Use of an ANN–Levenberg–Marquardt optimization for radiative hybrid nanofluids and heat transfer efficiency over non-isothermal wedge and cone surfaces with chemical reaction effects

1 Introduction

By combining nanoparticles of different materials, researchers can harness their benefits and create hybrid nanofluids that possess a more desirable balance of thermal conductivity, sustainability, and rheological properties than through a singular combination of nanoparticles [1, 2]. By virtue of their stability and thermal conductivity, hybrid nanofluids represent a very attractive fluid for a wide variety of industrial uses [3]. In numerous technical and industrial applications, the thermal Grashof number has an important impact on HNF across wedge and cone geometries. The mathematical investigation is of a laminar, steady, 2D, Darcy-Forchheimer flow related to a chemically-reacted HNF over vertically oriented porous wedge and cone geometries with non-isothermal and -isosolutal constraints, taking into account the thermal Grashof number and the impact of radiation absorption [4]. In addition, the shape of the heated surface will also change the complexity of these events. Due to their theoretical and practical importance, non-isothermal cones and wedges are two critical shapes to understand. Since boundary layers form over each of these slender bodies, the dynamics of flow across them are significantly different than those for flat plates or cylinders; hence, there are both unique challenges and opportunities for optimizing heat transfer.

In order to achieve active control of the flow and thermal characteristics of these systems, external forces must be utilized. Creating a magnetic field, which produces magnetohydrodynamic (MHD) effects, is a powerful non-intrusive method for achieving control of an electrically conductive fluid flow [5]. Use of the magnetic region produces Lorentz forces that can orient or suppress the flow, allowing for extremely precise control of heat and mass transfer rates.

The parameter is a measure of how easily fluid can enter/flow through porous media; thus, if a porous medium has a high permeability, the fluid will flow smoothly through it. Permeability is an intrinsic property of a porous medium and is therefore relevant for MHD analyses of fluid flow in packed beds, geological formations, and porous sheets. Darcy–Forchheimer is commonly the model of choice for modern analyses; it considers the viscous and inertial forces acted on by the porous structure at high flow velocities, thus resulting in a better representation of physical reality [6]. [7] numerically assessed Marangoni convection in both nanofluids and hybrid nanofluids as they flowed through a Darcy–Forchheimer porous medium, delivering a comprehensive examination of the corresponding transport characteristics. In a later investigation [8], the same authors applied the Levenberg–Marquardt feedforward algorithm (LMFA) in combination with a machine learning framework. The optimization was performed using a stochastic approach, making it especially appropriate for representing the inherent uncertainty and probabilistic nature of the underlying model. Furthermore, magnetic field effects interact with permeability; therefore, the total velocity/thermal distributions differ in MHD boundary-layer flow over porous stretching surfaces [9–11].

Thermal radiation is a major means of heat transfer at the elevated temperatures found in nuclear reactors, combustion chambers, or spacecraft re-entry vehicles as it also represents an equivalent or exceeding means of heat transfer relative to convection. Most researchers simplify their analyses by employing the linear model assumption for radiative heat transfer, as it significantly reduces mathematical complexity while still providing a reasonable approximation of energy transfer and temperature distribution within the fluid under moderate temperature differences [12, 13]. The kinetics of a reaction can be a source or a sink of energy and mass (species), adding a layer of complexity to the overall transport phenomena in liquid systems. Thermophoresis and Brownian motion are both physical processes that can be used to control the diffusion of particles in liquid systems; thermophoresis is the directed movement of particles in response to a temperature difference across a system (thermal gradient), and Brownian motion is the random (diffusive) motion of particles caused by the thermal motion of surrounding molecules. The ability to predict how particles behave in complex fluid systems is dependent on understanding these physical phenomena, which have significant implications for many fields. The impact of slip-on flow was first reported by Buongiorno, who explained how thermophoresis and Brownian moment affect nanofluids.

[14] examined how hybrid nanofluids flow over a stretched surface, taking into account both thermophoresis and Brownian motion as a whole. [15] coupled the impacts of thermophoretic and Brownian motion upon the flow. The thermophoresis diffusion effect on MHD flow via a shrinking sheet was demonstrated by [16]. Fiber extrusion and aerodynamic shaping are two examples of industrial operations that frequently involve wedge flow geometries due to the critical importance of controlling the behavior of the boundary layer. It was [17] who first proposed the concept of a thermal edge layer covering a wedge with constant injection or suction in forced flow. Using the Gauss–Lobatto method, [18] examined the different aspects of dissipative heat on a radiative micropolar HNF as it moves through a wedge surface. Theoretical and practical considerations of hybrid nanofluid flow over a shrinking/widening wedge were covered by [19]. This type of geometry is frequently encountered on extrusion procedures, on aerospace overlays, and in the designs of advanced heat exchangers. Heat transmission efficiency in such arrangements is affected by melting impacts, thermal radiation, and slip conditions, according to their findings. [20] explored MHD HNF flow on a radiative wedge, emphasizing its significance in engineering applications, where wedge-molded geometries frequently arise in convoluted thermal and electromagnetic constraints.

Researchers have paid much attention to the ubiquitous phenomena of fluid flow over a conical geometry because of the many practical uses for it in fields as diverse as heat transfer models of exchangers, reservoir-type geothermal, cooling systems of nuclear power reactors, and nuclear waste disposal flasks. Taking into account magnetic field and higher-order chemical interactions, [21] examined the function of slip phenomena in the radiative flow of a hybrid nanofluid over a porous cone. They also investigated thermal radiation, thermal conductivity, and magnetic field behavior. Most research in the past focused on isothermal surface environments. However, in many real-world scenarios, non-isothermal circumstances can be used for heating or cooling. [22] investigated the kinetics of heat transmission between a liquid and a non-isothermal surface. They assumed that the surface temperature would follow a power law distribution. An increase in heat transfer rate was noted as a result of the non-isothermal behavior resulting from the wall temperature constraint. Thereafter, the movement properties of power-law fluid flows past a non-isothermal plane were studied by [23]. Nanofluid transport over non-isothermal and -isosolutal geometries, such as plates, cones, and wedges, was examined by [24]. [25] examined non-isothermal wedge-cone mixed convection steady flow hybrid nanofluid across a permeable surface. Many have worked, for a variety of reasons, with ANNs as a tool to validate deterministic numerical simulations. Due to its quick convergence and high prediction accuracy, ANN-LM is used as a surrogate model to validate and forecast numerical findings [26, 27]. Using the bvp5c method to obtain numerical solutions for BVPs is considered a reliable computational technique for providing accurate numerical solutions of BVPs for many different types of systems. Using an ANN trained on data generated from bvp5c may not always lead to less computationally efficient results than the original numerical method; however, it will provide an ANN with the opportunity to learn the underlying flow behavior and reproduce dynamic behavior with greater adaptability and speed than would have been possible with the original numerical method [28]. A common method used for optimizing ANNs is the Levenberg–Marquardt (LM) method, which has rapid convergence and allows for efficient computation [29, 30]. [31] proposed a physics-informed neural network framework with constrained optimization to solve singular three-point boundary value problems, effectively handling the difficulties arising from singularities and interior boundary conditions that lead to complex solution behavior. [32] investigated the convective transport of a ternary hybrid nanofluid around a stretchable cylindrical vein for hyperthermia-based cancer treatment using a Python-based artificial intelligence framework integrated with a machine learning algorithm that incorporated the non-monotonic Swish activation function and adaptive moment (Adam) optimization. The successful application of ANN surrogate models trained on numerically-generated data by [33] for predicting nanofluid heat-transfer characteristics has motivated this study; it combines PINN solutions validated against RK4/BVP results with regression-based surrogate models to achieve rapid and accurate prediction of engineering quantities. [34] adopted a data-driven prediction framework based on an ANN, where the model was trained using the backpropagation algorithm together with Bayesian regularization to enhance the accuracy and robustness of the computed solutions.

The novelty of this study lies in its first wide-ranging analysis of A l 2 O 3 − C u / H 2 O HNF flow on non-isothermal wedge and cone geometries entrenched in a porous medium under the influence of thermal buoyancy (thermal Grashof number), magnetic field, thermal radiation, thermophoresis, Brownian motion, chemical reaction, and convective boundary conditions. While several studies have deliberated some of these effects individually, their collective impact on cone and wedge geometries has not been studied previously. Additionally, the LM-ANN model is extended to validate and predict the numerical solutions attained through the bvp5c method, thereby improving the reliability, predictive ability, and computational efficiency of the proposed analysis.

2 Structure of mathematical modeling

This study investigates the MHD HNF (A l 2 O 3 − C u / H 2 O) flow across a vertical, permeable, non-isothermal wedge/cone by means of numerical and statistical analyses. A hybrid nanofluid consisting of water and two separate nanoparticles—alumina and copper—comprises the heat transfer fluid. It could be argued that

• The tiny particles remain in a state of thermal equilibrium, exhibiting the same size and shape; and
• The tiny particles in dispersion and the base fluid possess the capacity to keep their temperatures equal.

Figure 1 illustrates the geometry along with the coordinate system of the two-dimensional flow, with the corresponding x and y axes oriented perpendicular to the cone/wedge surface, respectively. The boundary is assumed to undergo stretching with the velocity u w = ν f x l 2, the flow is considered to be present in the area where y is greater than or equal to zero, and the temperature and concentration distribution of the cone or wedge is defined as T w x = T ∞ + a x s 1 and C w x = C ∞ + b x s 2. In this context, T ∞ , C ∞, respectively, represent the ambient temperature and concentration, a and b are constants, and s 1 and s 2, respectively, signify the wall temperature and concentration parameters, where s 1 = 0 indicates an isothermal condition (i.e., a constant wall temperature). The cone/wedge surface is considered permeable, with v w representing the mass transfer velocity, where v w < 0 for suction and v w > 0 injection. The cone or wedge features a half-angle of α, with the radius of the cone defined as r = x sin ⁡ α. In the meantime, the convective heat transfer coefficient and radiative heat flux are represented by h f and q r correspondingly.

The leading equations for the fluid flow problem include the momentum, energy, and mass equations [25]:

∂ ∂ x r n u + ∂ ∂ y r n v = 0 , (1)

u ∂ u ∂ x + v ∂ u ∂ y = μ h n f ρ h n f ∂ 2 u ∂ y 2 + g ρ β T h n f ρ h n f T − T ∞ cos ⁡ α − σ h n f ρ h n f B 0 2 u − μ h n f ρ h n f u K 0 , (2)

u ∂ T ∂ x + v ∂ T ∂ y = k h n f ρ C p h n f ∂ 2 T ∂ y 2 − 1 ρ C p h n f ∂ q r ∂ y + τ D B ∂ T ∂ y ∂ C ∂ y + D T T ∞ ∂ T ∂ y 2 , (3)

u ∂ C ∂ x + v ∂ C ∂ y = D B ∂ 2 C ∂ y 2 + D T T ∞ ∂ 2 T ∂ y 2 − K * C − C ∞ . (4)

i When α ≠ 0 and n = 0, the geometry illustrates a wedge.

i i When α ≠ 0 and n = 1, the geometry illustrates a cone.

Boundary conditions:

u = u w ; v = v w ; − k h n f ∂ T ∂ y = h f T w − T ; C = C w a t y = 0 u → 0 ; T → T ∞ ; C → C ∞ a s y → ∞ . (5)

To streamline the examination, the subsequent similarity transformations are implemented. For the purpose of streamlining the study, the subsequent similarity transformations are implemented:

η = y l , u = ν f x l 2 f ′ η , v = − ν f l 1 + n f η , θ η = T − T ∞ T w − T ∞ , ϕ η = C − C ∞ C w − C ∞ , q r = − 4 σ * 3 k * ∂ T 4 ∂ y ; T 4 ≅ 4 T T ∞ 3 − 3 T ∞ 4 . . (6)

Non-linear ODEs:

A 1 f ‴ + A 2 A 3 λ ⁡ cos ⁡ α θ + 1 + n f f ″ − f ′ 2 − A 1 K + A 4 M f ′ = 0 , (7)

A 5 + 4 3 R d θ ″ + A 6 ⁡ Pr N t θ ′ 2 + N b θ ′ ϕ ′ − s 1 f ′ θ + 1 + n f θ ′ = 0 , (8)

ϕ ″ + N t N b θ ″ − S c R c ϕ + s 2 f ′ ϕ − 1 + n f ϕ ′ = 0 . (9)

Here A 1 = μ h n f μ f ; A 2 = ρ h n f ρ f ; A 3 = β h n f β f ; A 4 = σ h n f σ f ; A 5 = k h n f k f ; A 6 = ρ C p h n f ρ C p f .

The corresponding boundary conditions are transformed as

f 0 = S 1 + n , f ′ 0 = 1 , θ ′ 0 = − B i M 4 1 − θ 0 , ϕ 0 = 1 , f ′ ∞ = 0 , θ ∞ = 0 , ϕ ∞ = 0 . (10)

The non-dimensional variables employed in the leading equations are expressed as

magnetic parameter M = σ f l 2 B 0 2 ρ f ν f, thermal Grashof number λ = g β f T w − T ∞ l 2 ν f u w, porosity parameter K = l 2 K 0, Prandtl number Pr = μ f C p f k f, radiation parameter R d = 4 σ * T ∞ 3 k f k *, thermophoresis parameter N t = τ D T T w − T ∞ ν f T ∞, Brownian motion parameter N b = τ D B C w − C ∞ ν f, suction parameter S = − l v w ν f, Biot number B i = h f l k f, Schmidt number S c = ν f D B, and chemical reaction parameter R c = K * l 2 ν f. Table 1 presents the correlations for the thermophysical properties of the hybrid nanofluid, while Table 2 lists the experimental values of the base fluid and the two nanoparticles, namely copper and alumina.

TABLE 1. A l 2 O 3 − C u / H 2 O hybrid nanofluid thermophysical properties.

TABLE 2. A l 2 O 3, C u, and H 2 O thermo-physical properties.

The important physical parameters associated with the current problem are described below:

C f = μ h n f ρ f u w 2 ∂ u ∂ y y = 0 , N u x = − x k f T w − T ∞ k h n f ∂ T ∂ y y = 0 + q r y = 0 , S h x = − x C w − C ∞ ∂ C ∂ y y = 0 . (11)

The following are the non-dimensional representations of the physical values that result from replacing the similarity features from Equation 6 into Equation 11:

Re x 1 2 C f = A 1 f ″ 0 , Re x − 1 2 N u x = − A 4 + 4 3 R d θ ′ 0 , Re x − 1 2 S h x = − ϕ ′ 0 . (12)

3 Numerical procedure

An analytic solution is enormously challenging to achieve due to the coupled and nonlinear Equations 7–9 and the associated boundary constraints stated in Equation 10. The solver bvp5c is utilized to handle this difficulty. The higher-order Equations 7–9 are condensed to a set of first-order equations by constructing the essential substitutions; meanwhile, bvp4c is useful for a set of first-order ordinary differential equations:

f = y 1 , f ′ = y 2 , f ″ = y 3 , θ = y 4 , θ ′ = y 5 , ϕ = y 6 , ϕ ′ = y 7 .

Thence, the system of ordinary differential equations becomesThe boundary conditions are

y 1 ′ = y 2 y 2 ′ = y 3 y 3 ′ = A 1 K + A 4 M − A 2 A 3 λ cos ⁡ α y 4 − A 2 1 + n y 1 y 3 + A 2 y 2 2 A 1 y 4 ′ = y 5 y 5 ′ = A 6 ⁡ Pr s 1 y 2 y 4 − 1 + n y 1 y 5 − R d 1 + θ w − 1 y 4 4 θ w − 1 y 5 2 A 5 + 4 3 R d y 6 ′ = y 7 y 7 ′ = − S c 1 + n y 1 y 6 − s 2 y 2 y 6 − R c y 6 − N t N b A 6 ⁡ Pr s 1 y 2 y 4 − 1 + n y 1 y 5 − R d 1 + θ w − 1 y 4 4 θ w − 1 y 5 2 A 5 + 4 3 R d . (13)

y 1 0 = S 1 + n , y 2 0 = 1 , y 5 0 = − B i A 4 1 − y 4 0 , y 6 0 = 1 , y 2 0 = 0 , y 4 0 = 0 , y 6 0 = 1 . (14)

The similarity transformations employed here reduce the governing PDEs to a coupled nonlinear boundary-value problem while preserving the physical assumptions of the original model. Since the present investigation is confined to steady boundary-layer flow over stretching cone/wedge geometries, only the physically meaningful similarity solution is considered.

To assess the numerical accuracy of the solution obtained using MATLAB’s bvp5c solver for the cone geometry, an independent differential residual verification was performed. The differential residuals corresponding to the momentum, energy, and concentration equations were evaluated at 5,001 uniformly distributed points over the computational domain 0 ≤ η ≤ 5. For initial meshes of 50, 100, and 150 points, the maximum momentum, energy, concentration, combined, and boundary residuals are reported in Table 3. In all cases, the maximum combined residual remained well below the prescribed numerical tolerance of 10 − 6, while the maximum boundary residual was negligible. These results confirm that the numerical solutions satisfy the prescribed residual-verification criterion and demonstrate mesh-independent convergence.

TABLE 3. Mesh-independence and differential residual verification for the bvp5c numerical solution.

4 Outcomes and discussions

The leading nonlinear ODEs (11)–(14), organized by their consequent boundary constraints, are resolved utilizing the bvp5c shooting method executed in MATLAB. Numerical simulations are carried out for numerous parameter scales, together with the porosity parameter 1 ≤ K ≤ 5, magnetic field factor 0.1 ≤ M ≤ 1.0, thermal Grashof number 1 ≤ λ ≤ 5, Brownian motion parameter 0.1 ≤ N b ≤ 0.3, thermophoresis parameter 1 ≤ N t ≤ 3, Biot number 0.1 ≤ B i ≤ 0.14, thermal radiation factor 0.1 ≤ R d ≤ 0.6, wall temperature factor 0.1 ≤ s 1 ≤ 0.5, wall concentration factor 0.1 ≤ s 2 ≤ 1, Schmidt number 1.0 ≤ S c ≤ 1.4, and chemical reaction parameter 1.0 ≤ R c ≤ 2.0. The resultant velocity, temperature, and concentration contours are confirmed over graphical demonstrations.

Figure 2 illustrates that the cross-flow component weakens as the porosity of the medium upsurges, since the size of the swiftness distribution decreases as K increases. The porosity parameter appears through the drag term A 1 K f ′, which represents the resistance exerted by the porous medium. Figure 3 exemplifies the association between the thermal Grashof number and the momentum of the fluid. The former appears through the buoyancy term A 2 A 3 λ ⁡ cos ⁡ α θ . λ is the buoyant force owing to temperature differences divided by the viscous force in the fluid. The buoyant force, compared to viscous resistance, grows stronger as G r increases. The boundary layer fluid’s motion is accelerated by this enhanced buoyancy force, leading to a growth in the velocity field. Thus, the velocity profiles of the cone and the wedge improve with increasing thermal Grashof numbers. The influence of the magnetic field on the velocity curve is realized in Figure 4. The analysis demonstrates that when the values of the magnetic parameter M increase, the velocity profile for both the cone and wedge decreases. This trend is explained directly by the Lorentz-force term A 4 M f ′ appearing in the transformed momentum equation. When the magnetic effect is used, it forms the Lorentz force, which acts as an opposition to the motion of the hybrid nanofluid. This is the physical meaning of the term. Additionally, the frictional force among nanoparticles in a fluid is created as a result of an increase in Lorentz forces; these cause a decrease in particle speed, which in turn causes a decrease in velocity. Figures 5, 6 illustrate the result of the N b on velocity and temperature distributions for together wedge and cone geometries. The Brownian motion parameter appears through the coupling term N b θ ′ ϕ ′, representing thermal energy transport generated by the random motion of nanoparticles. As N b enhances, the velocity contour and the temperature distribution both increase. Physically, stronger random nanoparticle motion enhances the momentum transport and also augments heat transfer within the boundary layer. It is also noted that the cone geometry exhibits slightly higher velocity and temperature than the wedge due to axisymmetric flow effects. The inspiration of thermophoresis parameter N t on the thermal and solutal contours for the cone and wedge geometries is shown in Figures 7, 8. It appears through the nonlinear term N t θ ′ 2, while in the concentration equation it contributes through the coupling term − N t N b θ ″ . As N t increases, both temperature and concentration distribution increase. The temperature profile increases, which indicates a thicker thermal boundary layer due to enhanced nanoparticle transport driven by temperature gradients. The concentration profile increases, which leads to a thicker concentration boundary layer because stronger thermophoretic forces push nanoparticles away from the heated surface into the fluid. Physically, higher thermophoresis enhances nanoparticle migration from hot to cold regions, which elevates both thermal and mass transport within the boundary layer. The cone geometry exhibits slightly higher temperature and concentration distribution than the wedge. Figure 9 shows the temperature distribution’s fluctuation for various quantities of Biot number (B i). It appears in the thermal boundary condition θ ′ 0 = − B i A 5 1 − θ 0 . In both wedge and cone scenarios, it is noted that temperature fields at the border grow quickly as the Biot number increases. An increase in the thickness of the thermal boundary layer occurs as a result of convective heat exchange at the surface of the plate. Figure 10 demonstrates that the radiation parameter R d influences the distribution of heat at the HNF’s cone and wedge boundaries. The thermal field of the HNF is shown to be greater for the combination of the cone and the wedge as radiation parameter R d is enriched through the coefficient A 5 + 4 3 R d θ ″ in energy ODE. The increased thickness of momentum boundary layer and the increased quantity of energy kept in HNF nanoparticles are the results of an elevated radiation parameter. The outcome is an improved temperature distribution brought about by the higher radiation parameter. For both the cone and the wedge, Figure 11 shows the performance of the hybrid nanofluid temperature versus wall temperature parameter s 1, which looks through the convection term − s 1 f ′ θ . A greater estimate of s 1 is associated with poorer performance in the temperature profile. The wall temperature parameter also reduces the temperature outline; it impacts the boundary layer thickness. In both the cone and the wedge, Figure 12 displays the concentration graph’s fluctuations along with the estimated wall concentration parameter s 2 hybrid nanofluid. The parameter s 2 seems through S c s 2 f ′ ϕ . Boundary layer thickness declines as the wall concentration constraint rises. An increase in wall concentration parameter s 2 results in a less steep concentration outline. Figure 13 describes interrogation of the changes in fluid concentration as a result of variation in S c, denoting growth in the measures of Schmidt number S c; a noteworthy decline in fluid concentration can be concluded. This observation has the physical significance that a rise in values of S c lowers mass diffusivity, as the Schmidt number defines the ratio of momentum diffusivity to mass diffusivity. The Schmidt number multiplies the concentration transport terms. This further causes the concentration boundary layer to become thinner. The significance of R c on the concentration boundary layer is displayed in Figure 14. From the transformed concentration equation, the reaction parameter appears in the sink term S c R c ϕ, from which it can be noted that the concentration boundary layer boldness diminishes when there is a hike in the measures of R c.

Tables 4–6 evaluate the consequence of magnetic field, permeability of porous media, thermal radiation, Brownian motion, thermophoresis parameter, chemical reaction, and Schmidt number on the skin friction coefficient, Nusselt number, and Sherwood number of the HNF (A l 2 O 3 − C u / H 2 O) for the wedge and cone together. The Nusselt, Sherwood, and skin friction factors decay with the rise of the magnetic, Brownian motion, thermophoresis parameter, and porous media, while the opposite impression is shown for thermal radiation. Mass transfer rate increases with enhancement of the chemical reactivity and Schmidt number values.

TABLE 4. Skin friction coefficient values for M , K , R d , λ , N t , N b , S c , and R c.

TABLE 5. Nusselt number values M , K , R d , λ , N t , N b , S c , and R c.

TABLE 6. Sherwood number values M , K , R d , λ , N t , N b , S c , and R c.

To evaluate the skin friction coefficient, Nusselt number, and Sherwood number, the parameter values were fixed as ϕ 1 = ϕ 2 = s 1 = 0.1 , R c = M = N t = N b = R d = λ = S = B i = 0.5 , s 2 = S c = 1. In each computation, only one parameter was varied to examine its influence, whereas the remaining parameters were maintained at their prescribed values.

To demonstrate the numerical stability and convergence of the present computations, a sensitivity analysis was performed with respect to the finite computational boundary η ∞, the initial mesh density and the prescribed solver tolerances used in MATLAB’s bvp5c solver. The corresponding values of the skin friction coefficient C f, local Nusselt number N u x, and local Sherwood number S h x are presented in Tables 7–9. It is evident that the computed engineering quantities exhibit negligible variations with increasing computational boundary, finer initial meshes, and stricter solver tolerances. These results confirm that the numerical solution is mesh-independent, numerically stable, and converged with respect to the selected computational parameters.

TABLE 7. Effect of the computational boundary η ∞ on C f , N u x, and S h x.

TABLE 8. Mesh-independence study for C f , N u x, and S h x.

TABLE 9. Effect of solver tolerances on C f , N u x, and S h x.

To reveal the precision and reliability of the projected model, a comparison of Tables 10 and 11 is provided, highlighting the results for various values of the Pr ⁡ , s 1, α, and S alongside existing published data. These values confirm that the current solution aligns closely with the results available in the literature.

TABLE 10. Validation of results for − f ″ 0 when ϕ 1 = ϕ 2 = n = λ = R d = 0 and B i = 10 6.

TABLE 11. Validation of results for − θ ′ 0 when ϕ 1 = ϕ 2 = n = λ = R d = 0 and B i = 10 6.

5 Artificial neural networks

The flowchart above (Figure 15) shows the overall flow of the ANN-LMBPS model as implemented in this study by executing the LM optimization algorithm. Three layers (input, hidden, and output layer) are shown in Figure 16. ANNs provide a framework to process input data and determine outputs derived from them. While plotting Figures 17–21, the evasion values of the variables are chosen as follows: ϕ 1 = ϕ 2 = B i = 0.1, R d = λ = M = K = N t = N b = R c = 0.5 , α = π / 4 , S c = 0.2 . One parameter is varied at a time, while all remaining parameters are kept. The network undergoes training for 1,000 epochs. Figures 17 and 18 illustrate the ANN-LM training outcomes for the wedge geometry, and (19) and (20) illustrate the ANN-LM training outcomes for the cone geometry through performance curves, training-state plots, error histograms, and regression analyses corresponding to the parameters N t , and N b. Overall, the high degree of similarity between the final numerical solution determined using the bvp5c method and the predicted values provided by the LM-ANN model supports the validity, reliability, robustness, and accuracy of the prediction of the proposed hybrid computation model.

To further assess the predictive capability of the trained ANN model, the approximation error between the numerical solutions obtained using MATLAB’s bvp5c solver and the corresponding ANN predictions was evaluated using the root-mean-square Euclidean norm ε p = y B V P 5 C − y A N N. The approximation errors for representative values of N t and N b are presented in Tables 12, 13, while their corresponding variations are illustrated in Figures 21a,b. It is observed that the approximation error remains consistently of the order of 10 − 5 throughout the parameter range investigated. These small error values demonstrate excellent agreement between the ANN predictions and the numerical solutions, thereby confirming the robustness and predictive capability of the proposed ANN model.

TABLE 12. The approximation error for different values of Nt.

TABLE 13. The approximation error for different values of Nb

The computational efficiency of the proposed bvp5c–ANN framework was evaluated by comparing repeated bvp5c simulations with ANN inference for multiple parameter combinations. CPU execution time, memory consumption, and computational complexity were analyzed. While the bvp5c solver requires an iterative numerical solution for each parameter set, the trained ANN performs a single forward propagation, resulting in significantly reduced computational time and memory use. The quantitative comparison is presented in Table 14.

TABLE 14. Computational efficiency comparison of bvp5c and ANN.

6 Conclusion

Using mixed convection, thermal radiation, and chemical reaction as inputs, this study investigates how a magnetic field affects the fluid flow of HNF across a non-isothermal wedge and cone surfaces under suction/injection conditions. Here are the main findings.

• The momentum of the hybrid nanofluid rises for augmenting values of Grashof number while decreasing for permeability values.
• Temperature goes up with increasing values of magnetism, permeability, and thermal radiation, but fluid temperature drops with augmenting values of the wall temperature factor.
• Concentration decays with enhancing values of chemical reactivity and Schmidt factors.
• Elevated temperatures and slowed heat transfer rates are the results of large thermal radiation estimates.
• Parameters of chemical reaction and Schmidt number increase mass transfer rates for hybrid nanofluids in the wedge and cone geometries, respectively.
• Overall, the high degree of similarity between the final numerical solutions determined using the bvp5c method and the predicted values provided by the LM-ANN model supports the validity, reliability, robustness, and accuracy of prediction of the proposed hybrid computation model.

7 Limitations and future scope

Despite these significant results, limitations must also be noted. The study focuses on a steady-state, laminar, and incompressible flow using a power-law lubricant that does not adequately model the transient or turbulence effects found in real-world conditions. This theory does not include the effects of nanoparticle agglomeration, surface roughness, and thermal non-equilibrium, all of which significantly affect real heating and cooling dynamics. In addition, the influence of external forces such as electromagnetic fields or buoyancy-driven convection was not considered. These limitations may limit the significance of these findings, especially under complex industrial or natural fluid flow conditions. Further research could extend this research by incorporating time-varying flow conditions, turbulence models, and multi-phase interactions of nanofluids in order to improve the relevance of these results in many engineering and industrial applications. Artificial neural networks (ANNs) have become an important research area in fluid dynamics as researchers continue to look for new and improved computational techniques for highly complex physical phenomena using ANNs.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Glossary

u , v Velocity mechanisms along the x- and y-axis. (m/s) μ h n f Dynamic viscosity of hybrid nanofluid (kgm⁻¹s⁻¹) ρ h n f Density of HNF (kg/m³) ρ c p h n f Heat capacity of HNF (J/kg K) β h n f Thermal expansion coefficient D B Brownian diffusion coefficient (cm² s⁻¹) D T Thermophoresis diffusion coefficient u w Stretch velocity (m/s) n Power index ϕ 1 , ϕ 2 C u and A l 2 O 3 nanoparticle volume fraction a , b Constants s 1 , s 2 Wall temperature and concentration parameters k h n f Thermal conductivity of HNF (W/mK) l Characteristic length of cone/wedge r Radius of the cone α Half-angle q r Heat flux g Acceleration caused by gravity (m/s⁻¹) T Temperature σ * Stefan–Boltzmann constant h f Convective heat transfer coefficient M Magnetic parameter τ Ratio of nanoparticle heat capacity to base fluid heat capacity C f Skin friction coefficient N u x Nusselt number S h x Sherwood number T ∞ Ambient temperature C ∞ Ambient concentration λ Thermal Grashof number K Porosity parameter N t Thermophoresis parameter N b Brownian motion B 0 Strength of transverse magnetic field (kg s⁻² A⁻¹) R d Thermal radiation S c Schmidt number R c Chemical reaction parameter Pr Prandtl number S Suction parameter B i Biot number R e x local Reynolds number f Fluid h n f Hybrid nanofluids ∞ Free stream w At the surface

Footnotes

Edited by: Haris Hameed Mian, Norwegian University of ife Sciences, Norway

Reviewed by: Hamid Qureshi, Govt. Graduate College Asghar Mall, Pakistan

Aamir Farooq, Zhejiang Normal University, China