.

Objective

• To conceive and research the behaviour of a tangent-hyperbolic tetra-hybrid nanofluid () model over the stretching/shrinking wedge while considering a magnetic field, porous medium resistance, nonlinear thermal radiation, heat source/sink, and slip effects.

• To examine the velocity, temperature, entropy generation, and Bejan number responses to key physical parameters such as

• To compute and compare skin-friction coefficient and local Nusselt number, entropy generation and Bejan number for stretching and shrinking flows.

• To develop and train an ANN-based surrogate model capable of accurately predicting the skin-friction coefficient and Nusselt number using high-fidelity numerical data generated by the MATLAB bvp4c solver.

Novelty

• This study develops a unified tangent-hyperbolic tetra-hybrid nanofluid wedge-flow model that simultaneously incorporates nonlinear thermal radiation, porous-medium resistance, magnetic effects, slip mechanisms, and heat generation/absorption, enabling the investigation of their coupled influence on momentum, heat transfer, and entropy production.

• The work integrates tetra-hybrid thermophysical property relations with the nonlinear Rosseland radiation model, enabling examination of the interaction between enhanced nanoparticle-assisted thermal transport and nonlinear radiative heat transfer within a single formulation.

• A comprehensive analysis of entropy generation and the Bejan number is performed for the proposed tetra-hybrid non-Newtonian wedge-flow configuration, providing insights into thermal and frictional irreversibilities that remain limited in the existing literature.

• An ANN-based surrogate model is developed using high-fidelity bvp4c data to predict wall shear stress and heat-transfer characteristics, providing a computationally efficient prediction framework for complex thermal-fluid systems.

In addition to the numerical investigation, artificial neural networks (ANNs) are employed as surrogate models to capture the complex nonlinear relationship between the governing physical parameters and the resulting thermal-fluid characteristics. Using high-fidelity numerical data generated through the MATLAB bvp4c solver, the ANN framework is trained to predict the skin-friction coefficient and Nusselt number with high accuracy. The proposed numerical-AI approach combines the physical reliability of numerical simulations with the computational efficiency of machine learning, providing a rapid prediction tool for complex thermal transport systems involving tetra-hybrid tangent-hyperbolic nanofluids.

Mathematical model development

Interpretation of geometry and flow

A tetra-hybrid nanofluid composed of nanoparticles uniformly dispersed in water are considered in this study. The behavior of this high-performance nanofluid is examined under the following physical assumptions:

• The flow is assumed to be steady and incompressible, and it is assumed that the flow is affected by a magnetic field that is applied externally.

• The motion of the tetra-hybrid nanofluid is over a wedge-shaped surface, which is either stretching or shrinking, thereby controlling the boundary layer development.

• The model features heat generation inside the body, heat absorption, and thermal radiation, thus offering a quite realistic thermal environment for the study.

• The variable stands for the velocity of the wedge surface that can cause the nanofluid to stretch or shrink.

• The free-stream velocity is given by , where a is a positive constant. The wedge parameter m is responsible for the surface motion: m < 0: shrinking wedge, m> 0: stretching wedge, m = 0: static wedge.

• The wedge-flow similarity solution is possible only if the wedge index is within the range .

• T∞ is the temperature of the atmosphere, and Tw is the temperature of the melting surface, with the requirement that

• The wedge and flow field physical regions are shown in Fig. 1.

Fig. 1.

Boundary layer flow pattern across a wedge surface with a expanding and b contracting wedge configurations

The pseudoplastic tangent hyperbolic non-Newtonian fluid's Cauchy stress tensor is defined as. [32]

1

where m is the power law rheological index, which provides the additional stress tensor, Γ denotes time-reliant constant material, shows the shear rate for zero viscosity, and provides the shear rate for infinite viscosity and is represented by:

2

where the term is the invariant second strain tensor. Taken and in Eq. (1), since it is challenging to assess the shear thinning impact and the flow's infinite viscosity shear rate at the same time, Eq. (1) becomes

3

The governing continuity, momentum, and energy equations are adopted from the tangent-hyperbolic nanofluid flow formulations reported in Refs. [32–35]. In the present study, these equations are extended by incorporating tetra-hybrid nanofluid thermophysical property relations and nonlinear thermal radiation effects.

4

5

To account for thermal radiation effects, the Rosseland diffusion approximation is employed. Under this assumption, the radiative heat flux is expressed as

where and denote the Stefan -Boltzmann constant and the mean absorption coefficient, respectively. Retaining the nonlinear temperature dependence yields

6

Substituting the above expression into the energy equation leads to the nonlinear thermal radiation model used in the present study.

7

The appropriate boundary conditions are introduced to the flow dimensions [32]:

8

In this context, the terms , , , , denote the slip in velocity, thermal slip, and surface velocities, respectively. It is important to note that the second boundary condition in Eq. (7) represents the slip velocity condition. This condition describes a situation in which the fluid allows some movement, or "slip," at the boundary because it does not adhere strictly to the surface. The use of slip phenomena in microfluidics, transport through porous media, tribology, biomimetics, nanotechnology, and surface engineering must be emphasized in this context.

The dimensionless variables are first represented in the following forms before the solution method is developed [32, 35].

The similarity transformations are.

, .

, , .

The underlying governing Eqs. (4) through (7) can be transformed using the previously mentioned methods to yield,

9

10

Along with the boundary conditions are.

where .

f(0) = S, where at η = 0

11

The boundary conditions represent the physical behavior of the flow at the wedge surface and in the free stream. The velocity-slip condition accounts for partial slip between the fluid and the wedge surface, while the thermal-slip condition represents finite thermal resistance at the fluid–solid interface. The suction/injection parameter S models fluid flow through the porous wedge surface. As , the velocity and temperature approach the ambient free-stream conditions, indicating that the influence of the wedge surface vanishes outside the boundary layer.

The following dimensionless parameters are involved in the equations displayed above:

M (Magnetic parameter), (Porous medium parameter) We(Weissenberg number) Rd(Radiation parameter)

The engineering quantities are represented as

12

The wall shear stress and surface heat flux are represented by the symbols and respectively. Formally, these quantities are described as

13

Equation (10) can be expressed in dimensionless form.

14

15

The tetra-hybrid thermophysical relations are constructed by sequentially extending the conventional Maxwell and mixture models to account for the combined contributions of four distinct nanoparticles (Au,,,) dispersed in water.

Analysis of entropy generation

Entropy is the irreversible thermal energy caused by processes that take place while performing fluid movement. According to Bejan [26], the local entropy generation rate per unit volume can be expressed using the following relation:

16

The entropy generation expression given in Eq. (16) consists of contributions from two distinct sources of thermodynamic irreversibility. The first term represents thermal irreversibility, which arises from finite temperature gradients during heat transfer. The remaining terms account for fluid-friction irreversibility, including viscous dissipation, magnetic-field-induced Joule heating, and porous-medium resistance. Therefore, an increase in the temperature gradient intensifies thermal entropy generation, whereas stronger viscous, magnetic, and porous effects increase entropy production through frictional mechanisms. The relative contribution of these sources of irreversibility is subsequently quantified using the Bejan number.

Using the similarity transformation, the dimensionless form is

17

where denotes the temperature difference parameter and is the Brinkman number, and represent the entropy generation rate.

We introduce the Bejan Number, Be, the ratio of entropy generation via heat transfer to the total amount of entropy generated in the system is known as the Bejan number:

18

The Bejan number quantifies the relative contribution of heat-transfer irreversibility to total entropy generation. Values of (Be > 0.5) indicate that thermal irreversibility is the dominant source of entropy generation, whereas (Be < 0.5) signifies that fluid-friction irreversibility predominates. When (Be = 0.5), both heat-transfer and fluid-friction irreversibilities contribute equally to the total entropy production.

Numerical procedure

Subject to the boundary conditions in Eq. (7), Eqs. (8) and (9) constitute a coupled nonlinear system of ordinary differential equations. To facilitate the numerical solution, the governing equations are first transformed into an equivalent system of first-order ordinary differential equations.

The resulting boundary-value problem is solved using MATLAB's built-in bvp4c solver, a well-established collocation-based boundary-value solver, which is employed here as a numerical solution technique rather than as a novel methodology. An initial graded mesh generated using a hyperbolic-tangent distribution is adopted to provide a higher concentration of grid points near the wall, where the velocity and temperature gradients are significant. Unless otherwise stated, an initial mesh of 701 grid points is used over the computational domain.

The numerical computations are carried out using a relative tolerance and an absolute tolerance. During the solution process, bvp4c automatically performs adaptive mesh refinement to satisfy the prescribed error tolerances.

After each iteration, we check whether the velocity and temperature have effectively reached zero at the far boundary (i.e., the solution behaves as expected far away from the surface). The thermophysical properties of the base fluid and nanoparticles used in the present study are presented in Table 1 and the effective thermophysical property relations for the tetra-hybrid nanofluid are presented in Table 2.

Table 1.

The thermal properties of nanosolid (Au,,,) and base fluid (Water) [23, 36]

Properties	Water	Au
	997.1	19,282	4250	2220	3970
	4,179	129	686.2	745	765
	0.613	318	8.953	1.4	40
		4.1 × 106	2.4 × 106

Table 2.

Physical thermofluid features for tetra hybridized nanofluid [23, 36]

Properties	Tetra Hybrid nanofluid
Dynamic Viscosity
Density
Heat capacitance
Electrical conductivity
Thermal conductivity

If these conditions are not satisfied, the computational domain is automatically extended, and the previously converged solution is used as the initial guess for subsequent computations. This continuation strategy improves the robustness and convergence of the numerical solution for strongly nonlinear parameter combinations.

To ensure that the computed results are independent of the computational mesh, a grid-independence study was performed by considering several initial mesh densities while keeping all physical parameters fixed. The wall shear parameter and local Nusselt number exhibited negligible variation with further mesh refinement, confirming that the numerical solution is grid-independent. Based on this study, an initial mesh of 701 grid points was used for all subsequent computations to strike an appropriate balance between numerical accuracy and computational efficiency. The corresponding results are presented in Table 3.

Table 3.

Grid independence study for the present bvp4c solution

Initial mesh points	f''(0)	θ(0)
201	0.568842	0.143277	0.568289	0.429831
401	0.568842	0.143277	0.568289	0.429831
601	0.568842	0.143277	0.568289	0.429831
701	0.568842	0.143277	0.568289	0.429831
801	0.568842	0.143277	0.568289	0.429831
1001	0.568842	0.143277	0.568289	0.429831
1201	0.568842	0.143277	0.568289	0.429831

The equivalent first-order system together with the transformed boundary conditions is given as follows:

The first-order-based system was obtained

and the accompanying boundary conditions:

Numerical validation

To verify the accuracy and reliability of the present numerical procedure, the MATLAB bvp4c solver was first validated using the classical Falkner–Skan wedge-flow benchmark reported by Yacob et al. [37]. For this purpose, the well-established Falkner–Skan boundary-layer equations governing wedge flow, together with the corresponding boundary conditions.

As were solved using the same numerical procedure adopted in the present study. The computed values of the wall shear parameter (f''(0)) were then compared with the benchmark results reported by Yacob et al. [37], as presented in Table 3. The excellent agreement between the present computations and the published benchmark values confirms the accuracy and robustness of the bvp4c implementation. Following this verification, the validated numerical procedure was employed to solve the proposed tangent-hyperbolic tetra-hybrid nanofluid model. The excellent agreement observed in Table 3 confirms the accuracy and reliability of the present MATLAB bvp4c implementation for solving boundary-layer flow problems.

Artificial neural network (ANN) methodology

To develop a computationally efficient surrogate model for predicting wall transport quantities, a feed-forward Artificial Neural Network (ANN) was employed using numerical data generated by the MATLAB bvp4c solver. The ANN was trained to establish the nonlinear relationship between the governing physical parameters and the corresponding output responses, namely the skin-friction coefficient and local Nusselt number .

The dataset consisted of 2000 numerical samples covering the complete range of physical parameters considered in the present investigation. Prior to training, all input variables were standardized using the StandardScaler technique to improve numerical stability and accelerate convergence. The available dataset was randomly divided into training (70%), validation (15%), and testing (15%) subsets. The training set was used for network optimization, the validation set was employed to monitor model generalization and prevent overfitting, and the testing set was reserved for independent performance evaluation (as presented in Table 4).

Table 4.

Validation of the present MATLAB bvp4c solution for the classical Falkner–Skan wedge-flow problem through comparison with the benchmark results of Yacob et al. [37]

Wedge parameter, (m)	Yacob et al [37]	Present bvp4c
0.00000	0.469600	0.469600
0.09091	0.654979	0.654994
0.20000	0.802125	0.802126
0.33333	0.927653	0.927680
1.00000	1.232588	1.232588

The ANN architecture comprised an input layer, two hidden layers containing 64 and 32 neurons, respectively, and a single output neuron. The Rectified Linear Unit (ReLU) activation function was adopted in the hidden layers owing to its ability to capture complex nonlinear relationships, whereas a linear activation function was employed in the output layer for continuous-value prediction. Network training was performed using the Adam optimization algorithm with a learning rate of 0.001 and a batch size of 128. The mean squared error (MSE) was selected as the loss function. To enhance model robustness and mitigate overfitting, an early-stopping strategy with a patience value of 20 epochs was implemented, and the network parameters corresponding to the minimum validation loss were retained. The complete ANN architecture and training parameters adopted in the present study are summarized in Table 5.

Table 5.

ANN architecture and training parameters

Parameter	Value
ANN type	Feed-forward neural network
Dataset size	2000 samples
Training data	70%
Validation data	15%
Testing data	15%
Hidden layers	2
Neurons per layer	64, 32
Hidden activation	ReLU
Output activation	Linear
Optimizer	Adam
Learning rate	0.001
Batch size	128
Loss function	Mean Squared Error (MSE)
Early stopping	Patience = 20 epochs

The predictive capability of the ANN model was assessed through regression analysis, validation performance curves, and error-histograms. In addition to the coefficient of determination, the mean absolute error (MAE) and root mean square error (RMSE) were employed to quantify the prediction accuracy of the developed ANN framework. The overall prediction performance of the ANN model for the skin-friction coefficient and local Nusselt number is summarized in Table 6.

Table 6.

Machine learning prediction performance for and

Target	Best epoch	Best Val loss	-Test	MAE Test	RMSE Test
	463	0.000939	0.981477	0.023816	0.030853
	406	0.518495	0.998281	0.584761	0.868988

Results and discussion

In this study, the baseline values of the governing parameters are Pr=6.2, n=0.001, We=0.10, m=0.3, M=0.1, K=0.4, Br=0.001, , , , and λ =0.2. During the parametric study, only one governing parameter was varied while the remaining parameters were maintained at their baseline values. The selected parameter ranges represent moderate operating conditions commonly reported for tangent-hyperbolic nanofluid boundary-layer flows over wedge geometries, ensuring physically realistic and numerically stable solutions. These ranges were selected to capture the influence of fluid elasticity (We), wedge geometry , magnetic-field strength (M), porous-medium permeability , and nonlinear thermal radiation under practically attainable operating conditions without violating the boundary-layer assumptions. The selected ranges are consistent with previous investigations and are representative of engineering applications such as porous heat exchangers, electronic cooling, thermal management systems, and energy transport devices. The composition of the tetra-hybrid nanofluid is illustrated in Fig. 2. The overall computational procedure adopted in the present study is illustrated in Fig. 3 and the structure of the artificial neural network employed in the present study is illustrated in Fig. 4.

Fig. 2.

Tetra Hybrid Nanofluid

Fig. 3.

Algorithm flow chart

Fig. 4.

Structure of artificial neural networks

The numerical trends obtained in the present investigation are consistent with the previously published tangent-hyperbolic wedge-flow studies, particularly the baseline model reported by Hafeez et al. [33]. Similar trends in the velocity, temperature, wall shear stress, and heat-transfer characteristics are observed, confirming the reliability of the present numerical formulation. However, unlike the baseline study, which considered a hybrid nanofluid with melting heat transfer, the present work incorporates a tangent-hyperbolic tetra-hybrid nanofluid, nonlinear thermal radiation, entropy generation, Bejan number analysis, and an artificial neural network (ANN)-based surrogate model. These additional physical mechanisms provide a more comprehensive understanding of coupled momentum and thermal transport while enabling rapid and accurate prediction of the engineering quantities of interest.

According to Fig. 5, the Weissenberg number visibly increases the velocity of the tetra-hybrid nanofluid in both stretching and shrinking cases. The velocity curves shift upward with increasing We, indicating enhanced momentum transport within the boundary layer. This behaviour is attributed to the increasing influence of fluid elasticity as the fluid relaxation time becomes more significant relative to the characteristic flow time. As the Weissenberg number increases, the elastic memory of the fluid becomes more pronounced, allowing it to retain deformation for a longer duration. Consequently, momentum is transported more effectively away from the wall, increasing the momentum boundary-layer thickness. The upward displacement of the velocity profiles confirms that larger values of We facilitate fluid motion.

Fig. 5.

Effect of the Weissenberg number We on the velocity profile

Figure 6 shows that the temperature continues to increase for both flow configurations as the Weissenberg number (We) increases. A higher Weissenberg number indicates a longer relaxation time, which in turn results in weaker convective heat transfer from the surface. Since the fluid relaxes more slowly, heat is trapped within the boundary layer, causing it to thicken and increasing the fluid temperature. Owing to the enhanced elastic characteristics of the tetra-hybrid nanofluid, a larger We signifies stronger elastic memory, enabling the fluid to retain its deformation for a longer duration. Consequently, thermal energy is transported away from the wall less effectively, resulting in greater heat retention and an increase in the thickness of the thermal boundary layer. The rise of the temperature curves indicates that stronger elastic effects promote thermal energy storage within the fluid.

Fig. 6.

Effect of the Weissenberg number We on the temperature profile

According to Fig. 7, the velocity decreases as the wedge index (m) increases. A larger wedge index reduces the external free-stream acceleration along the wedge, thereby supplying less momentum to the boundary layer. This trend results from changes in the pressure-gradient characteristics of the external flow. The wedge index governs the pressure-gradient distribution along the wedge surface. As m increases, the pressure gradient is altered, reducing momentum transport within the boundary layer and consequently decreasing the fluid velocity. However, the reduction is comparatively smaller for the stretching case because the outward motion of the surface continuously supplies momentum to the adjacent fluid, partially compensating for the reduced free-stream acceleration. Overall, increasing m reduces the flow intensity throughout the boundary layer.

Fig. 7.

Effect of the wedge index m on the velocity profile

According to Fig. 8, the corresponding temperature profiles decrease as the wedge index increases. The weakening of the free-stream accelerating force enhances convective cooling at the wall, thereby enabling more efficient heat removal from the near-wall region. This results in lower temperatures across the boundary layer. The observed variation is closely associated with changes in convective heat transport due to the wedge geometry. As the wedge index increases, the pressure-gradient distribution modifies the momentum boundary layer, thereby enhancing the wall temperature gradient and promoting more efficient heat transfer from the surface to the fluid, thereby reducing the thermal boundary-layer thickness. The cooling effect is more pronounced in the stretching case because of outward advection. The fall of the curves indicates that the increase in leads to a decrease in the thermal field.

Fig. 8.

Effect of the wedge index m on the temperature profile

Figure 9 illustrates the effect of the magnetic parameter (M) on the velocity distribution for both stretching and shrinking wedge surfaces. It is observed that the velocity increases slightly with increasing values of (M) throughout the boundary layer. This behaviour is consistent with the present mathematical formulation, in which the magnetic interaction is coupled with the free-stream velocity, and wedge geometry. Under the selected operating conditions, the magnetic field enhances the effective momentum transfer from the external flow to the fluid, resulting in higher velocity profiles. Consequently, the boundary layer becomes slightly thicker, and the velocity increases with increasing (M). The effect is more pronounced for the shrinking surface, where the magnetic interaction contributes to stabilizing the near-wall flow.

Fig. 9.

Effect of the magnetic parameter M on the velocity profile

The temperature response to the magnetic field is represented in Fig. 10, which shows that temperature increases with increasing magnetic parameter (M). For the present mathematical model and the selected range of governing parameters, increasing M enhances the thermal energy retained within the boundary layer, resulting in higher fluid temperatures for both stretching and shrinking cases. This increases the thickness of the thermal boundary layer and promotes heat retention within the fluid. The thermal response is controlled by the magnetic damping effect on convective heat transport. The magnetic parameter characterizes the interaction between the applied magnetic field and the electrically conducting nanofluid. As M increases, the magnetic interaction modifies the thermal transport process, causing more thermal energy to remain within the boundary layer and thereby increasing the fluid temperature.

Fig. 10.

Effect of the magnetic parameter M on the temperature profile

Figure 11 explains the steady rise of temperature with the radiation parameter (Rd) for both wall movements. Intensified radiation brings more heat energy to the boundary layer and thus it expands. The observed enhancement is primarily due to the increased contribution of thermal radiation to the energy transport process. The radiation parameter represents the contribution of thermal radiation to the overall heat-transfer process. As (Rd) increases, more radiative energy is absorbed by the fluid, which supplements thermal energy transport within the boundary layer and increases the thermal energy stored within the fluid. Consequently, the thermal boundary-layer thickness increases, leading to higher fluid temperatures. This effect is more pronounced for shrinking flows, where the inward movement of the wall compresses the thermal layers and enhances heat retention. Furthermore, the high thermal conductivity of the tetra-hybrid nanoparticles further strengthens this trend.

Fig. 11.

Effect of the radiation parameter Rd on the temperature profile

Figure 12 shows the velocity changes at different permeability levels. When the permeability increases, the porous medium resistance decreases, and the fluid can move more freely. Hence, velocities increase, and the momentum boundary layers become thicker for both the stretching and shrinking flows. This phenomenon arises because the porous medium modifies the resistance encountered by the flowing fluid. As increases, the resistance offered by the porous medium decreases, allowing the fluid to move more freely and thereby enhancing momentum transport within the boundary layer. Consequently, the fluid velocity increases and the momentum boundary-layer thickness becomes larger.

Fig. 12.

Effect of the porous parameter on the velocity profile

Figure 13 shows that the temperature increases steadily with the increase in . The increase in temperature is directly related to the influence of porous-medium resistance on heat transport. The permeability parameter characterizes the ease with which the fluid passes through the porous medium. As increases, the resistance offered by the porous matrix decreases, allowing the fluid to move more freely. For the present model and the selected range of governing parameters, this enhanced flow is accompanied by an increase in the thermal energy retained within the boundary layer, resulting in higher fluid temperatures. Consequently, the thermal boundary-layer thickness increases for both stretching and shrinking cases.

Fig. 13.

Effect of the porous parameter on the temperature profile

Figure 14 underlines the prominent effect of the Brinkman number (Br) on temperature. A higher Br increases viscous dissipation, and therefore, more mechanical energy is converted into internal heat. The heat generated by viscous dissipation increases the fluid temperature and enlarges the thermal boundary layer, demonstrating the significant contribution of viscous heating to the overall thermal behavior as Br increases. The increase in temperature originates from the intensified viscous dissipation within the flow field. The Brinkman number represents the ratio of the heat generated by viscous dissipation to the heat transferred by conduction across the boundary layer. As Br increases, frictional heating becomes more pronounced, supplying additional thermal energy to the fluid while reducing the effectiveness of conductive heat removal. Consequently, the thermal boundary layer becomes thicker, leading to higher temperature profiles for both stretching and shrinking cases. The increased separation between successive temperature profiles further confirms that the influence of viscous dissipation remains consistent throughout the computational domain.

Fig. 14.

Effect of the Brinkman number Br on the temperature profile

The impact of the temperature-ratio parameter () is depicted in Fig. 15. For stretching surfaces (λ = + 0.2), the rise in results in an increase in wall temperature and, when combined with outward advection, leads to higher temperature profiles occur. In contrast, shrinking surfaces (λ = - 0.2) demonstrate a decline in temperature due to the fact that the inward wall movement intensifies the cooling, and the fluid layers are compressed. The opposite behaviors in these two cases highlight the role of wall movement in controlling heat transfer. The thermal behavior is governed by the wall-to-fluid temperature difference represented by the temperature-ratio parameter. It represents the relative difference between the wall and ambient temperatures. A larger value strengthens the thermal driving force at the wall, resulting in higher temperatures in the stretching case, whereas inward wall motion modifies the thermal boundary-layer development, resulting in the lower temperature profiles observed in the shrinking case.

Fig. 15.

Effect of the wall temperature parameter on the temperature profile

Figure 16 illustrates the influence of the magnetic parameter (M) on the entropy generation () for both stretching and shrinking configurations. It is observed that entropy generation increases progressively with increasing magnetic parameter. It is observed that entropy generation increases progressively with increasing magnetic parameter. The increase in entropy generation originates from the enhanced magnetic and viscous irreversibilities within the boundary layer. Thus, the increase in viscous and magnetic irreversibilities favours the conversion of mechanical energy to thermal energy, leading to an increase in total entropy generation. It is evident that entropy generation is more pronounced near the wall, where the velocity and temperature gradients are largest, and it gradually decreases away from the surface as boundary-layer effects fade. The smooth variation of the entropy-generation surface also indicates that the magnetic field consistently enhances thermodynamic irreversibility throughout the studied flow domain in both the stretching and shrinking cases.

Fig. 16.

Effect of the magnetic parameter M on the entropy-generation profile

Figure 17 illustrates the influence of the magnetic parameter (M) on the Bejan number () for both stretching and shrinking configurations. It is observed that the Bejan number increases progressively with increasing magnetic parameter. The variation in the Bejan number is governed by the redistribution of thermal and fluid-friction irreversibilities within the boundary layer. The magnetic field modifies the balance between heat-transfer and fluid-friction irreversibilities through the Lorentz force. As the magnetic parameter increases, the distribution of irreversibilities shifts toward thermal irreversibility, resulting in a larger contribution of heat-transfer irreversibility to the total entropy generation and, consequently, an increase in the Bejan number. The increase in the Bejan number is more pronounced near the wall, where the thermal gradients are strongest, whereas it gradually approaches a nearly uniform value away from the surface as the boundary-layer effects diminish. The smooth variation of the Bejan-number surface further demonstrates that the magnetic parameter consistently influences the thermodynamic irreversibility characteristics throughout the investigated flow domain for both stretching and shrinking cases.

Fig. 17.

Effect of the magnetic parameter M on the Bejan-number profile

Figure 18 illustrates the influence of the temperature-ratio parameter () on the entropy generation () for both stretching and shrinking configurations. It is observed that entropy generation increases progressively with increasing values of . The increase in entropy generation is associated with the larger thermal potential difference between the wall and the ambient fluid. The temperature-ratio parameter governs the thermal potential difference between the wall and the ambient fluid. An increase enhances the temperature gradient at the wall, thereby increasing conductive heat transfer and thermal irreversibility within the boundary layer. Consequently, a greater amount of thermal energy is dissipated via irreversible heat transfer, leading to higher entropy generation. The increase in entropy generation is more pronounced near the wall, where temperature gradients are highest, and it gradually decreases away from the surface as the thermal boundary-layer effects diminish. The smooth variation of the entropy-generation surface further suggests that increasing α₁ consistently enhances thermodynamic irreversibility throughout the investigated flow domain.

Fig. 18.

Effect of the temperature-ratio parameter on the entropy-generation profile

Figure 19 illustrates the influence of the temperature-ratio parameter () on the Bejan number () for both stretching and shrinking configurations. It is observed that the Bejan number increases progressively with increasing values of . The variation in the Bejan number is governed by the redistribution of thermal and fluid-friction irreversibilities within the boundary layer. The temperature-ratio parameter governs the thermal potential difference between the wall and the ambient fluid. An increase in strengthens the wall temperature gradient, thereby enhancing heat-transfer irreversibility relative to the total entropy generation. Therefore, the relative contribution of thermal irreversibility is more dominant and the values of the Bejan number are higher. The increase in the Bejan number is more pronounced near the wall, where the thermal gradients are strongest, while it gradually approaches a nearly uniform value away from the surface as the boundary-layer effects weaken. The smooth variation of the Bejan-number surface further demonstrates that increasing consistently influences the thermodynamic irreversibility characteristics throughout the investigated flow domain for both stretching and shrinking cases.

Fig. 19.

Effect of the temperature-ratio parameter on the Bejan-number profile

It is clear from Fig. 20 that entropy generation increases with the porous-medium parameter (). The increase in enhances the Darcy–Forchheimer resistance and consequently increases frictional heating and viscous dissipation. Physically, the porous-medium parameter characterizes the resistance offered by the porous matrix to fluid motion. As increases, the fluid experiences greater drag and requires additional mechanical energy to overcome the porous resistance. This energy is dissipated as heat through viscous friction, thereby increasing thermodynamic irreversibility and the total entropy generation. Consequently, both the stretching and shrinking flows evolve toward a more dissipation-controlled regime. The smooth variation of the entropy-generation surface further confirms that this behaviour persists throughout the investigated flow domain.

Fig. 20.

Effect of the porous-medium parameter on the entropy-generation profile

The corresponding Bejan number trend in Fig. 21 reflects this behaviour. It is observed that the Bejan number increases with the porous-medium parameter () for both stretching and shrinking cases. This evolution is governed by the influence of porous-medium resistance on the irreversibility balance. The porous medium parameter increases the resistance of the porous matrix, thereby modifying the balance between heat-transfer and fluid-friction irreversibilities. As increases, the relative contribution of heat-transfer irreversibility to the total entropy generation becomes larger, leading to higher values of the Bejan number. Consequently, the thermodynamic behaviour gradually shifts toward a heat-transfer-dominated irreversibility regime. The smooth variation of the Bejan-number surface further confirms that this behaviour remains consistent throughout the investigated flow domain for both stretching and shrinking configurations.

Fig. 21.

Effect of the porous parameter on the Bejan-number profile

Figure 22 shows that entropy generation grows with increasing Brinkman number (Br). Higher viscous dissipation implies greater frictional heating, driving both stretching and shrinking flows toward a viscous-irreversibility-dominated regime. The enhancement in entropy generation results from the increasing conversion of mechanical energy into thermal energy through viscous dissipation. The Brinkman number represents the ratio of heat generated by viscous dissipation to heat transferred by thermal conduction. As Br increases, a larger fraction of the mechanical energy is converted into internal thermal energy through fluid friction. This additional heat enhances thermodynamic irreversibility, thereby increasing the total entropy generation. The increase in entropy generation is more pronounced near the wall, where the velocity gradients and viscous effects are strongest, and gradually decreases away from the surface as the boundary-layer effects weaken. The smooth variation of the entropy-generation surface further confirms that increasing Br consistently enhances thermodynamic irreversibility throughout the investigated flow domain for both stretching and shrinking cases.

Fig. 22.

Effect of the Brinkman number Br on the entropy-generation profile

This observation is also verified in Fig. 23, where the Bejan number decreases with increasing Brinkman number (Br). The relative contribution of the heat-transfer irreversibility decreases due to the dominance of the dissipation-driven entropy, hence the Bejan-number values become lower. The decreasing trend of the Bejan number is a consequence of the growing dominance of viscous irreversibility. The Brinkman number is the ratio of the heat generated by the viscous dissipation to the heat transferred by thermal conduction. With the increase of the Br, more mechanical energy is converted into the internal thermal energy via viscous friction, thus increasing the fluid-friction irreversibility. As a consequence, the relative contribution of the heat-transfer irreversibility to the total entropy generation decreases, thus leading to a reduction of the Bejan number. The decrease is more pronounced near the wall, where viscous effects are strongest, and gradually diminishes away from the surface as the boundary-layer effects weaken. The smooth variation of the Bejan-number surface further confirms that increasing Br consistently shifts the thermodynamic behaviour toward a viscous-irreversibility dominated regime for both stretching and shrinking cases.

Fig. 23.

Effect of the Brinkman number Br on the Bejan-number profile

Figure 24 illustrates the combined influence of the magnetic parameter (M) and radiation parameter (Rd) on the local Nusselt number for the stretching case. The three-dimensional surface provides a comprehensive representation of the wall heat-transfer rate's coupled response to simultaneous variations in both governing parameters. It is observed that the local Nusselt number increases with increasing radiation parameter, whereas it decreases with increasing magnetic parameter. The coupled thermal response is governed by the competing influences of magnetic damping and radiative heat transfer. An increase in the radiation parameter enhances radiative heat transport, thereby strengthening the temperature gradient at the wall and consequently increasing the heat-transfer rate. Conversely, a stronger magnetic field intensifies the Lorentz force acting on the electrically conducting fluid, suppressing fluid motion, weakening convective heat transport, and consequently reducing the wall heat-transfer rate. The smooth, continuous variation of the surface further demonstrates that the combined effects of the magnetic and radiation parameters govern the thermal transport behavior across the entire investigated parameter range.

Fig. 24.

Combined effect of the magnetic parameter M and radiation parameter Rd on the local Nusselt number

Figure 25 illustrates the combined influence of the magnetic parameter (M) and radiation parameter (Rd) on the local Nusselt number for the shrinking case. The three-dimensional surface provides a comprehensive representation of the wall heat-transfer rate's coupled response to simultaneous variations in both governing parameters. It is observed that the local Nusselt number increases with increasing radiation parameter, whereas it decreases with increasing magnetic parameter. The combined influence of magnetic interaction and thermal radiation determines the wall heat-transfer characteristics. An increase in the radiation parameter enhances thermal energy transport, thereby increasing the temperature gradient at the wall and improving the heat-transfer rate. On the other hand, the Lorentz force on the electrically conducting fluid is increased by a stronger magnetic field, which suppresses the fluid motion and reduces convective transport, thereby reducing the wall heat transfer rate. The smooth, continuous variation of the surface also indicates that the thermal transport behavior is governed by the combined effects of the magnetic and radiation parameters in a stable, consistent manner across the entire range of parameters explored.

Fig. 25.

Combined effect of the magnetic parameter M and radiation parameter Rd on the local Nusselt number

Figure 26 illustrates the combined influence of the magnetic parameter (M) and porous parameter () on the local skin-friction coefficient for the stretching case. The three-dimensional surface provides a comprehensive representation of the coupled response of the wall shear stress to simultaneous variations in both governing parameters. It is observed that the local skin-friction coefficient increases with increasing values of both the magnetic and porous parameters. Physically, a stronger magnetic field intensifies the Lorentz force acting on the electrically conducting fluid, thereby increasing the velocity gradient at the wall and consequently enhancing the local skin-friction coefficient, although the fluid velocity within the boundary layer may decrease. Similarly, increasing the porous parameter increases the resistance imposed by the porous medium, resulting in greater momentum loss and higher wall shear stress. The smooth, continuous variation of the surface further indicates that the combined effects of the magnetic and porous parameters govern the wall-friction behavior over the entire investigated parameter range.

Fig. 26.

Combined effect of the magnetic parameter M and porous parameter on the local skin-friction coefficient

Figure 27 illustrates the combined influence of the magnetic parameter (M) and porous parameter () on the local skin-friction coefficient for the shrinking case. The three-dimensional surface provides a comprehensive representation of the coupled response of the wall shear stress to simultaneous variations in both governing parameters. It is observed that the local skin-friction coefficient increases progressively with increasing values of both the magnetic and porous parameters. The variation in skin friction arises from the combined influence of magnetic forces and porous-medium characteristics. A stronger magnetic field intensifies the Lorentz force acting on the electrically conducting fluid, thereby increasing the wall velocity gradient and, consequently, enhancing the local skin-friction coefficient, although the fluid velocity within the boundary layer is reduced. Similarly, increasing the porous parameter enhances the resistance of the porous medium, resulting in greater momentum loss and a corresponding rise in wall shear stress. The smooth, continuous variation of the surface further demonstrates that the combined effects of the magnetic and porous parameters govern wall friction behavior across the entire investigated parameter range.

Fig. 27.

Combined effect of the magnetic parameter M and porous parameter on the local skin-friction coefficient

The velocity-coloured streamline patterns for the stretching (λ>0) and shrinking (λ <0) cases are presented in Fig. 28. The streamlines illustrate the flow direction together with the distribution of velocity magnitude throughout the computational domain. For the stretching surface, the streamlines are smoothly distributed and gradually diverge away from the wall, indicating the continuous outward transport of momentum generated by the stretching motion. The streamline topology reflects the distinct momentum-transport mechanisms associated with stretching and shrinking surfaces. The outward movement of the surface continuously transfers momentum to the adjacent fluid, resulting in a well-developed momentum boundary layer, while the velocity decreases gradually with increasing distance from the wall. In contrast, the shrinking surface exhibits a pronounced convergence of streamlines near the wall, reflecting inward motion of the surface and compression of the fluid layers. This inward transport modifies the streamline structure and produces steeper velocity gradients in the near-wall region, whereas the color distribution indicates that the maximum velocity magnitude is concentrated near the wall and decreases progressively with increasing distance from the surface. Overall, the streamlined patterns clearly demonstrate the distinct momentum transport characteristics associated with stretching and shrinking surfaces.

Fig. 28.

Streamline patterns for stretching and shrinking configurations

According to Fig. 29, the Nusselt number increases with the radiation parameter Rd under both stretching and shrinking conditions. Enhanced radiative heat emission strengthens the overall heat-transfer rate, especially for stretching flows where outward motion supports thermal diffusion. Shrinking flows exhibit lower heat transfer due to reduced convection. Figure 30 indicates that skin friction increases with Rd. Radiative heating elevates near-wall temperature, reducing effective viscosity and steepening the wall-velocity gradient. This results in higher shear stress for both types of wall motion. Figure 31 shows that the heat-transfer rate remains nearly unchanged for all nanoparticle volume fractions . Since changes in do not significantly affect the wall-temperature gradient, the Nusselt number is almost constant in both stretching and shrinking cases. Thus, heat transfer is governed mainly by wall motion rather than particle concentration. Figure 32 presents the variation of skin-friction coefficient with . The effect is minimal for both stretching and shrinking surfaces. In stretching flows, the positive shear stress remains nearly unchanged, and in shrinking flows, the negative shear is likewise unaffected. This indicates that wall deformation, not particle loading, primarily controls surface friction.

Fig. 29.

The various values of Rd on

Fig. 30.

The various values of Rd on

Fig. 31.

The various values of ϕ on

Fig. 32.

The various values of ϕ on

From Fig. 33, it is evident that the Nusselt number declines when the magnetic parameter M becomes larger. The Lorentz force caused by a stronger magnetic field essentially impedes the fluid motion and thus the convective heat removal is decreased. The magnetic damping effect that is responsible for this change slows down heat transfer in both stretching and shrinking situations. Figure 34 shows that the skin-friction coefficient changes only slightly with M. The Lorentz force has limited influence on the near-wall velocity gradient, so shear stress remains nearly constant for both stretching and shrinking surfaces. Wall motion remains the dominant factor affecting friction. Figure 35 shows that the Nusselt number is only weakly affected by the Brinkman number Br. Although viscous dissipation rises with Br it does not significantly alter the wall-temperature difference, so heat-transfer rates remain nearly constant. Stretching surfaces continue to show higher Nusselt numbers than shrinking ones. Figure 36 indicates that skin friction likewise changes very little with Br. The increased internal heating does not meaningfully modify the near-wall velocity gradient, so both stretching and shrinking flows maintain nearly unchanged shear levels. This confirms that wall motion dominates frictional behaviour rather than variations in Br.

Fig. 33.

The various values of M on

Fig. 34.

The various values of M on

Fig. 35.

The various values of Br on

Fig. 36.

The various values of Br on

Figure 37 illustrates the best validation performance of the artificial neural network (ANN) model and the corresponding error histogram for predicting the reduced Nusselt number. The graphs of mean squared error (MSE) for training, testing, and validation sets gently slope downward with the number of epochs and thus show regular learning without overfitting. Around the 406th epoch, the network attains its minimum validation error. Most of the residuals are within ± 0.5 and are heavily concentrated around zero, indicating that the ANN predictions are in close agreement with the numerical data. The small error range is a strong indication that the model precisely accounts for the nonlinearity of heat transfer over the extensive range of the parameter combinations. Figure 38 exhibits the training history and error distribution of the ANN that predicts the skin-friction coefficient. The MSE curves for training, validation, and test sets decrease significantly at first and then are almost flat, thus convergence is steady and there is no overfitting. The least validation error can be found at the epoch number 463 approximately. The residuals in the error histogram are very close to zero which confirms the high predicting accuracy and strong generalization ability of the ANN for the nonlinear wall-shear characteristics.

Fig. 37.

Best validation and Histogram for Nusselt's number

Fig. 38.

Best validation and Histogram for Skin friction

Figure 39 portrays the training features of the ANN model aimed at the prediction of the reduced Nusselt number. With increasing epochs, the gradient goes down in a very smooth way indicating that the error is being corrected and the network weights are converging in a stable manner. The small and stable value of the parameter μ indicates that the training process proceeds with consistent parameter adjustment throughout the epochs. Just two validation checks can be seen being activated close to the 200th epoch, thus the possibility of overfitting is ruled out. All these features are indicative of the successful training of the ANN on the underlying nonlinear heat-transfer response and achieving a reliable convergence. Figure 40 depicts the training behaviour of the ANN utilised to determine the skin-friction coefficient. The target error is successively lowered as the gradient demonstrates a quick initial drop and then a slow flattening, which is indicative of the network weights being adjusted in a smooth manner. The damping parameter changes from 10⁻³ to approximately 5 × 10⁻⁷, thus indicating that the training process proceeds with consistent parameter adjustment throughout the epochs. The only two validation failures recorded, the model’s capability of reproducing the wall-shear behaviour without overfitting.

Fig. 39.

Gradient for Nusselt number

Fig. 40.

Gradient for skin friction

Figure 41 is a representation of the regression performance of the ANN model for the reduced Nusselt number. The data points from the training, validation, and test sets are very close to the reference line Y = T, which is indicative of the agreement between the ANN predictions and numerical solutions being almost perfect. The correlation coefficients R = 0.99994(training), 0.99965(validation), 0.99971(test), and 0.99987(overall) are the proof that the network is able to reproduce the nonlinear heat-transfer response with very high precision. The strong alignment of the three datasets also implies that the model has excellent generalization capability which means that the ANN can very well predict thermal behavior for parameter combinations that it has never seen before. Figure 42 depicts the regression results of the ANN model for predicting the skin-friction coefficient. The training data points are almost perfectly fitted to the reference line, as is evident from the high correlation coefficient R = 0.99966 value. The validation and test sets also show good consistency with their numerical targets as their correlation coefficients are 0.98693 and 0.98891, respectively. The overall correlation coefficient R = 0.99605 for the combined dataset is a testimonial of the ANN's capability of capturing the nonlinear shear-stress changes and at the same time showing stability in its predictions of new flow conditions.

Fig. 41.

Regression quadrant for Nusselt number

Fig. 42.

Regression quadrant for Skin Friction

Conclusion

An examination of the tetra-hybrid tangent-hyperbolic nanofluid with the help of a figure shows straightforward momentum, thermal, and entropy-related patterns due to the impact of the magnetic, porous, and radiative effects. The present study integrates tangent-hyperbolic tetra-hybrid nanofluid modeling, nonlinear thermal radiation, entropy generation analysis, and ANN-based surrogate prediction within a unified porous wedge-flow framework, enabling accurate and computationally efficient prediction of wall transport characteristics. The pictorial results of these effects highlight the dominant physical mechanisms governing transport behavior in both stretching and shrinking flows. The main findings are:

• The velocity keeps on going up with a higher Weissenberg number (We) because of the more powerful elastic stretching, while the Temperature also goes up, as the weaker convection heats the boundary layer, thus the heat is trapped there.

• An artificial neural network (ANN)-based surrogate modeling framework was developed to predict heat-transfer and entropy-related properties of tetra-hybrid tangent-hyperbolic nanofluid flow over stretching and shrinking wedge surfaces under the influence of nonlinear thermal radiation, a magnetic field, and porous medium resistance.

• High-quality numerical data generated by the bvp4c solver was used to train and validate the ANN model.

• The numerical analysis shows that flow elasticity, magnetic effects, porous resistance, and nonlinear thermal radiation all significantly influence temperature, velocity, entropy production, and the Bejan number.

• The developed ANN model achieved excellent predictive accuracy, with test values of 0.9815 for and 0.9983 for , demonstrating excellent agreement with the numerical solutions.

• The trained neural network's strong correlation with the numerical findings for the skin-friction coefficient and Nusselt number validated its value as a computationally efficient surrogate model.

• The well-established ANN-assisted framework, which is highly scalable across various non-Newtonian fluids and multi-nanoparticle configurations, enables rapid prediction of nonlinear heat transport behavior in complex engineering systems.

• The present findings provide useful design guidelines for advanced thermal management systems, electronic cooling devices, porous heat exchangers, energy transport systems, and other engineering applications requiring enhanced heat transfer with reduced thermodynamic irreversibility.

Limitations of the present study

The present model applies to steady, two-dimensional laminar boundary-layer flow of a tangent-hyperbolic tetra-hybrid nanofluid over a porous wedge. It is not directly applicable to turbulent, unsteady, or three-dimensional flow configurations, nor does it account for nanoparticle agglomeration, Brownian motion, thermophoresis, or chemical reactions. Therefore, caution should be exercised when extending the present findings to complex flow conditions beyond these assumptions.

List of symbols

Specific heat

k

Thermal conductivity

M

Magnetic field parameter

EG

Entropy generation number

\Pr

Prandtl number

Heat generation parameter

Reynolds number

Volumetric rate of entropy generation

Temperature at wall

u,v

Components of velocity

We

Weissenberg number

Volume fraction of

Volume fraction of Au

Volume fraction of

Volume fraction of

Magnetic field strength

Brinkman number

Be

Bejan number

Porous parameter

Permeability parameter

Local Nusselt number

Skin friction coefficient

Rd

Radiation parameter

m

Wedge index parameter

T

Fluid temperature

Ambient temperature

Free stream velocity

x,y

Coordinates in Cartesian space

Greek letters

μ

Dynamic viscosity

υ

Kinematic viscosity

Wall temperature ratio

Surface shear stress

θ

Dimensionless temperature

ρ

Density

σ

Electrical conductivity

Ω

Angle of the wedge

η

Non-dimensional similarity factor

ψ

Stream function

Temperature difference parameter

α

Thermal diffusivity

Dimensionless velocity slip parameter

Dimensionless thermal slip parameter

λ

Stretching/Shrinking parameter

Subscripts

1

Nanoparticle

2

Au Nanoparticle

3

Nanoparticle

4

Nanoparticle

f

Base fluid

nf

Nanofluid

hnf

Hybrid nanofluid

Thnf

Tri hybrid nanofluid

Tethnf

Tetra hybrid nanofluid

Superscripts

(*)

Special/nondimensional parameter as defined in the formulation

( )

Differentiation with respect to η

Appendix A. Numerical solution procedure

First-order system

For implementation in MATLAB bvp4c, the transformed nonlinear governing equations were converted into an equivalent first-order system by introducing the following variables:

The first-order based system was obtained.

,

and the accompanying boundary conditions:

MATLAB bvp4c implementation

The first-order boundary-value problem described in Appendix A.1 was solved numerically using MATLAB's bvp4c solver based on a fourth-order collocation method with adaptive mesh refinement. The computations were carried out over the computational domain () using an initial graded mesh. The solver was implemented with a relative tolerance of

), an absolute tolerance of(), and an adaptive mesh refinement strategy that automatically adjusts the number of grid points during the solution process. After each solution, the far-field boundary conditions were verified, and the computational domain was enlarged whenever necessary until the prescribed convergence criterion was satisfied.

Representative numerical result

To facilitate verification and reproducibility of the numerical implementation, Table 7 presents a representative numerical solution corresponding to the baseline parameter values Pr=6.2, n=0.001, We=0.10, m=0.3, M=0.1, K=0.4, Br=0.001, , , , and λ =0.2.

Table 7.

Representative numerical solution obtained using the MATLAB bvp4c solver for the baseline parameter values

Quantity	Numerical value
f''(0)	0.568842
-θ'(0)	0.143277
	0.568289
	0.429831

Author contributions

D. Krishnan: Conceptualization, Investigation, Writing – original draft, Validation, Visualization. P. Durgaprasad: Validation, Writing – review & editing, Supervision.

Funding

Open access funding provided by Vellore Institute of Technology. The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Data availability

Numerical datasets generated during this study available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Consent to publish

The authors consent to the publication of this manuscript.