Levenberg–Marquardt Analysis of MHD Hybrid Convection in Non-Newtonian Fluids over an Inclined Container

Abstract

This work aims to explore the magnetohydrodynamic mixed convection boundary layer flow (MHD-MCBLF) on a slanted extending cylinder using Eyring–Powell fluid in combination with Levenberg–Marquardt algorithm–artificial neural networks (LMA-ANNs). The thermal properties include thermal stratification, which has a higher temperature surface on the cylinder than on the surrounding fluid. The mathematical model incorporates essential factors involving mixed conventions, thermal layers, heat absorption/generation, geometry curvature, fluid properties, magnetic field intensity, and Prandtl number. Partial differential equations govern the process and are transformed into coupled nonlinear ordinary differential equations with proper changes of variables. Datasets are generated for two cases: a flat plate (zero curving) and a cylinder (non-zero curving). The applicability of the LMA-ANN solver is presented by solving the MHD-MCBLF problem using regression analysis, mean squared error evaluation, histograms, and gradient analysis. It presents an affordable computational tool for predicting multicomponent reactive and non-reactive thermofluid phase interactions. This study introduces an application of Levenberg–Marquardt algorithm-based artificial neural networks (LMA-ANNs) to solve complex magnetohydrodynamic mixed convection boundary layer flows of Eyring–Powell fluids over inclined stretching cylinders. This approach efficiently approximates solutions to the transformed nonlinear differential equations, demonstrating high accuracy and reduced computational effort. Such advancements are particularly beneficial in industries like polymer processing, biomedical engineering, and thermal management systems, where modeling non-Newtonian fluid behaviors is crucial.

1. Introduction

MHD is also known as magnetohydrodynamics, and it is that part of physics that studies the behavior of electrically conducting fluids under the influence of magnetic fields. These fluids include plasmas, liquid metals, saltwater, and electrolytes [1]. MHD is based on principles derived from fluids and electromagnetism, based on the Navier–Stokes and Maxwell equations. The basic idea is based on the coupling between the velocity of the fluid and the electromagnetic fields, with an important role played by fluid conductivity [2]. An arising component of MHD is the Lorentz force, which tends to arise from the interaction of currents in the fluid with either outside or created magnetic fields. If present, it can change the velocity of the fluid and form unusual flow fields [3]. However, the conducting fluid can alter the field distribution within the magnetic field, giving rise to a pair of Lorenz equations that characterize the interplay between the two. The MHD framework is widely applied in astrophysics to address a range of questions regarding solar flares, stellar winds, and the behavior of accretion disks around black holes [4]. In engineering, magnetic fields are used to control temperature plasmas used in devices like tokamaks in the field of fusion [5]. Magnetic fields are also used in MHD generators, which convert fluid kinetic and thermal energy to electrical energy without mechanical turnover, and MHD pumps, which move liquid metals in industries [6]. MHD flows can be laminar or turbulent, depending on the flow structure and the effect of the magnetic field. The magnetic interaction force is expressed in terms of nondimensional parameters such as the Hartmann number, which compares the magnetic force with the viscous force. Thermal effects are commonly included in the MHD configuration; the temperature gradients affect conductivity and fluids [7]. The subject continues to be an active field of research and development as more parameters become available to solve more complicated MHD problems through advances in computational techniques, which are essential for governing natural and industrial phenomena involving viscous fluids with magnetization. MHD generators are advanced devices that convert current to electricity or produce electricity by making use of the flow of an electrically conducting fluid through the influence of the magnetic field. Originally known as an efficient replacement for turbine-generator combinations, MHD generators have evolved through the twentieth century. First invented as high-power thermoelectric converters using plasma for direct conversion of thermal energy to electricity, these have been developed further into modular power systems for harvesting heat from various power sources.

Thermal stratification is a situation whereby a fluid is divided into layers depending on the temperature, with the hot fluid at the top and the cold one at the bottom [8]. This is common in different conditions, for example, in the sea or a lake or in any work where heat is incorporated with a liquid to build steady temperature stratifications [9]. The layered structure may affect heat and mass transfer rates with associated fluid mixing patterns. Mixed convection occurs between forced convection (when the flow motion is induced by an external agency) and free convection (which is caused by a temperature gradient). The mode of heat transfer in which heat is transferred by both the motion of the fluid and the buoyancy-produced flow gives the complex behavior of the fluid, especially in the thermal stratification region where these two mechanisms dominate the efficiency of heat transfer [10]. This study explores neural computing systems for analyzing thermally stratified mixed convective micropolar fluids, integrating the effects of thermal diffusion and nanofluid interactions. The focus is on heat transfer dynamics over a heated sheet. Advanced neural network models are used to optimize and simulate the system’s behavior under various thermal conditions [11]. Pan [12] examines particle dynamics and turbulence modification in an unstable shear, particle-laden turbulent flow through a horizontal channel using DNS with LPTPM. Barman [13] discusses thermal-wind Rossby waves, and Ozmidov-Proudman states to analyze partial stable stratification of fluids with respect to the physical effect of core-mantle thermal coupling for generating the geomagnetic field using fluid motion in the lower mantle of the Earth. In this research work [14], a numerical analytical analysis of the flow field and heat transfer behavior in the stagnation zone of a mixed conventional fluid containing thermally stable ternary hybrid nanofluids is reported. The analysis takes into consideration magnetohydrodynamics and velocity slip, as well as a permeable sheet that could either stretch or shrink. In non-Newtonian fluids, the functionality of the stretching motions by means of squeeze driving is identified as the key to enhancing the mechanical reliability of a number of industrial processes, including crude oil extraction, nuclear reactor cooling, food processing, production of electronic chips and other materials, groundwater contamination treatment, and manufacturing of plastic goods [15]. Thermal stratifications are essential for many reservoir ecosystems, and their formation and destruction are controlled mainly by meteorological and hydrological conditions. Nonetheless, the impact and importance of these critical environmental factors and processes in reservoir regions with different water depths are relatively poorly understood [16].

The Eyring–Powell model is a non-Newtonian fluid model for fluids that exhibit shear-thinning or shear-thickening behavior. They include the effects of both viscosity and the extent to which the viscosity changes with shear stress and are, therefore, applicable for use in fluids such as polymer solutions or slurries [17]. As used in airflow simulation, heat generation/absorption pertains to the various quantities of heat supplied or extracted in a system owing to chemical reactions that occur, changes of phases, and mechanical friction of the fluid [18]. Flowing heat adds temperature to fluids, while absorption reduces the heat from the thermal systems. This model assists in studying the thermal impacts in the non-Newtonian fluids, which may be useful in technology or production industries where such materials are used [19]. The study by Rikitu [20] is encouraged to analyze the influences of major flow parameters such as magnetic field, Joule heating, variable dynamic viscosity, porous medium dissipation, viscous dissipation, thermal radiation, heat sources, and convective heating on the mixed convection flow of Eyring–Powell Cu–water nanofluid through an inclined micro-channel porous Darcy–Forchheimer medium. Nonlinear, radiative, mixed convective boundary layer flow of non-Newtonian Eyring–Powell fluid (EPF) over a porous stretching wedge is analyzed using partial numerical analysis, while the effects of suction/injection, viscous dissipation, activated energy, and ohmic heating are taken into account [21]. The organization of the article [22] is, therefore, informed by a comprehensive sensitivity analysis of the enhanced thermal transport in Eyring-Powell nanofluid flow over a radiating permeable convective Riga plate immersed in a permeable environment. Furthermore, under the context of the heat flux conditions, the study benefits from the integration of a non-uniform heating/cooling source. Sagheer’s [23] work transcribes a two-dimensional numerical model towards the investigation of the flow of an Eyring-Powell/hybrid nanofluid with variable temperature and velocity. The investigation compares the fluid flow over a stretching sheet through a permeable medium incorporating a suspension of hybrid nanoparticles of copper oxide and magnetite nanoparticles dispersed in ethylene glycol (EG). The approach in [24] is aimed at providing a distinct perspective for bioconvection due to gyrotactic microorganisms inside a nanofluid incorporating non-constant heat generators/sources, temperature and space-dependent viscosity, and Joule dissipation. A three-dimensional Eyring–Powell magneto-radiative nanofluid flow over a porous stretched sheet under convective surface conditions and new mass flux deliveries are analyzed for their physical limitations. The central purpose of the study by [25] is to examine the effects of mass diffusion and thermal radiation on the mass and heat transfer characteristics of magnetohydrodynamic Powell–Eyring fluid over an absorptive flat sheet. The aim of this work [26] is to assess the flow characteristics of Powell–Eyring nanomaterial fluid over a nonlinear stretching sheet with porosity and MHD. In particular, the effects of viscous dissipation and chemical reactions are included in the study, which makes the work unique.

The Levenberg–Marquardt algorithm (LMA) is widely applied to solve nonlinear least squares problems where the model is highly nonlinear or the data contain a large amount of noise [27]. Frequently, the Levenberg–Marquardt method is used when the problem requires fitting the data to the chosen model and utilizing the momentaneous optimizing from the gradient descent and Gauss–Newton methods simultaneously [28]. LMA oscillates between the steepest descent approach as well as the Gauss–Newton approach by analyzing the characteristics of the error surface. This feature, in particular, enables LMA to rapidly develop, especially in cases where the model involved has many parameters. When used in artificial neural networks (ANNs), it helps find the best network weights and biases in the course of training the network [29]. ANNs are made of layers of neuron cells that are interlinked; the cells feed data received as input through the calculation to make an overall prognosis. The LMA enables the alteration of the network’s parameters or weights and bias to minimize the error between the predicted and actual output and is thereby used with ANN training tasks such as classification, regression, and forecasting [30]. By improving the convergence and avoiding the problem of minimum traps, LMA enables the performance of ANNs to be boosted [31]. LMA is used in many applications instead of other gradient-based optimization algorithms due to its ability to offer a fine compromise between convergence speed and accuracy. It is especially useful in the areas of parametric modeling in which accurate forecasts are highly desirable, such as fluid dynamics, image processing, and time series forecasting. ANN associated with the Levenberg–Marquardt (LM) training algorithm provides a precise mathematical model for performance prediction. ANN obtains complex solutions concerning data, and the LM algorithm trains the network by minimizing the prediction error [32]. Thus, the present hybrid system helps improve accuracy and speed for the prediction in various applications. They are particularly applicable for forecasting in cases where the system has nonlinear dependencies and unclear dynamics [33]. In the article by Shoaib [34], a mathematical intelligence-based numerical simulation with the Levenberg–Marquardt algorithm for propagated artificial neural networks (LMA-BANNs) is applied to study the nonlinear radiation stagnation point flow of a nanofluid cross system (NRS-CNFS) over a stretching surface. Asghar’s [35] research findings seek to implement adaptive Levenberg–Marquardt backpropagation (LMB) neural networks to solve quantum calculus models by finding the numerical solutions to singular functional delay differential equations of third, fourth, and fifth order. The usage of artificial neural networks (ANNs) is on the rise because of the capability they demonstrate in solving higher nonlinear mathematical problems. In such complex areas as fluid dynamics, biochemical computations, and bioinformatics, ANNs provide a rather useful computing paradigm. It is, therefore, the intention of this article to apply the Levenberg–Marquardt technique added to backpropagation intelligent neural networks (LM-BPINNs) to analyze the undiscovered phenomena of heat transfer by the dispersion of nanoparticles [36]. The approximations obtained with this algorithm, in particular, gradient descent optimization, are then compared with those that result from the use of the Levenberg–Marquardt optimization method (LMOM) with the same groups of datasets. The approximations obtained with this algorithm, in particular, gradient descent optimization, are then compared with those that result from the use of the Levenberg–Marquardt optimization method (LMOM) with the same groups of datasets. The three orthogonal databases randomly generated from MATLAB’s Neural Network Toolbox were normalized and then gradually fed into the artificial neural network GUI designed and created by the researchers [37]. This work aims to explore the heat transfer in melting heat in magnetohydrodynamic Casson nanofluid flow over a porous stretching sheet while considering viscous dissipation, radiation, and full slip conditions [38]. The purpose of the work in [39] is to investigate the heat transmission by melting in the magnetohydrodynamic Casson nanofluid flow over a porous stretching sheet under viscous dissipation, radiation, and full slip effect.

The Eyring–Powell fluid is a non-Newtonian fluid model characterized by its foundation in molecular kinetic theory rather than empirical relations. It exhibits shear-thinning behavior, where viscosity decreases with increasing shear rate, and can approximate Newtonian behavior at both low and high shear rates. This model is particularly useful for describing fluids that do not conform to simpler models like the power law or Bingham plastics. Due to its ability to capture complex rheological behaviors, the Eyring–Powell fluid model is widely applied in engineering and industrial processes involving non-Newtonian fluids [40].

A numerical analysis is conducted to examine the heat and mass transfer of a Smoluchowski–Nield constrained ternary hybrid nanofluid involving three-dimensional Maxwell slip flow analysis [41]. To improve thermal system optimization, studies have been conducted on the entropy evaluation of Casson hybrid nanofluids flowing in both directions in stalled domains [42]. The study models nanoparticle parameters for alumina and copper characteristics in order to investigate unsteady laminar flow under magnetic circumstances. A recent study used a nanofluid with random motion and thermal migration properties to describe the time-dependent three-dimensional MHD tangent-hyperbolic fluid properties of an extended sheet [43].

2. Mathematical Modeling

Flow Study

This paper investigates the effects of a steady 2D incompressible boundary layer flow of Eyring–Powell fluid over an inclined stretching cylinder. The flow field is investigated, taking into account the impact of thermal stratification. Herein, it is presumed that the temperature in the region close to the cylinder surface is higher than that of the fluid medium. The rheological equality of state for the Eyring–Powell fluid model is given as [44]:

$$\mathrm{\Gamma }=\mu +{\displaystyle \frac{1}{\beta {\gamma }^1}}sin{h}^{-1}\left({\displaystyle \frac{{\gamma }^1}{c}}\right)A_1,$$

(1)

$${\gamma }^1=\sqrt{{\displaystyle \frac{1}{2}}tr{\left(A_1\right)}^2}$$

(2)

The second-order calculation for the $sin{h}^{-1}\left(\right)$ function is calculated as [44]:

$$sin{h}^{-1}\left({\displaystyle \frac{{\gamma }^1}{c}}\right)\approxeq {\displaystyle \frac{{\gamma }^1}{c}}-{\displaystyle \frac{{{\gamma }^1}^3}{6c^3}},where\left|{\displaystyle \frac{{\gamma }^1}{c}}\right|<<1.$$

(3)

Figure 1 shows the physical configuration of the proposed model. The steady two-dimensional equations for the Eyring–Powell fluid flow under the boundary layer approximation and using standard notation are presented as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathit{\partial }\left(ur\right)}{\mathit{\partial }x}}+{\displaystyle \frac{\mathit{\partial }\left(vr\right)}{\mathit{\partial }r}}& =0,\hfill \\ \hfill u{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }x}}+v{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }r}}& =\left(v+{\displaystyle \frac{1}{\beta \rho c}}\right){\displaystyle \frac{{\mathit{\partial }}^{2}u}{\mathit{\partial }{r}^2}}-{\displaystyle \frac{1}{2\beta c^3\rho }}{\left({\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }r}}\right)}^2{\displaystyle \frac{{\mathit{\partial }}^{2}u}{\mathit{\partial }{r}^2}}+{\displaystyle \frac{1}{r}}\left(v+{\displaystyle \frac{1}{\beta \rho c}}\right){\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }r}}-{\displaystyle \frac{1}{6\beta r\rho c^3}}{\left({\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }r}}\right)}^3\hfill \end{array}$$

(4)

$$\begin{array}{c}\hfill -{\displaystyle \frac{\sigma B_0^2}{\rho }}u+g{\beta }_T(T-T_{\infty })cos\alpha .\end{array}$$

(5)

$$v{\displaystyle \frac{\mathit{\partial }T}{\mathit{\partial }r}}+u{\displaystyle \frac{\mathit{\partial }T}{\mathit{\partial }x}}={\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }r}}{\displaystyle \frac{{\alpha }^{\prime }}{r}}\left(r{\displaystyle \frac{\mathit{\partial }T}{\mathit{\partial }r}}\right)+{\displaystyle \frac{Q_0}{\rho c_p}}(T-T_{\infty }).$$

(6)

The cylinder’s axial line is aligned with the x-axis, while the radial direction, perpendicular to the axial line, is designated as the r-axis. Within this framework, v, u, $\nu$, T, g, ${\beta }_T$, $c_p$, $\rho$, $Q_0$, and ${\alpha }^{\prime }$ denote the velocity components along the r and x directions, the kinematic viscosity, temperature, gravitational acceleration, thermal expansion coefficient, specific heat capacity at constant pressure, fluid density, heat generation/absorption parameter, and thermal diffusivity, respectively. Furthermore, $A_1$, c, $tr$, $\beta$, and $\mu$ signify the first Rivlin–Ericksen tensor, fluid parameters, trace, fluid compressibility factor, and dynamic viscosity, respectively. The endpoint conditions are given as:

$$\begin{array}{cc}\hfill U\left(x\right)=u(x,r)={\displaystyle \frac{U_0}{L}}x,u(x,r)\to 0\mathrm{as}r\to \infty \mathrm{and}v(x,r)=0\mathrm{at}r=R,& \\ \hfill T_w\left(x\right)=T_0+{\displaystyle \frac{bx}{L}}\mathrm{at}r=R,T_w\left(x\right)=T_0+{\displaystyle \frac{cx}{L}}r\to \infty .\end{array}$$

(7)

To derive the elucidation for Equations (5) and (6) under the boundary circumstances given in Equation (7), we apply the following alterations:

$$\begin{array}{c}\hfill u={\displaystyle \frac{U_{0}x}{L}}{F}^{\prime }\left(\eta \right),v=-{\displaystyle \frac{R}{r}}\sqrt{{\displaystyle \frac{U_{0}v}{L}}}F\left(\eta \right),\eta ={\displaystyle \frac{r^2-R^2}{2R}}{\left({\displaystyle \frac{U_0}{vL}}\right)}^{{\displaystyle \frac{1}{2}}},\end{array}$$

(8)

$$\begin{array}{c}\hfill \psi ={\left({\displaystyle \frac{U_{0}v{x}^2}{L}}\right)}^{{\displaystyle \frac{1}{2}}}F\left(\eta \right),,T\left(\eta \right)={\displaystyle \frac{T-T_{\infty }}{T_w-T_0}}.\end{array}$$

(9)

where $T_{\infty }\left(x\right)$, $T_w\left(x\right)$, $T_0$, L, $U_0$, $F\left(\eta \right)$, and $F^{\prime }\left(\eta \right)$ denote the prescribed variables, ambient temperature, surface temperature, reference temperature, reference length, free stream velocity, dimensionless variable, and the velocity of fluid on an inclined extending cylinder, respectively. The prime symbol indicates that differentiation concerning $\eta$, and b, c are positive constants. The stream function $\psi$, which fulfills the continuity equation (Equation (4)), is defined as:

$$v=-\left({\displaystyle \frac{\mathit{\partial }\psi }{\mathit{\partial }x}}\right){\displaystyle \frac{1}{r}},u={\displaystyle \frac{1}{r}}\left({\displaystyle \frac{\mathit{\partial }\psi }{\mathit{\partial }r}}\right).$$

(10)

By Substituting Equations (8) and (9) into Equations (5) and (6), we get: [44]

$$\begin{array}{cc}\hfill F{F}^{″}-{\left(F^{\prime }\right)}^2+(1+2K\eta )(1+M)F^{‴}+2K(1+M)F^{″}-{\displaystyle \frac{4}{3}}\lambda MK(1+2K\eta ){\left(F^{″}\right)}^3& \\ \hfill -M\lambda {(1+2K\eta )}^2{\left(F^{″}\right)}^2{F}^{‴}-{\gamma }^2{F}^{\prime }+{\lambda }_{T}T\left(\eta \right)cos\alpha =0,\end{array}$$

(11)

$$2K{T}^{\prime }+(1+K\eta )T^{″}+Pr(F{T}^{\prime }-F^{\prime }T-F{\delta }_1+\delta T)=0,$$

(12)

The boundary conditions are

$$F\left(0\right)=0,F^{\prime }\left(0\right)=1,T=1-{\delta }_1,F^{\prime }\to 0,T\to 0.$$

(13)

where K, $\gamma$, $\lambda$, M, ${\lambda }_T$, ${\delta }_1$, $Pr$, $\delta$, denote the curvature parameter, magnetic field parameter, mixed convection parameter, fluid parameters, thermal stratification parameter, Prandtl number, and heat absorption/generation parameter correspondingly.

3. Solution Methodology

To analyze the heat transfer rate of the magnetohydrodynamic mixed convection boundary layer flow (MHD-MCBLF) over a slanted stretching cylinder in the presence of an Eyring–Powell fluid, a similarity transformation is introduced. This transformation reduces the real governing partial differential equations to a number of ordinary differential equations. These ODEs are then solved numerically using the bvp4c solver in MATLAB R2013B and used as a reference database for the analysis that follows. For the same purpose, an artificial neural network (ANN) is also used as shown in Figure 2, with the Levenberg–Marquardt algorithm (LMA) for the proximal solution. MHD-MCBLF generates another data set for the flow dynamics by modeling the ANNs using intelligent MATLAB technology. The ANN is designed to optimize the learning process, with the dataset partitioned into three segments: it can be divided for training at 80.

4. Results and Discussion

Table 1. Scenario 1: Effect of $\gamma$ on $F^{\prime }\left(\eta \right)$ via $LMA-ANN$.

Cases	MSE			Perfom	Gradient	Mu	Epoch
	Training	Validation	Testing
1	2.8634 × ${10}^{-9}$	4.0120 × ${10}^{-9}$	3.2855 × ${10}^{-9}$	2.86 × ${10}^{-9}$	1.00 × ${10}^{-7}$	1.00 × ${10}^{-8}$	327
2	5.9889 × ${10}^{-10}$	9.9859 × ${10}^{-10}$	1.2361 × ${10}^{-9}$	5.68 × ${10}^{-10}$	1.14 × ${10}^{-6}$	1.00 × ${10}^{-9}$	289
3	3.9696 × ${10}^{-11}$	5.5535 × ${10}^{-10}$	1.6394 × ${10}^{-10}$	3.97 × ${10}^{-11}$	9.90 × ${10}^{-89}$	1.00 × ${10}^{-10}$	674
4	1.2154 × ${10}^{-9}$	5.4755 × ${10}^{-9}$	1.4029 × ${10}^{-9}$	1.22 × ${10}^{-9}$	9.85 × ${10}^{-8}$	1.00 × ${10}^{-9}$	301

and

Table 2. Scenario 2: Impact of ${\delta }_1$ on $\theta \left(\eta \right)$ via $LMA-ANN$.

Cases	MSE			Perfom	Gradient	Mu	Epoch
	Training	Validation	Testing
1	2.3170 × ${10}^{-9}$	2.1973 × ${10}^{-9}$	3.2828 × ${10}^{-9}$	2.32 × ${10}^{-9}$	9.95 × ${10}^{-8}$	1.00 × ${10}^{-8}$	319
2	3.4683 × ${10}^{-9}$	6.2595 × ${10}^{-9}$	5.2422 × ${10}^{-9}$	3.472 × ${10}^{-9}$	9.992 × ${10}^{-8}$	1.002 × ${10}^{-8}$	284
3	3.2036 × ${10}^{-9}$	3.35202 × ${10}^{-9}$	3.62122 × ${10}^{-9}$	3.102 × ${10}^{-9}$	9.962 × ${10}^{-8}$	1.002 × ${10}^{-8}$	285
4	3.4141 × ${10}^{-9}$	4.81472 × ${10}^{-9}$	3.42196 × ${10}^{-9}$	3.41 × ${10}^{-9}$	9.92 × ${10}^{-8}$	1.00 × ${10}^{-8}$	308

present the results of the different situations and their corresponding values. Table 1 is provided to study the relation between the magnetic fluid parameter $\left(\gamma \right)$ and the velocity distribution $F^{\prime }\left(\eta \right)$. Table 2 is provided to study the influence of the thermal stratification parameter ${\delta }_1$ on the temperature distribution $\theta \left(\eta \right)$. The mean squared error is the most widely used performance measure that discourages large differences between predicted and actual values. It plays an informative role as an overall measure of model performance, most relevant where regression tasks are at play, including neural networks. A lower value of MSE indicates better performance, as MSE is the measure of the discrepancy between predicted values and actual values of a model. During the training of the neural network structure, derivatives of weights and biases of the loss function are calculated with a view to the updates. This gradient information is used in optimization algorithms, which include the Levenberg–Marquardt method, to produce better model parameters. While modifying weights, the Levenberg–Marquardt algorithm uses a damping factor (mu) with a crucial role in the optimization procedure. The epochs quantify the number of passes made by the training dataset in the training phase of the neural network. Deciding on the right number of epochs is indeed vital so that learning takes place optimally without overfitting. These measures, such as MSE, are performed independently for training data, validation data, and test data. The training data set is used to adjust the particulars of the model; the validation data set helps in tuning to the other particulars and prevents overfitting, while the test data set gives an overall performance of the model on unseen data.

Figure 3a–e further presents the fluctuation of mean squared error (MSE), error distribution histograms, state transition analysis of error, regression plots, and the comparison of LMA-ANN with the reference solution for the determined magnetic parameter $\gamma$ for the velocity profile $F^{\prime }\left(\eta \right)$. Figure 3a illustrates how the MSE of the velocity profile $F’\left(\zeta \right)$ changes during the training, retention, and testing stages, showing more error variation during the training process. Figure 3b shows the state transition results as a way of depicting the dynamic behavior of the system. The best training performance is subject to epoch 327 with the minimum MSE $0.01\times {10}^{-9}$. Before we proceed, the grade is $9.9\times {10}^{-8}$ and $\mu$ is $1\times {10}^{-8}$. Figure 3c shows the error histogram, which provides a minute insight into the error map related to the velocity profile that is essential for judging the credibility of the model and the potential areas that need enhancement. Figure 3d is the regression analysis, which demonstrates the extent to which the proposed model matches the dataset for the velocity profile. Figure 3e also represents the results obtained by LMA-ANN with the reference solution of $\gamma$ and shows a good correlation. This is reflected in the chosen MSE within the range of ${10}^{-4}$, which proves the high accuracy of the chosen model. All of these figures together give an evaluation of the model and show how close the approach is to the reference solution. Similarly, Figure 4a–e shows the changes with the mean squared error (MSE), state transition, error histograms, and regression analysis between LMA-ANN and the reference solution to the parameter ${\delta }_1$ associated with the temperature profile $\theta \left(\eta \right)$. Another thing shown in Figure 4a is that the epoch at which training achieves the best performance on ${\delta }_1$ in $\theta \left(\eta \right)$ is epoch 319, with an almost minimum MSE of $2.19\times {10}^{-9}$. This points to highly credible and reliable solutions. Thus, the validation phase enhances the acknowledgment of the model across scenarios once more. Figure 4b shows data at the state transition level, such as gradient, epoch, and validation checks. The findings indicate that there is a ranking $9.94\times {10}^{-8}$ and $\mu =1\times {10}^{-8}$ that both point towards the effectiveness of the optimization process at epoch 319. In Figure 4c, the error histograms of ${\delta }_1$ in $\theta \left(\eta \right)$ are depicted to analyze the error density per scenario—valuable information for the estimation of model efficiency. Regression graphs in Figure 4d show a graphical representation of the model’s forecast and the actual output. Regression coefficients (R) closer to 1 represent high correlation; hence, a high model representation and values closer to 0 represent less correlation. Figure 4e shows similar LMA-ANN predictions against the reference solution for all cases. The highlighted visualizations corroborate a high level of consistency and accuracy, hence proving the efficiency of the proposed approach. These figures provide a complete analysis of model accuracy and identify the degree of accuracy and gauge constancy for solving ${\delta }_1$ in connection with temperature distributions.

The effect of the magnetic field parameter on the velocity profile is depicted in Figure 5. In particular, with the enhancement of the magnetic field parameter $\gamma$, there is a reduction in the velocity profile. Physically, this happens because an increase in $\gamma$ increases the Lorentz force on the fluid, and this is a resistive force, which therefore reduces the fluid velocity. As a result, the velocity of the given fluid tends downward due to the presence of this resistance. Figure 6 analyzes the velocity profile depending on the thermal stratification parameter ${\delta }_1$. The above results indicate a prevalent dysfunction and a weak one with better values of ${\delta }_1$ that cause the velocity profile to reduce. This behavior is primarily due to the reduction in surface-to-temperature driving force associated with the convective processes of the cylinder. In other words, with greater thermal stratification, the overall thermal gradient is then decreased to limit the convective flow and inherently lower the velocity of the fluid. These results pinpoint the roles of magnetic and thermal influences in affecting the behavior of fluids by presenting a factor of resistance through different physical processes.

Figure 7 describes the dependence of the mixed convection parameter (${\lambda }_T$) on the velocity profile. This indicates that with increasing ${\lambda }_T$, the velocity of the fluid is also expected to improve. This rise is due to an improvement in the intensity of the thermal buoyancy force, which drives the fluid and improves its velocity. Figure 8 focuses on a discussion related to the fluid parameter (M) and its impact on velocity. Increasing the value of M results in increased values of fluid velocity, as evident in the analysis. This is because M is inversely proportional to the viscosity of the fluid. An increase in parameter M decreases viscosity, which implies that the fluid is more easily deformable. This results in a higher deformation rate, in other words, fluid velocity, at any point in the geometry. Figure 9 below shows the influence of the curvature parameter (K) on the velocity profile. From the above work, it can be observed that as the velocity K increases, the velocity also increases. In physical terms, the quantity K is inversely proportional to the radius of curvature. In turn, as the area K becomes greater, the radius of the cylinder becomes smaller, increasing the minimum contact surface area of the fluid and the cylinder. Compared to a large surface area, this smaller area of the lung can reduce the amount of resistance to the flow of fluid in the body. Thus, these figures illustrate the impact of important previously simulated parameters of ${\lambda }_T$, M, and K on the velocity profile. Compared to ${\lambda }_T$, M decreases the viscous resistance, and K minimizes the contact resistance, collectively contributing to an overall increase in the velocity of the fluid under certain circumstances.

In Figure 10, the effect of the mixed convection parameter ${\lambda }_T$ on the temperature profile is shown. It is observed that as the mixed convection parameter ${\lambda }_T$ increases, the temperature profile decreases. This is because higher values of ${\lambda }_T$ enhance thermal buoyancy forces, which in turn increase the heat transfer rate, leading to a reduction in the temperature profile. Figure 11 illustrates the influence of the thermal stratification parameter ${\delta }_1$. It is noted that as the temperature ${\delta }_1$ increases, the temperature profile decreases. This occurs because of a reduction in the temperature difference between the surface of the cylinder and the surrounding fluid, which causes a drop in the temperature profile. Figure 12 reveals that the temperature profile increases with higher values of the curvature parameter K. Since temperature is related to average kinetic energy, increasing the curvature of the cylinder leads to a higher fluid velocity, which increases the kinetic energy and subsequently raises the temperature. It is important to note that the fluid temperature decreases near the surface of the cylinder but increases farther away from it. In summary, these figures show how mixed convection, thermal stratification, and cylinder curvature each influence the temperature profile through changes in heat transfer, temperature differences, and kinetic energy.

5. Conclusions

In conclusion, this study investigates the magnetohydrodynamic mixed convection boundary layer flow (MHD-MCBLF) over a slanted stretching cylinder using Eyring–Powell fluid. Coupled with the Levenberg–Marquardt algorithm for artificial neural networks (LMA-ANN). The model considers various thermal and fluid properties, including thermal stratification, heat generation/absorption, geometry curvature, magnetic field intensity, and the Prandtl number. By transforming partial differential equations into coupled nonlinear ordinary differential equations, the study uses a computational approach to solve the MHD-MCBLF problem under two different cases: a flat plate and a cylinder with curvature. The study demonstrates that the LMA-ANN solver, combined with regression analysis, mean squared error evaluation, histograms, and gradient analysis, provides an effective tool for solving complex thermofluid interactions in reactive and non-reactive phases. The results reveal that the velocity profile increases with higher values of the mixed convection parameter ${\lambda }_T$, curvature parameter K, and fluid parameter M, while it decreases with higher magnetic field parameter $\gamma$ and thermal stratification parameter ${\delta }_1$. The temperature profile increases with larger values of the curvature parameter K, but decreases with the mixed convection parameter ${\lambda }_T$ and thermal stratification parameter ${\delta }_1$. Overall, the study highlights the significant effects of these parameters on the velocity and temperature profiles, providing valuable insights for predicting thermofluid behavior in magnetohydrodynamic systems.