Intelligent neural network–based framework for Maxwell nanofluid flow with Soret–Dufour coupling and nonlinear thermal radiation

Abstract

This study examined the optimization of the Levenberg–Marquardt backpropagation approach, which uses artificial neural networks, to calculate the interacting features of Soret and Dufour effects on convective Maxwell nanofluid flow through a porous surface with nonlinear thermal emission. The governing system of partial differential equations was first converted into ordinary differential equations, which were then simulated using the Levenberg–Marquardt algorithm and the numerical method BVP4c. Additionally, the consistency and stability of the technique were ensured. Lie symmetry group transformations were used to convert the partial differential equations into ordinary differential equations. Neural networks may also be used to map temperature, velocity, and concentration characteristics from input to output. Various profiles against the porosity parameter, Dufour number, and Soret number are explored. The numerical results depict thermal radiation, and a heat source were found to increase the heat output of the flow. It is also intriguing to note that the skin friction, Sherwood number, and Nusselt number are strongly influenced by the porosity and Dufour factors. Nonetheless, the present findings could have applications in the fields of nuclear waste disposal, steel industries, heat exchangers, cooling applications, and petroleum reservoirs, among others.

Introduction

With the industrial advancement of nanotechnology toward enhanced performance of advanced thermal systems, such advancement is triggered by the need for alternative energy sources. Nanofluids, traditionally described as nanometer-sized suspensions of ultra-fine nanoparticles in a base fluid-like medium, have been developed to increase heat exchange in diverse applications. Water (H₂O), ethylene glycol (EG), and engine oil are the most common materials in the preparation of nanocomposites. Water is widely available, cheap, and renewable but has relatively low thermal conductivity. While these fluids are compelling, much thicker fluids have a better ability to sustain heat transfer; however, these fluids are generally considered environmentally unfriendly. Concoctions made of water or ethylene glycol blended with nanoparticles are more frequently employed as automotive coolants in order to enhance engine efficiency in a relatively safe manner. Electronic cooling technologies are used in high-performance computing systems for microprocessor circuits operating at power levels as high as 100–300 W/cm². The concept of nanofluid was initially established in 1995 by Choi and Eastman [1]. The Buongiorno model used in the study of Rana et al. [2] proposed a non-homogeneous framework in which seven factors are recognized as influential toward the thermal enhancement of nanofluids, with thermophoresis and Brownian motion contributing the greatest share. Effects of high-temperature gradients, thermal radiation, and viscous dissipation, in addition to Brownian motion, thermophoresis, and gyrotactic microorganisms refinement in the Buongiorno model, were examined by Madhukesh et al. [3]. Consequently, the search for practical nanofluid applications has inclined many researchers in this field in recent years, including Khan et al. [4], Mishra et al. [5], Rao et al. [6], and Hussain et al. [7]. Numerous relevant studies have been reported in the literature [8–11].

The involvement of Soret and Dufour phenomena has lately gained expeditious prominence due to their significance in modern engineering and scientific applications. Such effects are pertinent in petroleum reservoirs, heat exchangers, steel making, cooling systems, and even nuclear waste management. These two effects are of less importance in the dynamics of double-diffusive studies because they are relatively insignificant in magnitude. The Soret effect is known as mass flux in the mass conservation equation due to the temperature gradient, and the Dufour effect is energy flux in the energy equation due to the mass gradient. Filahi et al. [12] investigated the effects of cross diffusion on two parallel porous plates through analytical solutions verified numerically by the finite difference method. The authors discovered variations in parallel flow behavior governed by buoyancy ratio and diffusion-thermo effect. In an investigation by Hasnaui et al. [13], the impacts of heat sources and thermo-diffusion were numerically simulated, producing findings that the Soret effect is reinforced in the presence of mean concentration parameter R, while it diminishes in its absence. Other analyses concerning the (SD) effects are captured in the studies by Bidemi and Ahmad et al. [14], along with Williams and Yabo et al. [15]. Several relevant studies have been reported in the literature [16–19].

Mass/heat transformation through extended surfaces still provides a hot research domain in numerous engineering and industrial applications. Knowledge of heat transmission in non-Newtonian materials is relevant to thermal engineering processes, metal spinning, packed bed systems, glass fiber production, hot rolling, etc. Zaid et al. [20] worked on free convective flow, heat and mass transmission in a permeable channel via the cubic B-spline collocation method. Hayat et al. [21] provided an analysis of the behavior of Eyring-Powell liquid under thermal and solutal effects. Mahanthesh et al. [22] focused on mass/heat analysis induced by chemical reaction in stationary and moving surfaces, while Selimefendigil et al. [23] studied heat/mass transfer configurations of fluid flow in a channel. Ahmed et al. [24] employed a finite difference scheme to study microstructural effects and numerical simulations of heat and mass transfer in a viscous fluid. The notion of nonlinear thermal radiation has gained important momentum nowadays since it is of great significance from an engineering point of view, particularly for improving product quality during industrial processing. Power generation plants, rocket propulsion, gas turbines, etc., come under these applications. Khan et al. [25] studied the effects of nonlinear Rosseland approximation on a Walter-B nanofluid with Brownian motion, thermophoresis, and convective constraints. It was found that a higher Schmidt number reduces nanoparticle concentration. Several relevant studies have been reported in the literature [26–33].

Ilias et al. [34] considered MHD heat transfer on a stretching sheet, showing how heat transport and fluid flow affect the surface. Jha and Samaila [35] focused on the importance of nonlinear Rosseland approximation and thermophoretic features via vertically positioned channels. They observed that increased heat flux raises fluid temperature, which ultimately increases velocity and concentration near the permeable surface. The Lie symmetry group method has been widely adopted by researchers ostensibly in an attempt to explore a wide variety of convective boundary problems arising in specific flow scenarios, especially in fluid mechanics, aerodynamics, meteorology, plasma physics, and chemical engineering. The scaling symmetry groups are generalizations that reduce two independent variables into one independent variable, which can be handled with ease. Several authors [36–41] have employed the Lie symmetry group method.

Neural network advancements persisted into the 20th century. New neural networks that might function as “storage elements” were independently presented by Teuvo Kohonen and James Anderson in 1972. Throughout the decade, Stephen Grossberg focused on self-organization networks. In 1982, American physicist John Hopfield proposed the use of a dynamical process to explain the operation of a variety of feedback networks that may be utilized to store information. The backpropagation error method, first proposed by David Rumelhart and James McClelland in 1986, is the second most significant concept that served as the foundation for the development of ANNs in the 1980s. Neural networks have evolved since the introduction of these two concepts [41]. In the present world, neural networks, also referred to as artificial neural networks, are crucial for tackling problems in every area of engineering and research. A system that learns by connecting neurons or nodes in a layered structure that mimics the organization of the human brain is called an adaptive network, or ANN. Because neural networks learn from the analyzed data they receive as input, they may be trained to recognize patterns, make future predictions, and manage data. These days, ANNs are widely employed for nanocomposite flow problems. Tahir et al. [42, 43], Haq et al. [44], and Mehmood et al. [45] focused on ANN-based modeling for predicting complex nanofluid and bioconvective transport phenomena. Shahzadi et al. [46] examined the scope of machine learning techniques for exploring the flow of nanofluids using artificial neural networks. Shafiq et al. [47] used an ANN model to explain the convective heat transfer of nanofluids via a moving needle. The effects of the Dufour and Soret coefficients were investigated, and highly accurate ANN model was developed to predict heat transfer performance.

From the aforementioned literature, it is observed that limited studies have addressed neural network modeling of nanofluid flow in conjunction with the Lie symmetry approach. In particular, previous works such as Ahmad et al. [40] focused on Maxwell nanofluid flow using Lie group transformations without incorporating Soret–Dufour effects and nonlinear thermal radiation simultaneously. Although these studies provided important numerical results, they did not explore intelligent predictive frameworks for highly nonlinear transport characteristics of such flow systems. Similarly, studies by Williams and Yabo [15] considered Soret–Dufour effects, while ANN-based investigations by Shafiq et al. [47] did not combine Lie symmetry analysis with data-driven modeling. This indicates that the incorporation of Lie symmetry-based reduced formulations with neural-network prediction remains insufficiently explored in the available literature. Moreover, the simultaneous effects of convective Maxwell nanofluid flow through porous media under nonlinear thermal radiation and cross-diffusion effects with an ANN framework have not been addressed.

Therefore, the present study extends existing work by incorporating the combined effects of nonlinear thermal radiation and Soret–Dufour mechanisms in Maxwell nanofluid flow over a porous stretching surface. Furthermore, a Levenberg–Marquardt-based artificial neural network is developed to provide an efficient predictive framework alongside the numerical BVP4c solution. This hybrid approach enables a more comprehensive analysis of heat and mass transfer characteristics in complex flow systems. The findings of this study have potential practical applications in cooling technologies, heat exchangers, polymer processing, petroleum transport, thermal management systems and other engineering systems where non-Newtonian nanofluid flow through porous media is involved.

Mathematical modelling

The present investigation examines the combined aspects of Soret and Dufour on flowing convective Maxwell nano-composite over a stretchable porous parallel plate having nonlinear radiation. The nanomaterial’s movement is induced by thermophoresis and Brownian motion and experience Lorentz force due to magnetic field. As illustrated in Fig. 1, the flow of the Maxwell nanofluid is checked to be along the x-axis.

Fig. 1.

Problem model

The flow is driven by the following controlling expressions:

1

2

3

4

With associated conditions

5

Non-dimensional variables to dimensionless Eqs. (1)–(5) below.

6

Heat flux expression is stated as

7

The expansion of is done based on Taylor’s series to algorithmically linearize Eq. (7), leading to

8

The invoking of Eqs. (6–8) into Eqs. (1–5) yields dimensionless equations as

9

10

11

12

Boundary conditions are:

13

In the momentum equation, the convective acceleration terms represent the transport of momentum due to fluid motion, while the viscous diffusion term accounts for internal resistance due to fluid viscosity. The additional Maxwell fluid term includes the relaxation time effect which describes the viscoelastic nature of the fluid. The porous medium resistance is modeled through the Darcy term, which opposes the fluid motion. The magnetic field effect introduces a Lorentz force that acts as a resistive body force which reduces the velocity of the conducting fluid.

Stream functions can be defined as:

14

Continuity equation mentioned in Eq. (9) is automatically satisfied and by incorporating Eq. (14) on Eqs. (10–13), we obtain:

15

16

17

Boundary conditions are:

18

The numerical solution procedure is illustrated in Fig. 2.

Fig. 2.

Flow chart

Lie group transformation

The Lie symmetry group method is used to transform the governing partial differential equations into ordinary differential equations. The detailed derivation steps are provided in Appendix A for brevity.

Using the symmetry transformations, the following similarity variables are obtained by Ahmed et al. [40]:

After resolving Eqs. (A4, A5) the following similarity is obtained.

19

By applying Eq. (A6) upon Eqs. (15–18) we obatin

20

21

22

Transformed BCs are:

23

Engineering quantities

The following definitions of the relevant engineering physical interest of practical significance are Nusselt number, Skin Friction, and Sherwood Number:

24

The primary structural arrangement of the neural network model put out in this paper is seen in Fig. 3.

Fig. 3.

Neural network architecture

Numerical procedure

Soret-Dufour convective Maxwell nanoliquid flow over a porous thermal emission was analyzed. The influence of relevant non-dimensional flow parameters is shown in tables and graphs. The governing nonlinear system represented by Eqs. (20)–(22), together with the boundary conditions (23), was solved numerically in the MATLAB environment using bvp4c, which is based on a finite scheme along with mesh refinement. As the physical domain is semi-infinite, it is converted to a finite interval to ensure that further increases in the domain will not affect the numerical results. The initial mesh consisted of uniformly distributed 200 points, and the solver automatically refined the mesh until the required accuracy was achieved. The relative and absolute tolerances for the solver were adjusted as and .

Validation of results

The findings of the validation investigation, which was carried out in Table 1, were in great agreement with Ahmad et al. [40] and Hayat et al. [21].

Table 1.

Estimations of for and

Sr. No.	M
		Hayat et al. [33]	Ahmad et al. [25]	Present results
1	0.0	-1.00000	-1.00000	-1.0000000
2	0.5	-1.22474	-1.22474	-1.2247425
3	1.0	-1.41421	-1.41421	-1.4142109

Results and discussion

Velocity profile

The influence of the relevant flow parameters on the velocity distribution is illustrated in Figs. 4, 5, 6 and 7. As shown in Fig. 5, an increase in the porosity parameter enhances the velocity profile. Physically, a larger permeability corresponds to a more open porous medium, which reduces the resistance offered by the porous matrix and allows the fluid to move more freely. Consequently, the momentum boundary layer becomes thicker. Figure 4 shows that increasing the material parameter reduces the velocity field. This behavior shows that stronger viscoelastic effects increase the effective resistance within the fluid and results in decreasing fluid velocity. The effect of the suction parameter  S on the velocity profile is shown in Fig. 6. It is observed that increasing  S decreases the velocity throughout the boundary layer. Physically, suction removes fluid from the surface and leads to reduces to lowering momentum boundary layer. Finally, Fig. 7 visualize the effect of the magnetic parameter on the velocity profile. Increasing the magnetic parameter strengthens the Lorentz force, which acts as a resistive force against the fluid motion. As a result, the velocity decreases and the momentum boundary layer becomes thinner.

Fig. 4.

Impacts of K upon

Fig. 5.

Impacts of β upon

Fig. 6.

Variations of S upon

Fig. 7.

M impacts upon

Thermal field

Figure 8 illustrates the variation of temperature with respect to the governing heat source parameter where it is observed that an growing values of heat source parameter enhances the temperature profile. Physically, this indicates an increase in internal thermal energy within the system, which leads to rise in fluid temperature. Figure 9 shows a similar increasing trend, where the temperature rises with an increase in the dimensionless temperature ratio parameter. This behavior is attributed to the fact that the parameter depicts a source of internal energy, which contributes to heating of the fluid.

Fig. 8.

Features of Q upon θ (η)

Fig. 9.

Response of C_T upon θ (η)

Concentration profile

A representative view of the impact of the governing parameter on the concentration distribution is shown in Fig. 10. It is observed that an increase in the Soret parameter leads to growing concentration profile. This behavior can be linked to strengthening of the thermal gradient, which enhances energy transport within the fluid and results in higher concentration field. Presents the effect of the chemical reaction parameter on the concentration profile. It is observed that increasing significantly decreases the concentration distribution. Physically, stronger chemical reaction enhances the consumption of species within the fluid, which reduces the concentration boundary layer thickness. The impact of the Lewis number on concentration is illustrated in Fig. 11. It is noted that as the Lewis number increases from 0.2 to 0.5, the concentration boundary layer thickness decreases, which results in decline of concentration away from the surface. This behavior arises because the Lewis number represents the ratio of thermal diffusivity to mass diffusivity; therefore, higher values of the Lewis number correspond to lower mass diffusivity.

Fig. 10.

Response of S_r upon ϕ (η)

Fig. 11.

Curves of upon ϕ (η)

Fig. 12.

Response of Le upon ϕ (η)

Physical quantities

The numerical results for the skin friction coefficient, local Sherwood number, and local Nusselt number are presented in Table 2. It is observed that increasing values of the porous medium material parameter and the Brownian motion parameter effectively reduce the magnitude of skin friction, indicating a decrease in surface drag. Furthermore, the mass transfer rate is significantly enhanced with increasing values of the chemical reaction parameter and the Brownian motion parameter, while higher values of the Soret number lead to decrease in diffusion. These variations highlight the sensitivity of the boundary layer characteristics to the governing thermophysical parameters, where enhanced particle motion and stronger chemical reaction effects improve the concentration gradients at the wall.

Table 2.

Influences of various parameters on

Influences of on
Sr. No.
1	0.25	-2.0404172	0.4770451	0.06787561
2	0.75	-2.0404098	0.4772341	0.0677241
3	0.95	-2.0403961	0.4773937	0.0676093
Influences of on
Sr. No.
1	0.3	-2.0395241	0.4727492	0.0536472
2	0.6	-2.0382354	0.4665294	0.0328861
3	0.9	-2.0370051	0.4606731	0.0127852
Influences of K on
Sr. No.	K
1	0.2	-2.7717593	0.4736601	0.0622291
2	0.4	-2.1787341	0.4762109	0.0667282
3	0.6	-1.9428761	0.4776941	0.0687153
Influences of on
Sr. No.
1	0.2	-2.0404163	0.4865672	0.0676642
2	0.6	-2.0404052	0.4738741	0.0679438
3	0.8	-2.0404052	0.4675429	0.0680841
Influences of γ on
Sr. No.	γ
1	1.5	-2.0404172	0.5771952	0.0660351
2	2.0	-2.0404279	0.6616682	0.0645248
3	4.5	-2.0403982	0.9227062	0.0599452
Influences of Nbon
Sr. No.	Nb
1	0.5	-2.0402463	0.4877251	0.0647843
2	1.2	-2.0399076	0.4913973	0.0592871
3	2.5	-2.0391061	0.4933672	0.0462680

Artificial neural network (ANN) outcomes

To enhance the predictive capability of the present study, an artificial neural network (ANN) model based on the Levenberg–Marquardt backpropagation algorithm is developed as a data-driven surrogate of the numerical solution. The dataset used for training the ANN is generated from the numerical solutions obtained via the BVP4c solver for different values of the governing dimensionless parameters. The input layer of the network consists of the independent variable while the output layer represents the corresponding flow quantities, including velocity, temperature, and concentration profiles.

The ANN architecture consists of one input layer, two hidden layers, 10 neurons in each layer, and one output layer. Each hidden layer contains 10 neurons with nonlinear sigmoid as an activation function. The dataset is divided into training, validation, and testing subsets in the ratio of 70%, 15%, and 15%, respectively, to secure proper learning and generalization of the model.

The performance of the ANN model is examined using mean squared error (MSE) and regression analysis. The ANN model is trained using the dataset generated from the numerical BVP4c solution. Although the primary focus is on training and performance evaluation, the trained network fundamentally learns the nonlinear mapping between input parameters and flow variables. Therefore, it can be used as a predictive tool to approximate the flow behavior with high accuracy.

The model fastly converges around 50 epochs, as the MSE plot demonstrates. Fast dropping MSE estimations at this early stage suggest significant learning and adaptability to the training data. The rapid learning scenario shows that the model has successfully predicted and incorporated correlations and patterns in the dataset. The MSE exhibits a gradual decline after the initial sharp decline, suggesting gradual advancement toward peak performance. Even when progress is modest, the MSE steadily declines, indicating that model is setting accuracy and picking up new knowledge as it goes via epochs. Figure 13(a) explain that it reaches its minimum value of around epoch 592. According to Figs. 14, 15 and 16(a), the optimum validation performance occurs at epochs 407, 756, and 444, when the mean square error (MSE) drops to its lowest values of , and , respectively. This illustrates the model’s strong learning and generalization capabilities by showing how effectively it can extrapolate to novel data. The MSE consistently diminishes over time, as Figs. 13(a)–16(a) demonstrate. It should be mentioned that the model works best when the MSE is lowered to during epochs 592, 407, 756, and 444, as shown in Figs. 13(a), 14(a), 15(a), and 16(a), respectively. The (NN)’s high learning, convergence, and generalization abilities are demonstrated by the MSE curves in Fig. 13(a)–16(a). Despite the early learning speed increase and minor variations, the model is commended for its capacity to assimilate training data and extrapolate to unknown data, consequently achieving acceptable validation performance.

Fig. 13.

Design LMS-NNA plots for the CMN_SD fluid model’s velocity vesus β solution: a MSE results, b transition state, c histogram for error, d fitting curves, and e regression.

Fig. 14.

Design LMS-NNA plots for the CMN_SD fluid model’s velocity versus K solution: a MSE results, b transition state, c histogram for error, d fitting curves, and (e) regression

Fig. 15.

Design LMS-NNA plots for the CMN_SD fluid model’s Temperature profile versus Q solution: a MSE results, b transition state, c histogram for error, d fitting curves, and e regression

Fig. 16.

Design LMS-NNA plots for the CMN_SD fluid model’s Concentration profile versus solution: a MSE results, b transition state, c histogram for error, d fitting curves, and e regression

The training features of a supervised back propagated neural network are depicted in Fig. 13(b)-16(b). Initially exhibiting fast learning, the gradient then starts to decline as the network gets closer to convergence. At epoch 592, the gradient indicates that performance is almost perfect. The learning rate (Mu) steadily decreases throughout training to lessen the likelihood of overfitting; this phenomenon is known to as “learning rate decay.” The learning rate at epoch 592 is , which suggests that parameter changes occur less often. The diminishing trends in both gradient and learning rate demonstrate the convergence towards a solution in order to lessen the likelihood of overfitting. The gradient reached are at 592, at epochs 407, at epochs 756, and at 444, and the rates of learning or Mu values are , , and as illustrated by Figs. 13, 14, 15 and 16(b) accordingly.

A Levenberg-Backpropagated Neural Network (LBPNN) generates an error histogram with a less value of mean error and classified distribution of errors. This indicates a noteworthy accomplishment in the total objective value projection. The majority of forecasts are nearest to the actual values, as indicated by the histogram’s minor right skew, which concentrates a higher percentage of errors in the lower error bars. The observed distribution pattern indicates that the network can generate precise and accurate predictions. However, comprehension would be enhanced with further details on the network architecture, particular task and training data. The LBPNN is often successful at finding patterns and making precise predictions, as seen by the low mean error number and error distribution throughout the bins. In Figs. 13(c) to 16(c), the Error Histograms (EHs) graphs show mean value respectively. Back propagating neural networks are utilized to examine curve fitting errors (Figs. 13, 14, 15 and 16(d)). The picture displays the absolute error between the output and target estimations for each of the three datasets i.e. training, test and validation. These curves, which span from 0 to 1, correspond to objective functions. Figures 10(d), 11(d) illustrate that the error values fall within the intervals. The model’s performance is considered sufficient because the errors are nearly negligible. However, matching the test and validation data with the training data reveals that the error size is a little larger in the latter two. The curves represent the predicted output of the neural network as a function of the initial output component. The curve’s consistent downward trajectory, which closely follows the goal values, is shown by the Targets line.

The curves in Fig. 13(e) to 16(e) illustrate the back propagated neural network’s exceptional efficiency when used to regression tasks. The model is highly effective in identifying the basic link between objective and output variables, as seen by the training data’s smallest range. The test and validation data show a somewhat wider range of values while still being within the goal values, recommending that the extrapolation to additional data was successful. The error bars in each data set display the standard deviation of faults, which is mostly linked in the region surrounding the mean estimation. As the R-squared values approach unity, the model successfully describes the variance of the objective variable and a sizable portion of the variability of the target variable.

Key remarks

In this experimental investigation of the interaction of Soret and Dufour aspects on convective Maxwell nanofluid over a porous stretching plane taking account of nonlinear thermal radiation. The results have been discussed in detail in for significant distributions. One of the most powerful optimization tools is the LM approach, which allows one to approximate the solutions to a set of nonlinear expressions. Our computing architecture is based on the numerical approach, which can reliably solve higher-order differential equations. Using convective Maxwell nanofluid flow through porous stretching sheet that had both thermophoresis and Brownian motion effects, we solve the fluid flow governing equations using the LM approach in this article. We then utilize the results to composite the ideal estimations of important variables. Our findings have consequences beyond theoretical advances. We present insights from our neural network technique and computational framework that might be employed to optimize fluid systems in the real world. The breadth of the computer technique can be expanded in future studies to include more intricate fluid dynamics and other physical elements. Through the use of innovative neural network topologies and training techniques, we may improve our surrogate models’ accuracy and efficacy. Our research provides a strong foundation for further investigations and improvements in computational methods in CMN_DS, paving the way for upcoming innovations and breakthroughs in fluid dynamics research and applications. Around, the ideal compatibility is reached, suggesting a successful simulation of the steady accuracy of the ANN-LMBAO method. Below are some noteworthy findings that provide a quick illustration of the parameters influence:

• The training features of a (LBPNN) minimize the possibility of overfitting by exhibiting rapid early learning follow up by a dropping gradient as the network converges.

• Regression tasks are successfully completed by (BPNN), which identify fundamental relationships between output and objective variables with suitable variance explanation and precise predictions.

• The network’s Error Histograms (EHs) exhibit low mean error levels, indicating its exceptional prediction performance.

• Increase in temperature ratio gives an increase in thermal profile , similarly rise in enhances concentration .

• The two parameters K and increase , , and .

• Larger γ and Le reduce the concentration profile.

• Increase in Concentration Index means increase in temperature profile.

Future scope

Future studies may extend the present problem by considering other non-Newtonian nanofluid models and additional physical effects such as Arrhenius-based chemical reactions, variable magnetic fields, and variable thermophysical properties. The proposed Levenberg–Marquardt-based neural network framework may also be modified using advanced deep learning techniques and physics-informed neural networks for improved prediction accuracy. Furthermore, experimental validation of the proposed model could be helpful in broadening its practical applications in thermal management and industrial transport systems.

Appendix A

A1

We obtained the following by integrating Eq. (19) into Eqs. (15–18), where b is the real number and ɛ is the group’s parameter:

A2

A3

A4

For equations (20) through (22) the invariant of the method established by equating the powers of e:

A5

The invariance of BC’s becomes:

A6

Focus on the results of Eq. (23) the group changes (19) become: when applied to Eq. (24)

A7

Additionally, subject to extensions of the Taylor series, we have:

A8

The following resulted by changing Eq. (A2), since

A9

Now, using differential Eq. (A3) as a guide, we get:

A10

A11

Nomenclature

symbol

Parameter name

M

Magnetic parameter [-]

k

Dimensional permeability [m²]

$\beta$

Maxwell parameter [-]

Sr

Soret number [-]

$\gamma$

Dimensionless chemical reactions parameter [-]

D_f

Dufour number [-]

$\rho$

Density parameter [kgm^-3]

T_w

Temperature at the wall [K]

Pr

Dimensionless Prandtl number [-]

$C_p$

Specific heat [Jkg^-1K^-1]

σ₁

Stefan-Boltzmann constant [Wm^-2K^-4]

σ

Electrical conductivity [Sm^-1]

$Nt$

Thermophoresis parameter [-]

T_f

Temperature of the fluids [K]

$\lambda$

Dimensional relaxation parameter [s]

Nb

Brownian motion parameter [-]

K_T

Dufour energy density coefficient [Jm^-3]

c_s

Characteristics concentration [kgm^-3]

N

Dimensionless Radiation parameter [-]

T_∞

Ambient temperature [K]

k_f

Thermal conductivity of the fluid [Wm^-1K^-1]

ϕ

Dimensionless concentration [-]

θ

Dimensionless temperature [-]

C_w

Concentration at the wall [kgm^-3]

C_∞

Ambient concentration [kgm^-3]

C_T

Dimensionless temperature ratio [-]

$Pe$

Peclet number [-]

K

Dimensionless porosity parameter [-]

Le

Lewis number [-]

v

Kinematic viscosity [m²s^-1]

$Q$

Internal heat generation [-]

Bi

Biot number [-]

R

Dimensional chemical reactions parameter [s^-1]

S

Dimensionless suction/injection parameter [-]

K_c

Soret energy density [Jkg^-1]

Q_o

Dimensional Heat source parameter [Wm^-3]

Author contributions

Muhammad Ishaq (M.I.) conceived the study, developed the mathematical model, performed the numerical simulations, implemented the artificial neural network framework, analyzed the results, and wrote the original manuscript draft. Muhammad Bilal Ashraf (M.B.A.) contributed to methodology development, mathematical analysis, validation of numerical results, and manuscript revision. Abayneh Kebede Fantaye (A.K.F.) contributed to formal analysis and critical review of the manuscript. Hafedh Mahmoud Zayani (H.M.Z.) contributed to software validation, data interpretation, and manuscript editing. Sultan Alshehry (S.A.) contributed to investigation, visualization, and manuscript review. Ghada A. Alsawah (G.A.A.) contributed to interpretation of results and manuscript revision. Nidhal Ben Khedher (N.B.K.) supervised the research work, contributed to conceptualization, and critically reviewed the manuscript. All authors reviewed and approved the final version of the manuscript.

Data availability

The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Clinical trial statement

Not applicable. This study is not a clinical trial.

Competing interests

The authors declare no competing interests.