Numerical and intelligent neuro-computational modelling with Fourier’s energy and Fick’s mass flux theory of 3D fluid flow through a stretchable surface

Abstract

Current work explores the intricacies of magnetohydrodynamic and mix convectional boundary-layer flow concerning couple stress Casson nanofluid (CSCNF) dynamics via a 3D stretchable surface. The addition of active and passive control mechanisms for nanoscales creates an innovative dimension to the exploration. Remarkably, the analysis incorporates the influence of non-Fourier and non-Fickian heat and mass flux, alongside the effects of thermophoresis and Brownian diffusion, to systematically investigate the heat and mass transportation phenomena. The governing equations (PDEs) describing the MHD-3DCSCNF model are converted into a set of ordinary differential equations to facilitate the ANN analysis. Employing the bvp4c technique, a dataset is systematically generated for the back propagation artificial neural network with Levenberg–Marquardt Algorithm (BANN-LMA) through four distinct scenarios. Via accurate testing, validation, and training, the BANN-LMA produces estimated results for the MHD-3DCSCNF problem. The performance validation of BANN-LMA is executed through several metrics, involving the mean squared error, error histogram and regression analysis. The training process, characterized by minimizing the MSE through a gradient descent methodology with optimized weights, exhibits a compelling correlation R = 1, between the target and network output. Furthermore, the consistent convergence observed highlights the method robustness and reliability.

KEYWORDS:

• Regression
• active and passive controls
• Forchheimer flow
• ANN with Levenberg–Marquardt backpropagation
• bvp4c technique

1. Introduction

An important and cutting-edge technique to artificial intelligence is exemplified by artificial neural networks (ANNs). ANNs are remarkably adaptable, dynamically responding to the data that flows through the network while learning, whether it comes from within or outside the system. Artificial neural networks make use of the Back-Propagation approach for concurrent training to improve the performance of Multilayer Perceptron networks. This approach stands out as the most common, effective, and understandable framework for complex multi-layered networks. Backpropagation is a supervised learning technique that follows the gradient of the error curve to reduce network errors. Notably, Paul Werbos first introduced this algorithm in 1974. Rumelhart and Parker just revived it. In feedforward multilayer neural networks, backpropagation is now a frequently used learning technique. The Levenberg–Marquardt methodology offers computational solutions to a wide range of fluid flow problems in various technical disciplines by introducing a novel and convergent stability method for ANNs. Newtonian and non-Newtonian fluid systems have been the subject of recent research, which used the Levenberg–Marquardt artificial neural network (LMA-NN) for experimentation. To examine the impact of radiative heat transport on the Williamson fluid flow model, (Shoaib, Raja, et al., 2022) reported a ground-breaking use of the LMA-NN. By considering mixed convectional and slip impact between rotating disks, (Zhao et al., 2023) used LMA-NN to systematically examine the dynamics of Ree-Eyring fluid flow. Furthermore, led to a deeper knowledge of the fluid, (Asif Zahoor Raja et al., 2022) used the computational intelligence algorithms based on LMA-NN to learn more about ferrofluid flow, incorporating the effects of both magnetic forces and nonlinear thermal radiation effects. Recent studies (Dai et al., 2023; Karimmaslak et al., 2021; Z. Shah et al., 2022; B. K. Sharma et al., 2023; Shoaib, Kainat, et al., 2022; X. Zhang et al., 2022; Zuhra et al., 2022) that use the ANN approach to analyse heat transport mechanisms in various geometries while taking a variety of characteristics are presented in the literature.

In thermodynamic processes, irreversibility is caused by several phenomena, including fluid opposition, radiative heating, biological repetition, energy transmission, chemical changes, and electric and magnetic propagation. Various mechanisms such as energy transportation and movement of liquid inherently demonstrate irreversibility, supporting enhanced entropy and the loss of favourable properties. In these processes, the mechanism of entropy production remains constant during closed isothermal processes, but in case of irreversible, the entropy production increases. Such distinction is helpful in identifying reversible and irreversible mechanisms. When liquid flow experiences a drop in pressure, entropy is produced in any thermodynamic system. Bejan (1980, 2013) first investigated the idea of irreversibility in convection liquid flow because of the viscous effects of thermal transport. Tayebi et al. (2020) highlighted the numerical analysis of entropy production in the flow field of a nanofluid in presence of convectional heating. Khan et al. (2020) explained on the statistical modelling of slip flow in a micropolar liquid. Hayat et al. (2020) evaluated entropy-optimized flow with melting heat exchange. The rate of entropy production and nonlinear radiated thermal fluxes over a deformable surface with changeable properties were investigated by Nasir, Berrouk, Aamir, et al. (2023). Entropy improvement in high-temperature surfaces, including nanoparticles, was studied by Boujelbene et al. (2023). In the domain of literature, intriguing and contemporary flow-related challenges linked to the generation of entropy are explored in references (Mandal & Pal, 2023; Rafique et al., 2023; Sakkaravarthi & Reddy, 2023).

The quality of energy transmission becomes fundamental as distinct areas inside the same body undergo thermal variations. Fourier first proposed the idea of coolant temperature conduction, and Catteneo later improved it by adding a relaxation time to address the paradox of heat conduction. The Cattaneo–Christov heat flux model (CCHFM), which incorporates the notion of relaxation time, was created because of Christov’s further development of the Fourier model. The CCHFM was used by Nayak et al. (2022) to investigate heat transmission in hybrid nanofluids and the subsequent entropy formation. The CCHFM was used by Gowda et al. (2021) to examine heat transmission in a nanofluid flow. To study the movement of chemically reactive micropolar carbon nanotube-blood nanofluid through a constriction channel, (Shaw, 2021) applied the idea of CCHFM mechanism. Using the CCHFM, (Gowda et al., 2023) investigated the flow of liquid nanomaterial over a surface. The effect of the CCHFM on the flow of ferromagnetic Maxwell nanofluids across an extended geometry was explained by F. Shah et al, (2023).

The phenomenon of thermal radiation refers to the transmission of thermal energy between two bodies in the absence of physical contact. Thermal radiation is used by pyrometers and infrared cameras to measure and regulate temperature in a variety of sensing processes. Thermal radiation plays a dynamic role in a wide range of engineering applications, especially at high temperatures. Convective heat transfer was examined by Jayaprakash et al. (2022) together with radiative heat transmission. The impact of a homogenous heat source/sink on the flow of hybrid nanofluids was explored by Soumya et al. (2022). In order to account for heat radiation (Nasir, Berrouk, Gul, et al., 2023), described the behaviour of nanofluids over a surface. The effects of heat sources and radiation on the convective flow of a nanoliquid in a cavity were discussed by Gul et al. (2023). Thermal radiation has great uses in the field of medical sciences in addition to industrial and engineering domains. As a result, (Hsu et al., 2017) evaluated potential application in the field of medicine, providing illumination on how thermal radiation from the human body might result in both heating and cooling effects. The vital importance of maintaining a healthy body temperature as a basic requirement for living is emphasized by their research. In a similar fashion, (Wehinger & Flaischlen, 2019) established frameworks for thermal radiation in computational fluid dynamics, (Petela, 2010) concentrated on industrial thermodynamics, (Navarro et al., 2021; Wang et al., 2023) investigated thermal radiation within fluid dynamics. All the authors researched the novel and advanced uses of thermal radiation in various contexts.

The properties of mixed convective flow across an extended surface have numerous technological and industrial applications. When gravitational forces are at work, the effect of free convection is more prominent. The mechanisms of fluid flow and energy transmission are impacted by both the strained and buoyant forces. Thermal buoyancy, which affects the rate of heat movement in manufacturing environments, results from variations in the heat on the expanding surface. Applications of this phenomenon include precautionary stoppage of nuclear cooler systems, environmental fluxes, heaters, photovoltaic panels, conditioning of electrical gadgets, oceanic mechanics, low-velocity heat transfer mechanisms, precooler mechanisms, vehicle demisters, and more. According to (Patil et al., 2020), mixed convective nanoparticle transport along an exponentially expanded porous surface was influenced by suction/injection. Mixed convective radiative flow over an inclined permeability surface was explored by Moradi et al. (2013). Time-dependent magneto Williamson nanoparticle flow through a stretchy surface was evaluated by Das et al. (2022) considering velocity slips, nonlinear mixed convection, and radiative heat effects. In contrast to the focus on their Newtonian behaviour, the impact of nanofluids’ non-Newtonian properties on their heat transfer rate has not been adequately investigated. Many nanofluids display non-Newtonian characteristics, even though a lot of study has been devoted to employing Newtonian nanofluids to improve energy transmission (Elboughdiri et al., 2023; Rasool et al., 2023; J. Sharma et al., 2023; Wakif, 2023; Wakif et al., 2022; K. Zhang et al., 2023).

To the author’s knowledge, most of the literature in the area of Casson couple-stressed nanofluids concentrates on convective boundary conditions rather than active and passive control nanoparticle boundary conditions for examining thermal and concentration distributions. The analysis of the double-diffusive Cattaneo–Christov energy flux in couple stress nanofluids using the theory from reference (Hayat et al., 2017) while simultaneously considering passive and active control conditions has also not been the subject of any published study. Additionally, this theoretical and numerical studies have the potential to advance the field of bio-fluid dynamics and have applications in several engineering fields, including photovoltaic conversion, nuclear reactor implementation, cooling systems, and electronic generators. So, according to the authors, it helps to investigate thermal radiation, magnetohydrodynamic effects, and Darcy-analysis Forchheimer’s of couple stress nanofluids’ three-dimensional mixed convection flow. Because of this, authors feel it is appropriate to investigate the effects of passive and active control on magnetohydrodynamics, thermal radiation, Darcy-analysis Forchheimer’s of three-dimensional mixed convection flow of couple stress nanofluid with non-heat Fourier’s flux, and non-mass Fick’s flux theory. The couple stress nanofluid flow of magnetic, mix convectional, and double-diffusive Cattaneo–Christov energy flux within porous media is investigated in this research using the BANN-LMA approach. The importance of the constructed ANN procedure is emphasized through a comparison of convergence and accuracy measures, employing performance analysis, regression, correlation, histograms, and the Mean Squared Error. The reference dataset for the ANNs is established using the bvp4c technique for various model parameter variations. Additionally, using graphs and tables, the effects of various parameters on velocity, temperature, and concentration fields are thoroughly explored and explained.

2. Mathematical formulation

Here, a mathematical model for steady, incompressible Couple stress nanofluid flowing through a nonlinear extending sheet submerged in a porous media created by Darcy and Forchheimer is represented. In this analysis the surface is traced at $z=0$ and the fluid flow is produced as the extending of sheet surface in $x-$ and $y-$directions. Also, we assume that the starching velocity along $x-$direction is $U_w=a(x+y{)}^n$ and along $y-$direction is $V_w=b(x+y{)}^n$(Gowda et al., 2023; Ramzan et al., 2013). Where$a,b$ and $n$ are positive constants. Its also suggested that $T_w-$constant surface and $T_{\mathrm{\infty }}-$ambient temperature such that $T_w-T_{\mathrm{\infty }}>0$. Furthermore, we settled that $C_w-$constant surface and $C_{\mathrm{\infty }}-$ambient temperature such that $C_w-C_{\mathrm{\infty }}>0$. In this effort, the Buongiorno nanofluid model and Cattaneo–Christov heat and mass fluxes are also used (Figure 1).

Figure 1. Physical model of the problem.

2.1. System assumptions and limitations

In the current flow system, we have implemented the following assumptions and constraints:

• Three-dimensional flow

• Time independent flow

• Incompressible fluid

• Couple stress fluid model

• Dorcy Forchheimer porous medium

• Cattaneo–Christov heat and mass fluxes models

The velocity, temperature, and concentration fields for three-dimensional steady incompressible fluid flow can be represented as $V=\left[u\right(x,y,z),v(x,y,z),w(x,y,z\left)\right],T=T(x,y,z)$ and $C=C(x,y,z)$. For an incompressible couple-stress nanofluid, the fundamental equations of conservation of mass, motion, heat, and concentration are (Gowda et al., 2023; Ramzan et al., 2013; Zuhra et al., 2022): (1) $\begin{array}{rl}& \mathrm{\nabla }\cdot \overrightarrow{V}=0,\end{array}$(1) (2) $\begin{array}{rl}& {\rho }_f\left(\overrightarrow{V}\cdot \mathrm{\nabla }\overrightarrow{V}\right)-\mathrm{\nabla }\cdot \tau +{\mu }^{\times }{\mathrm{\nabla }}^4\overrightarrow{V}-{\rho }_{f}F=0,\end{array}$(2) (3) $\begin{array}{rl}& (\rho c{)}_f\left(\overrightarrow{V}\cdot \mathrm{\nabla }T\right)+\mathrm{\nabla }\cdot \overrightarrow{q}\\ & -(\rho c{)}_{nf}\left[D_B\left(\mathrm{\nabla }C\cdot \mathrm{\nabla }T\right)+\frac{D_T}{T_{\mathrm{\infty }}}{\left(\mathrm{\nabla }T\right)}^2\right]=0,\end{array}$(3) (4) $\begin{array}{rl}& \left(\overrightarrow{V}\cdot \mathrm{\nabla }C\right)+\mathrm{\nabla }\cdot \overrightarrow{J}-\left[D_B\left({\mathrm{\nabla }}^{2}C\right)+\frac{D_T}{T_{\mathrm{\infty }}}{\left(\mathrm{\nabla }T\right)}^2\right]=0\end{array}$(4)

The heat flux $\left(\overrightarrow{q}\right)$ and mass flux $\left(\overrightarrow{J}\right)$ model is presented as follows (F. Shah et al., 2023): (5) $\begin{array}{r}\begin{array}{l}\overrightarrow{q}+\left({\displaystyle \frac{\mathrm{\partial }\overrightarrow{q}}{\mathrm{\partial }t}}+\overrightarrow{q}\cdot \mathrm{\nabla }\overrightarrow{V}+(\mathrm{\nabla }\cdot \overrightarrow{V})\overrightarrow{q}+\overrightarrow{V}\cdot \mathrm{\nabla }\overrightarrow{q}\right)\\ {\mathrm{\Psi }}_E=-k\left(\mathrm{\nabla }T\right)\\ \overrightarrow{J}+\left({\displaystyle \frac{\mathrm{\partial }\overrightarrow{J}}{\mathrm{\partial }t}}+(\mathrm{\nabla }\cdot \overrightarrow{V})\cdot \overrightarrow{J}+\overrightarrow{J}\cdot \mathrm{\nabla }\overrightarrow{V}+\overrightarrow{V}\cdot \mathrm{\nabla }\overrightarrow{J}\right))\\ {\mathrm{\Psi }}_C=-D_B\left(\mathrm{\nabla }C\right)\end{array}\}\end{array}$(5) where ${\mathrm{\Psi }}_E-$relaxation time heat flux and ${\mathrm{\Psi }}_C-$ relaxation time mass flux. Equation reduces classical Fouriers and Fick’s law for ${\mathrm{\Psi }}_E=0={\mathrm{\Psi }}_C$(Shaw, 2021; Zuhra et al., 2022). Since the fluid flow in this suggested model is mass-conserving, or steady, at $\mathrm{\nabla }.\tilde{V}=0$, Equation (5) become. (6) $\begin{array}{r}\begin{array}{l}\overrightarrow{q}+\left(\overrightarrow{V}\cdot \mathrm{\nabla }\overrightarrow{q}+\overrightarrow{q}\cdot \mathrm{\nabla }\overrightarrow{V}\right){\mathrm{\Psi }}_E=-k\left(\mathrm{\nabla }T\right)\\ \overrightarrow{J}\left(\overrightarrow{J}\cdot \mathrm{\nabla }\overrightarrow{V}+(\mathrm{\nabla }\cdot \overrightarrow{V})\overrightarrow{J}+\overrightarrow{V}\cdot \mathrm{\nabla }\overrightarrow{J}\right){\mathrm{\Psi }}_C=-D_B\left(\mathrm{\nabla }C\right)\end{array}\}\end{array}$(6)

For the isotropic and incompressible flow of a Casson fluid, the rheological equation of state is as follows (J. Sharma et al., 2023; Wang et al., 2023; Zuhra et al., 2022). (7) $\begin{array}{r}{\tau }_{ij}=\begin{array}{l}2\left({\tilde{p}}_y/\sqrt{2\pi }+{\tilde{\mu }}_B\right){\omega }_{ij},\pi >{\pi }_c\\ 2\left({\tilde{p}}_y/\sqrt{2{\pi }_c}+{\tilde{\mu }}_B\right){\omega }_{ij},\pi <{\pi }_c\end{array}\}\end{array}$(7) Here $\pi -$ is the product of the component of deformation rate with itself, where $\pi ={\omega }_{ij}{\omega }_{ij}$ and ${\omega }_{ij}$ is the $\left(i,j\right)\text{th}$ component of the deformation rate, ${\pi }_c-$ is the critical value of this product based on the non-Newtonian model, ${\tilde{\mu }}_B-$ is the plastic dynamic viscosity of the non-Newtonian fluid and ${\tilde{p}}_y-$ is the yield stress of the fluid (Sakkaravarthi & Reddy, 2023), but for Casson fluid we supposed ${\pi }_c<\pi$ and ${\tilde{p}}_y=\frac{\sqrt{2\pi }{\mu }_B}{\text{β}}$, the dynamic viscosity could be described as (8) $\begin{array}{r}\mu ={\mu }_B+\frac{{\tilde{p}}_y}{\sqrt{2\pi }},\end{array}$(8)

We obtain by inserting the value of ${\tilde{p}}_y$ (9) $\begin{array}{r}\mu ={\mu }_B\left(1+\frac{1}{\text{β}}\right).\end{array}$(9)

The high order system of partial differential equations formed from the controlling problem are as follows, based on the above-mentioned explanations (Gowda et al., 2023; Ibrahim et al., 2022; Ramzan et al., 2013; Shaw, 2021; Zuhra et al., 2022): (10) $\begin{array}{rl}& \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }v}{\mathrm{\partial }y}+\frac{\mathrm{\partial }w}{\mathrm{\partial }z}=0,\end{array}$(10) (11) $\begin{array}{rl}& u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+w\frac{\mathrm{\partial }u}{\mathrm{\partial }z}={\upsilon }_f\left(1+\frac{1}{\text{β}}\right)\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}-\frac{{\sigma }_f}{{\rho }_f}{B}^{2}u\\ & -\frac{{\upsilon }_f}{K}\left(1+\frac{1}{\text{β}}\right)u-F_b{u}^2-{\upsilon }_f^{\ast }\frac{{\mathrm{\partial }}^{4}u}{\mathrm{\partial }{z}^4}\\ & +g{\text{β}}_T(T-T_w)+g{\text{β}}_C(C-C_w),\end{array}$(11) (12) $\begin{array}{rl}& u\frac{\mathrm{\partial }v}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }v}{\mathrm{\partial }y}+w\frac{\mathrm{\partial }v}{\mathrm{\partial }z}={\upsilon }_f\left(1+\frac{1}{\text{β}}\right)\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{z}^2}\\ & -\frac{{\sigma }_f}{{\rho }_f}{B}^{2}v-\frac{{\upsilon }_f}{K}\left(1+\frac{1}{\text{β}}\right)v-F_b{v}^2-{\upsilon }_f^{\ast }\frac{{\mathrm{\partial }}^{4}v}{\mathrm{\partial }{z}^4},\end{array}$(12) (13) $\begin{array}{rl}& u\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }T}{\mathrm{\partial }y}+w\frac{\mathrm{\partial }T}{\mathrm{\partial }z}+{\mathrm{\Psi }}_E{\chi }_E\\ & -\frac{1}{{\left(\rho C_p\right)}_f}\left(k_f-\frac{16{\sigma }^{\ast }{T}_{\mathrm{\infty }}^3}{3k^{\ast }{k}_f}\right)\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{z}^2}-\tau (\frac{D_T}{T_{\mathrm{\infty }}}{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\right)}^2\\ & +D_B\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }z}.\frac{\mathrm{\partial }C}{\mathrm{\partial }z}\right))-\frac{{\sigma }_f{B}^2{u}^2}{{\left(\rho C_p\right)}_f}=0,\end{array}$(13) (14) $\begin{array}{rl}& u\frac{\mathrm{\partial }C}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }C}{\mathrm{\partial }y}+w\frac{\mathrm{\partial }C}{\mathrm{\partial }z}+{\mathrm{\Psi }}_C{\chi }_C-D_B\frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }{z}^2}\\ & -\frac{D_T}{T_{\mathrm{\infty }}}\left(\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{z}^2}\right)=0.\end{array}$(14) where $u$ and $v$ (velocities components), $\text{β}$ (Casson parameter), ${\sigma }_f$ (electrical conductivity), ${\rho }_f$ (density), $B$ (magnetic field), $K$ (porous medium permeability), $F_b=C_b/x\sqrt{K}$ (porous medium inertia coefficient), $C_b$ (Drag force), ${\mathrm{\Psi }}_E$ and ${\mathrm{\Psi }}_C$ (thermal and concentration relaxation time coefficient), ${\alpha }_f=k/{\left(\rho c_p\right)}_f$, $k$(thermal diffusivity and conductivity), $\tau ={\left(\rho c_p\right)}_p/{\left(\rho c_p\right)}_f$ (heat capacities ratio), $(\rho c_p{)}_f$ (effective heat capacity of fluid), $D_B$ (Brownian diffusivity), $D_T$ (thermophoretic diffusion coefficient).

2.2. Boundary conditions

The relevant boundary conditions are as follows (Elboughdiri et al., 2023; Ibrahim et al., 2022; Rasool et al., 2023): (15) $\begin{array}{r}\begin{array}{l}\begin{array}{l}\text{At }z=0\\ w=0,u=U_w,v=V_w,\\ -k\left({\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }z}}\right)=\left(T_f-T\right)h_f,\\ C=C_w\to \text{ For active Control of }\varphi \\ D_B\left({\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }z}}\right)=-{\displaystyle \frac{D_T}{T_{\mathrm{\infty }}}}\left({\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }z}}\right)\to \\ \text{For passive Control of }\varphi \end{array}\}\\ \begin{array}{l}\text{at}z\to \mathrm{\infty }\\ T\to T_{\mathrm{\infty }},C\to C_{\mathrm{\infty }},u\to 0,v\to 0\end{array}\}\end{array}\}\end{array}$(15)

The arising terms ${\chi }_E$ and ${\chi }_C$ from Equations (13) and (14) are estimated as: (16) $\begin{array}{r}\begin{array}{l}{\chi }_E={\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }z}}\left(u{\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }x}}+v{\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }y}}+w{\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }z}}\right)\\ +{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }y}}\left(u{\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }x}}+v{\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }y}}+w{\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }z}}\right)\\ +{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }x}}\left(u{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}}+v{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}+w{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }z}}\right)\\ +u^2{\displaystyle \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{x}^2}}+2uv{\displaystyle \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }x\mathrm{\partial }y}}+2wv{\displaystyle \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }y\mathrm{\partial }z}}\\ +v^2{\displaystyle \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}}+2wu{\displaystyle \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }z\mathrm{\partial }x}}+w^2{\displaystyle \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{z}^2}},\\ {\chi }_C={\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }z}}\left(u{\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }x}}+v{\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }y}}+w{\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }z}}\right)\\ +{\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }y}}\left(u{\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }x}}+v{\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }y}}+w{\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }z}}\right)\\ +{\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }x}}\left(u{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}}+v{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}+w{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }z}}\right)+\\ 2uv{\displaystyle \frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }x\mathrm{\partial }y}}+u^2{\displaystyle \frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }{x}^2}}+2wv{\displaystyle \frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }y\mathrm{\partial }z}}\\ +v^2{\displaystyle \frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }{y}^2}}+2wu{\displaystyle \frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }z\mathrm{\partial }x}}+w^2{\displaystyle \frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }{z}^2}}.\end{array}\}\end{array}$(16)

Considering the appropriate similarity transformations as (Ibrahim et al., 2022): (17) $\begin{array}{r}\begin{array}{l}\xi ={\left({\displaystyle \frac{a\left(n+1\right).{\left(x+y\right)}^{(n-1)}}{2{\upsilon }_f}}\right)}^{0.5}z,\\ \left(C_w-C_{\mathrm{\infty }}\right)\varphi \left(\eta \right)=C-C_{\mathrm{\infty }}\to \\ \text{For active control of }\varphi ,\\ C_{\mathrm{\infty }}\varphi \left(\eta \right)=C-C_{\mathrm{\infty }}\to \\ \text{For passive control of }\varphi ,\\ \left(T_f-T_{\mathrm{\infty }}\right)\mathrm{\Theta }\left(\eta \right)=T-T_{\mathrm{\infty }},\end{array}\}\end{array}$(17) where the velocity components for present investigations are represented as (18) $\begin{array}{r}\begin{array}{l}u=a{F}^{\mathrm{\prime }}\left(\xi \right)(x+y{)}^n,\text{ v}=a{G}^{\mathrm{\prime }}\left(\xi \right)(x+y{)}^n,\\ w=-{\left({\displaystyle \frac{a{\nu }_f\left(n+1\right){\left(x+y\right)}^{(n-1)}}{2}}\right)}^{0.5}\\ \left(\left(F+G\right)+\left({\displaystyle \frac{n-1}{n+1}}\right)\xi \left(G^{\mathrm{\prime }}+F^{\mathrm{\prime }}\right)\right),\end{array}\}\end{array}$(18)

The aforementioned equations can be substituted in governing partial differential equations which transform them into couplet ODEs of the following form: (19) $\begin{array}{rl}& \left(1+\frac{1}{\text{β}}\right)F^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}+F^{\mathrm{\prime }\mathrm{\prime }}\left(F+G\right)-\frac{2n}{1+n}{F}^{\mathrm{\prime }}\left(F^{\mathrm{\prime }}+G^{\mathrm{\prime }}\right)\\ & -\frac{2}{1+n}{M}^2{F}^{\mathrm{\prime }}-K^{\ast }{F}^{\left(5\right)}-\frac{2}{1+n}\mathrm{\Lambda }\left(1+\frac{1}{\text{β}}\right)F^{\mathrm{\prime }}\left(\eta \right)\\ & -\frac{2}{1+n}{F}_r{F^{\mathrm{\prime }}}^2+\lambda \left(\mathrm{\Theta }+N\varphi \right)=0,\end{array}$(19) (20) $\begin{array}{rl}& \left(1+\frac{1}{\text{β}}\right)G^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}-\frac{2n}{1+n}\left(F^{\mathrm{\prime }}+G^{\mathrm{\prime }}\right)G^{\mathrm{\prime }}+\left(F+G\right)G^{\mathrm{\prime }\mathrm{\prime }}\\ & -\frac{2}{1+n}{M}^2{G}^{\mathrm{\prime }}-K^{\ast }{G}^{\left(5\right)}\\ & -\frac{2}{1+n}{F}_r{G}^{\mathrm{\prime }2}-\frac{2}{1+n}\mathrm{\Lambda }\left(1+\frac{1}{\text{β}}\right)G^{\mathrm{\prime }}=0,\end{array}$(20) (21) $\begin{array}{rl}& \left(\frac{1}{Pr}+\frac{4}{3}Rd\right){\mathrm{\Theta }}^{\mathrm{\prime }\mathrm{\prime }}+\left(F+G\right){\mathrm{\Theta }}^{\mathrm{\prime }}\\ & +\frac{2}{1+n}{M}^{2}Ec{F}^2+{\text{β}}_T{\delta }_1^{\ast }+Nt{\mathrm{\Theta }}^{\mathrm{\prime }2}+Nb{\mathrm{\Theta }}^{\mathrm{\prime }}{\varphi }^{\mathrm{\prime }}=0,\end{array}$(21) (22) $\begin{array}{rl}& {\varphi }^{\mathrm{\prime }\mathrm{\prime }}+\frac{Nt}{Nb}{\mathrm{\Theta }}^{\mathrm{\prime }\mathrm{\prime }}+Pr.Le\left(F+G\right){\varphi }^{\mathrm{\prime }}\\ & +PrLe{\text{β}}_C{\delta }_2^{\ast }+\frac{2}{\left(1+n\right)}Pr.Le{\chi }_1\varphi =0.\end{array}$(22)

With transform boundary conditions: (23) $\begin{array}{r}\begin{array}{l}\begin{array}{l}\text{At}\xi =0,F=G=0,{\mathrm{G}}^{\mathrm{\prime }}=\omega ,\\ F^{\mathrm{\prime }}=1,{\mathrm{\Theta }}^{\mathrm{\prime }}=-\gamma \left(1-\mathrm{\Theta }\right),\\ \varphi =1,\to \text{ For active control of}\varphi ,\\ Nb.{\varphi }^{\mathrm{\prime }}+Nt.{\mathrm{\Theta }}^{\mathrm{\prime }}=0,\to \text{For passive control of}\varphi \end{array}\}\\ \text{at}\xi =\mathrm{\infty },\mathrm{\Theta }\to 0,\varphi \to 0,F^{\mathrm{\prime }}\to 0,G^{\mathrm{\prime }}\to 0.\end{array}\}\end{array}$(23) where the term ${\delta }_1^{\ast }$ and ${\delta }_2^{\ast }$ are (24) $\begin{array}{r}\begin{array}{l}{\delta }_1^{\ast }=\left(n-1\right)F^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\Theta }}^{\mathrm{\prime }}\left(F+G\right)+\left(n-1\right)\left(F+G\right)G^{\mathrm{\prime }}{\mathrm{\Theta }}^{\mathrm{\prime }}\\ +\left({\displaystyle \frac{\left(n-1\right)\xi }{\left(1+n\right)}}{\displaystyle \frac{{\left(n-1\right)}^2\xi }{2\left(1+n\right)}}\right){G^{\mathrm{\prime }}}^2{\mathrm{\Theta }}^{\mathrm{\prime }}-\left(1+n\right)\\ \left({\displaystyle \frac{1}{2}}{F}^2+FG+{\displaystyle \frac{1}{2}}{G}^2\right){\mathrm{\Theta }}^{\mathrm{\prime }\mathrm{\prime }}\\ +\left({\displaystyle \frac{\left(n-1\right)\xi }{\left(1+n\right)}}+{\displaystyle \frac{{\left(n-1\right)}^2\xi }{2\left(1+n\right)}}-{\displaystyle \frac{2n\left(n-1\right)\xi }{\left(1+n\right)}}-{\displaystyle \frac{2n}{\left(1+n\right)}}\right)\\ {F^{\mathrm{\prime }}}^2{\mathrm{\Theta }}^{\mathrm{\prime }}+\left(\left(n-1\right)\xi +\xi \right)F{F}^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\Theta }}^{\mathrm{\prime }}+\left(\left(n-1\right)\xi +\xi \right)G{F}^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\Theta }}^{\mathrm{\prime }}\\ \text{ + }(-{\displaystyle \frac{2n}{\left(1+n\right)}}+2{\left(n-1\right)}^2\xi +{\displaystyle \frac{2\left(n-1\right)\xi }{\left(1+n\right)}}\\ -{\displaystyle \frac{{\left(n-1\right)}^2\xi }{\left(1+n\right)}}-{\displaystyle \frac{2n\left(n-1\right)\xi }{\left(1+n\right)}})G^{\mathrm{\prime }}{F}^{\mathrm{\prime }}{\mathrm{\Theta }}^{\mathrm{\prime }},\end{array}\}\end{array}$(24) (25) $\begin{array}{r}\begin{array}{l}{\delta }_2^{\ast }=\left(n-1\right)F^{\mathrm{\prime }}{\varphi }^{\mathrm{\prime }}\left(F+G\right)+\left(n-1\right).\left(F+G\right)G^{\mathrm{\prime }}{\varphi }^{\mathrm{\prime }}\\ +\left({\displaystyle \frac{\left(n-1\right)\xi }{\left(1+n\right)}}+{\displaystyle \frac{{\left(n-1\right)}^2\xi }{2\left(1+n\right)}}\right){G^{\mathrm{\prime }}}^2{\varphi }^{\mathrm{\prime }}\\ -\left(1+n\right).\left({\displaystyle \frac{1}{2}}{F}^2+FG+{\displaystyle \frac{1}{2}}{G}^2\right){\varphi }^{\mathrm{\prime }\mathrm{\prime }}\\ +\left({\displaystyle \frac{\left(n-1\right)\xi }{\left(1+n\right)}}+{\displaystyle \frac{{\left(n-1\right)}^2\xi }{2\left(1+n\right)}}-{\displaystyle \frac{2n\left(n-1\right)\xi }{\left(1+n\right)}}-{\displaystyle \frac{2n\xi }{\left(1+n\right)}}\right)\\ F^2{\varphi }^{\mathrm{\prime }}+\left(\left(n-1\right)\xi +\xi \right)F{F}^{\mathrm{\prime }}{\varphi }^{\mathrm{\prime }}+\left(\left(n-1\right)\xi +\xi \right)G{F}^{\mathrm{\prime }\mathrm{\prime }}.{\varphi }^{\mathrm{\prime }}\\ \text{ + }(-{\displaystyle \frac{2n.}{\left(1+n\right)}}+2{\left(n-1\right)}^2\xi +{\displaystyle \frac{2\left(n-1\right)\xi }{\left(1+n\right)}}\\ -{\displaystyle \frac{{\left(n-1\right)}^2\xi }{\left(1+n\right)}}-{\displaystyle \frac{2n\left(n-1\right)\xi }{\left(1+n\right)}})G^{\mathrm{\prime }}{F}^{\mathrm{\prime }}{\varphi }^{\mathrm{\prime }}.\end{array}\}\end{array}$(25) where the several model parameters obtained after transformation are presented in Table 1.

Table 1. Details regarding the dimensions and arrangement of the proposed network.

Index	Description	Index	Description
Learning Algorithm	Levenberg–Marquardt	Generated data	bvp4c scheme
Hidden neuron	20–30	Validation	10-fold
Input grid	312	Training performance	70%
Validation test	15%	Testing measure	15%
Fitness values MSE	0	Maximum epochs	1000
Sample selection	Random	Layers (hidden, output, input)	Single
Stopping standard	Default	Max Mu performance	10^9
Mai Mu performance	0.1	Minimum values of the gradient	${10}^{-5}$

The engineering aspects for the current analysis are $Nu$ and $Sh$ are given by Gowda et al. (2023), Ibrahim et al. (2022), Ramzan et al. (2013) and Zuhra et al. (2022): (26) $\begin{array}{r}Nu=\frac{x{q}_w}{k\left(T_f-T_{\mathrm{\infty }}\right)}\text{ , }Sh=\frac{x{p}_w}{D_B\left(C_f-C_{\mathrm{\infty }}\right)}\end{array}$(26) Here terms $q_w\to$ surface heat flux and $p_w\to$ mass flux mathematically stated as: (27) $\begin{array}{r}{q}_w={-k_f.\frac{\mathrm{\partial }T}{\mathrm{\partial }z}|}_{z=0}\text{ ,}{\text{p}}_w={-D_B.\frac{\mathrm{\partial }C}{\mathrm{\partial }z}|}_{z=0}\end{array}$(27)

In non-dimensionless form is (28) $\begin{array}{r}N{u}_x=-\sqrt{{Re}_x}{\left(\frac{n+1}{2}\right)}^{1/2}{\mathrm{\Theta }}^{\mathrm{\prime }}\left(0\right),\\ S{h}_x=-\sqrt{{Re}_x}{\left(\frac{n+1}{2}\right)}^{1/2}{\varphi }^{\mathrm{\prime }}\left(0\right).\}\end{array}$(28)

When the mass flux is equal to zero, then the Sherwood number will also equal zero. (29) $\begin{array}{r}R{e}_x=\frac{U_w\left(x+y\right)}{{\nu }_f}.\end{array}$(29) Equation (29) signifies the Local Reynolds number.

3. Solution methodology

3.1. Formation of data set

Assuming and organizing the relevant information for training the BANN-LMA or for resolving fluid mechanics related problems is referred to as the process of constructing a dataset for projected solutions. Such data set operates as the foundation for training and evaluating the performance of several algorithms, containing neural networks. The choice of representative models, data preparation, and ensuring data integrity and quality are all carefully considered. The dataset’s extensiveness and structure play a crucial role in assisting the model to generalize and produce dependable results. Consequently, the creation of a well-crafted dataset stands as a critical initial phase when utilizing neural networks for successful solutions. With regards to addressing the nonlinear couple stress Casson fluid model (CSCFM), the strategies and techniques presented here are characterized into the following three parts. The classification of present nonlinear CSCFM fluid model contains three categories, specifically velocity field, temperature field, and concentration field. Within each classification, there are two distinct cases. The first phase offers the principal data for the numerical solver bvp4c. This solver is working to establish the dataset based on final couplet system of equations. This consists of transforming higher-order nonlinear ODEs into first-order ODEs while containing the boundary conditions. The succeeding phase presents a suggested computational framework, BANN-LMA, which involves elements like the backpropagation procedure, layer layout, and network plan. The concluding phase offers guidelines for applying the suggested BANN-LMA methodology (Asif Zahoor Raja et al., 2022; Z. Shah et al., 2022; Shoaib, Raja, et al., 2022) to address the tasks of the nonlinear CSCFM. The resulting higher-order ODEs after transformation are (Ibrahim et al., 2022): (30) $\begin{array}{rl}& \begin{array}{l}F={\pi }_1,F^{\mathrm{\prime }}={\pi }_2,F^{\mathrm{\prime }\mathrm{\prime }}={\pi }_3,F^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}={\pi }_4,F^{iv}={\pi }_5,\\ G={\pi }_6,G^{\mathrm{\prime }}={\pi }_7,G^{\mathrm{\prime }\mathrm{\prime }}={\pi }_8,\\ G^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}={\pi }_9,G^{iv}={\pi }_{10},\mathrm{\Theta }={\pi }_{11},{\mathrm{\Theta }}^{\mathrm{\prime }}={\pi }_{12},\\ \varphi ={\pi }_{13},{\varphi }^{\mathrm{\prime }}={\pi }_{14}\end{array}\}\end{array}$(30) (31) $\begin{array}{rl}& {\pi }_5^{\mathrm{\prime }}={\displaystyle \frac{1}{K^{\ast }}}\left\{\begin{array}{l}{\displaystyle \frac{2n}{1+n}}{\pi }_2\left({\pi }_2+{\pi }_7\right)-\left(1+{\displaystyle \frac{1}{\text{β}}}\right)\\ {\pi }_4-{\pi }_3\left({\pi }_1+{\pi }_6\right)+{\displaystyle \frac{2}{1+n}}{M}^2{\pi }_2\\ +{\displaystyle \frac{2}{1+n}}\mathrm{\Lambda }\left(1+{\displaystyle \frac{1}{\text{β}}}\right){\pi }_2\\ +{\displaystyle \frac{2}{1+n}}{F}_r{{\pi }_2}^2-\lambda \left({\pi }_{11}+N{\pi }_{13}\right)\end{array}\right\},\end{array}$(31) (32) $\begin{array}{rl}{\pi }_{10}^{\mathrm{\prime }}& =\frac{1}{K^{\ast }}\left\{\frac{2n}{1+n}\right({\pi }_2+{\pi }_7){\pi }_7-\left(1+\frac{1}{\text{β}}\right){\pi }_9\\ & -\left({\pi }_1+{\pi }_6\right){\pi }_8+\frac{2}{1+n}{M}^2{\pi }_7\\ & +\frac{2}{1+n}{F}_r{{\pi }_7}^2+\frac{2}{1+n}\mathrm{\Lambda }\left(1+\frac{1}{\text{β}}\right){\pi }_7\},\end{array}$(32) (33) $\begin{array}{rl}{\pi }_{12}^{\mathrm{\prime }}& =\frac{-1}{\left(\frac{1}{Pr}+\frac{4}{3}Rd\right)}\left\{\right({\pi }_1+{\pi }_6){\pi }_{12}+\frac{2}{1+n}{M}^{2}Ec{{\pi }_1}^2\\ & +{\text{β}}_T{\delta }_1^{\ast }+Nt{{\pi }_{12}}^2+Nb{\pi }_{12}{\pi }_{14}\},\end{array}$(33) (34) $\begin{array}{rl}{\pi }_{14}^{\mathrm{\prime }}& =-\{\frac{Nt}{Nb}{{\pi }^{\mathrm{\prime }}}_{12}+Pr.Le({\pi }_1+{\pi }_6){\pi }_{14}+PrLe{\text{β}}_C{\delta }_2^{\ast }\\ & +\frac{2}{\left(1+n\right)}Pr.Le{\chi }_1{\pi }_{13}\}.\end{array}$(34)

With transform boundary conditions: (35) $\begin{array}{r}\begin{array}{r}\begin{array}{r}{\pi }_1\left(0\right)={\pi }_6\left(0\right)=0,{\pi }_7\left(0\right)=\omega ,\\ {\pi }_2\left(0\right)=1,{\pi }_{12}\left(0\right)=-\gamma \left(1-{\pi }_{11}\left(0\right)\right),\\ {\pi }_{13}\left(0\right)=1,\to \text{ For active control},\\ Nb.{\pi }_{14}\left(0\right)+Nt.{\pi }_{12}\left(0\right)=0,\to \\ \text{For passive control}\end{array}\}\\ {\pi }_{11}\left(\mathrm{\infty }\right)\to 0,{\pi }_{13}\left(\mathrm{\infty }\right)\to 0,\\ {\pi }_2\left(\mathrm{\infty }\right)\to 0,{\pi }_7\left(\mathrm{\infty }\right)\to 0.\end{array}\}\end{array}$(35)

3.2. Creating artificial neural network

In programming courses, artificial neural networks (ANNs) are a crucial tool. Artificial neural networks, which were originally utilized in the middle of the 20th century, gained popularity because of their improved ability to handle complex nonlinear functions (Dai et al., 2023; Shoaib, Raja, et al., 2022). The most well-known multilayer perceptron (MLP) systems are used in artificial neural network (ANN) models (Zhao et al., 2023). The neural network is made up of (i) weights, (ii) input/output, (iii) activation mechanism, and hidden layer. Backpropagation feed-forward (BPFF) is the most popular approach used in multilayer perceptron networks (Shoaib, Kainat, et al., 2022). The network then creates a relationship with at least one hidden layer, where the weighted connection’s structure is doing critical computations. Towards the end of training, error eventually decreases. Four ANN models were created in the current study to forecast the movement of a CSCFM through an expanding medium utilizing the Cattaneo–Christov and Darcy–Forchheimer models. Weights described as an input limitation in the involvement of neurons in the hidden layer as well as the prediction values collected in shape of system performance, gradient and histogram investigation at the output layer of the artificial neural networks model developed for the prediction of $\text{β},\omega ,\gamma ,Ha,Rd,Fr$ and $Nt$against $f\left(\eta \right),g\left(\eta \right)$ and $\theta \left(\eta \right)$. In Figure 2(a,b) we see the basic framework for setting up the proposed ANN models. Only 15% of the input used in the developed ANN models is intended for use in the testing process, 15% is intended for use in the validation stage, and 70% is intended for use in ANN model training. A MLP network model was designed to estimate the variables $\text{β},\omega ,\gamma ,Ha,Rd,Fr$ and $Nt$against $f\left(\eta \right),g\left(\eta \right)$ and $\theta \left(\eta \right)$ using 410 data sets from 8 distinct data components. 349 samples were used for training in this analysis, whereas 61 data sets were used for testing and validation.

Figure 2. (a) The structure of signal neural network model, (b) Construction of neural network for CSCFM.

3.3. ANN optimization

The sigmoid function, whose expression is $y\left(w_i\right)=\frac{1}{1+e^{-w_i}},w_i$ is the weighted sum of input connection has been included in the present research as the transformation function /activation function of the MLP network. Among the most popular transformation functions is the sigmoid function. Its precise performance on nonlinear regression issues in various applications leads to its adoption in MLP networks. During the training phase, errors were evaluated using a performance measure. In the present research, the determination MSE and R² (correlation coefficient) have been used to assess the network’s accuracy. Which is define as (Asif Zahoor Raja et al., 2022; Z. Shah et al., 2022; Shoaib, Raja, et al., 2022; Zhao et al., 2023): (36) $\begin{array}{r}\text{MSE}=\frac{1}{N}\sum _{k=1}^N{\left(T_k-A_k\right)}^2,\end{array}$(36) (37) $\begin{array}{r}R=\sqrt{\left(\frac{\sum _{k=1}^N{\left(T_k\right)}^2-\sum _{k=1}^N{\left(T_k-A_k\right)}^2}{\sum _{k=1}^N{\left(T_k\right)}^2}\right)},\end{array}$(37)

4. Results and discussion

Using MATLAB ‘bvp4c’ numerical solver, the nonlinear differential equations and their related boundary conditions are solved numerically. The original differential equations are changed into first-order form to facilitate this procedure. By changing the values of the physical model parameters, as shown in Table 2, several scenarios are investigated. Figure 3 depicts the full computing process of current analysis.

Figure 3. Working flowchart for current analysis.

Table 2. Non-dimensional model parameters.

Parameter	Symbol	Formula
Ratio parameter	$\omega$	$\frac{b}{a}$
Lewis number	$Le$	$\frac{{\alpha }_f}{D_B}$
Thermophoresis parameter	$Nt$	$\frac{D_T{\left(\rho c_p\right)}_p\left(T_f-T_{\mathrm{\infty }}\right)}{{\left(\rho c_p\right)}_f{T}_{\mathrm{\infty }}{\nu }_f}$
Prandtl number	$Pr$	$\frac{{\nu }_f}{{\alpha }_f}$
Time relaxation parameter of temperature	${\text{β}}_T$	$a{\mathrm{\Psi }}_E(x+y{)}^{n-1}$
Time relaxation parameter of concentration	${\text{β}}_C$	$a.{\mathrm{\Psi }}_C(x+y{)}^{n-1}$
Inertia parameter	$F_r$	$F_b\left(x+y\right)$
Biot number	$\gamma$	$\frac{h}{k}.\sqrt{\frac{2{\nu }_f}{a\left(n+1\right)}}$
Hartman number	$M$	$\sqrt{\frac{{\sigma }_f{B}_0^2{(x+y)}^{1-n}}{a{\rho }_f}}$
Eckert number	$Ec$	$\frac{{\stackrel{⌢}{U}}_w^2}{c_p\left(T_{\mathrm{\infty }}-T_f\right)}\text{, }$
Porosity parameter	$\mathrm{\Lambda }$	$\frac{{\nu }_f{\left(x+y\right)}^{1-n}}{aK}$
Radiation variable	$Rd$	$\frac{4T_{\mathrm{\infty }}^3{\sigma }^{\ast }}{k{k}^{\ast }}$
Brownian motion factor	$Nb$	$\frac{D_B{\left(\rho c_p\right)}_p\left(C_w-C_{\mathrm{\infty }}\right)}{{\nu }_f{\left(\rho c_p\right)}_f}\to \text{For active control of}\varphi$
		$\frac{D_B{\left(\rho c_p\right)}_p\left(C_{\mathrm{\infty }}\right)}{{\nu }_f{\left(\rho c_p\right)}_f}\to \text{For passive control of}\varphi$

4.1. Effect of model parameters on heat and concentration profile

Following the implementation of the MATLAB ‘bvp4c’ numerical solver to acquire the solution of the nonlinear ODEs, the solution is drawn to show the presence of numerous significant factors, including the $M,\gamma ,K^{\ast },\text{β},Nt,Nb,Pr$ and $\omega$, on the $\mathrm{\Theta }\left(\xi \right)$ and $\varphi \left(\xi \right)$ fields. Results are obtained for both passive and active control of nanoparticles. Figure 4(a,b) displays the variation in $\mathrm{\Theta }\left(\xi \right)$ and $\varphi \left(\xi \right)$ for varying values of $M$. An enhancement in $M$ signifies an augmentation in the $\mathrm{\Theta }\left(\xi \right)$ and $\varphi \left(\xi \right)$ profiles for both the active and passive control cases. The magnetic field, which lowers the liquid’s movement and increases the heat and concentration profiles, causes the body force to diminish as would be expected. Additionally, from Figure 4(a) the $\mathrm{\Theta }\left(\xi \right)$ distribution for passive control is smaller than it is for active control. Figure 4(b) illustrates how the $M$ affect the $\varphi \left(\xi \right)$ profile. By increasing the $M$, the $\varphi \left(\xi \right)$ profile is enhanced for both active and passive controller scenarios. Figure 4(c) shows the effect of the $\gamma$ stimulation on $\mathrm{\Theta }\left(\xi \right)$ field. To produce stronger convection, the $\gamma$ can be elevated. As a result, $\mathrm{\Theta }\left(\xi \right)$ field and thermal layer thickness rise under control situations. Figure 4(d) is developed to demonstrate how the $\gamma$impacts the $\varphi \left(\xi \right)$ profile. In this situation, the $\varphi \left(\xi \right)$ profile shows a propensity to rise in both active and passive nanoparticle control conditions. The effect of $Nb$ on the $\mathrm{\Theta }\left(\xi \right)$ and $\varphi \left(\xi \right)$ field is shown in Figure 4(e,f) for both cases. The collision of fluid atoms or molecules will result in an arbitrary motion known as the Brownian motion of suspended (pendulous) particles, which will enlarge the boundary layer in both active and passive control scenarios, as shown in Figure 4(e). Figure 4(f) demonstrates that choosing a higher $Nb$ parameter value results in a weaker $\varphi \left(\xi \right)$ field for both types of nanoparticles (Table 3).

Figure 4. Effect of various parameters on $\mathrm{\Theta }\left(\xi \right)$ and $\varphi \left(\xi \right)$ profile. (a) Effect of $M$ on $\mathrm{\Theta }\left(\xi \right)$. (b) Effect of $M$ on $\varphi \left(\xi \right)$. (c) Effect of $\gamma$ on $\mathrm{\Theta }\left(\xi \right)$. (d) Effect of $\gamma$ on $\varphi \left(\xi \right)$. (e) Effect of $Nb$ on $\mathrm{\Theta }\left(\xi \right)$. (f) Effect of $Nb$ on $\varphi \left(\xi \right)$.

Table 3. Variation in different parameters of ANN for various scenarios.

	Model parameters
Cases	$M$	$\gamma$	$K^{\ast }$	$\text{β}$	$Nt$	$Nb$	$\lambda$	$Pr$	Scenario
1	0.2	0.3	0.5	0.3	0.5	0.2	0.3	1.2	I
2	0.4	0.3	0.5	0.3	0.5	0.2	0.3	1.2
3	0.6	0.3	0.5	0.3	0.5	0.2	0.3	1.2
1	0.2	0.3	0.5	0.3	0.5	0.2	0.3	1.2	II
2	0.2	0.5	0.5	0.3	0.5	0.2	0.3	1.2
3	0.2	0.7	0.5	0.3	0.5	0.2	0.3	1.2
1	0.2	0.3	0.3	0.3	0.5	0.2	0.3	1.2	III
2	0.2	0.3	0.6	0.3	0.5	0.2	0.3	1.2
3	0.2	0.3	0.9	0.3	0.5	0.2	0.3	1.2
1	0.2	0.3	0.5	0.1	0.5	0.2	0.3	1.2	IV
2	0.2	0.3	0.5	0.4	0.5	0.2	0.3	1.2
3	0.2	0.3	0.5	0.8	0.5	0.2	0.3	1.2

4.2. Analysis of arterial neural network model results

To implement the suggested solution methodology, reference solutions in the form of the numerical data set produced via bvp4c against the $F^{\mathrm{\prime }}\left(\xi \right),\mathrm{\Theta }\left(\xi \right)$ and $\varphi \left(\xi \right)$ are delivered to the neural network input. The neural network is trained with 75% of the points, with the remaining 15% being used for testing and validating the neural network. Figure 2(a,b) depicts the solution technique design and bi-layer architecture. A visual comparison of a neural network’s output to the output of a fixed domain can be seen in time-series charts. These graphical representations are crucial for recognizing the consistency between the desired and actual values during the network’s training, validation, and testing phases. Figure 5(a) displays the transient stability for case (1) in scenario (I). An error histogram, which breaks down the full error range into 20 bins, is used to analyse the dispersion of errors. The total number of errors for the network’s training, validation, and testing stages is shown as a vertical line for each error range. The reliability and correctness of the solution outcomes are indicated by the presence of maximum error values near to the zero-error line is $-7.5\times {10}^{-6}$. The error distribution for case (1) in scenario (I) is displayed in Figure 5(b). Another helpful technique for evaluating the consistency between desired and actual values for the training, validation, and testing phases separately is regression analysis. This evaluation can be done mathematically or visually. A straight line is used to represent the reference solution in graphical representations, while BANN-LMA results are represented by little rings or spots. An $R=0$ denotes a lack of correlation between the targeted and obtained results, whereas an $R=1$ indicates outstanding correlation. A regression plot for case (1) in scenario (I) is shown in Figure 5(c). When analysing the quality and reliability of the machine learning model, the MSE (mean square error) is an essential aspect. MSE is calculated numerically by averaging the squares of the discrepancies between the values that were produced and a benchmark solution. A lower MSE value indicates greater accuracy and reliability in the findings, demonstrating the dependability of the solution technique. The MSE plot for case (1) in scenario (I) over the training, validation, and testing data sets is shown in Figure 5(d) after 445 epochs in total time 28s with MSE is $7.7741\times {10}^{-10}$.

Figure 5. Performance results variations of Case 1, scenario I.

For scenarios II, III, and IV, the subfigures of Figures 6–8 contain time-series response plots (Figures 6(a), 7(a), 8(a)), error histograms (Figure 6(b) with maximum error values near to the zero-error line is −7.11$\times {10}^{-6}$, 7(b) with −1.5$\times {10}^{-6}$, 8(b) with −9.6$\times {10}^{-7}$), regression graphs (Figures 6(c), 7(c) and 8(c)) and MSE plots (Figure 6(d) after 336 epochs in total time 22s with MSE is 1.804$\times {10}^{-9}$, 7(d) after 180 epochs in total time 17s with MSE is 2.6915$\times {10}^{-10}$, 8(d) after 78 epochs in total time 21s with MSE is 3.7068$\times {10}^{-11}$) respectively.

Figure 6. Performance results variations of Case 1, scenario II.

Figure 7. Performance results variations of Case 1, scenario III.

Figure 8. Performance results variations of Case 1, scenario IV.

4.3. Analysis of comparison of suggested and ref solutions with absolute errors

In comparison to the reference dataset of CSCFM, specifically the parameter $K^{\ast }$, Figure 9 shows the resultant graphs and evaluation of the variation in $F^{\mathrm{\prime }}\left(\xi \right)$ using BANN-LMA. We investigate the effect of the $K^{\ast }$ on $F^{\mathrm{\prime }}\left(\xi \right)$in Figure 9(a). The behaviour of $K^{\ast }$ depends on the couple stress viscosity, which acts as a barrier to flow and increases fluid density, which in turn increases fluid motion. Figure 9(a) demonstrates clearly how fluid velocity inside the flow zone accelerates as value of $K^{\ast }$ increases. An assessment of $F^{\mathrm{\prime }}\left(\xi \right)$ s absolute errors is shown in Figure 9(b), with the absolute errors lying within the range 10⁻⁴ and 10⁻⁸. While observing Figure 10(a,b), it becomes evident that $F^{\mathrm{\prime }}\left(\xi \right)$ profile experience a reduction as the value of $M$increases, and this is accompanied by absolute error ranges between 10⁻³ and 10⁻⁷. This decline in $F^{\mathrm{\prime }}\left(\xi \right)$ profile is mostly because of the Lorentz force produced by the applied magnetic field. It’s significant that the Lorentz force resists the motion of liquid, preceding to a reduction in its overall motion. Furthermore, such force creates additional heat, identified as Joule heating, within the system. Therefore, the employment of higher value of $M$ in the system results in an expansion in $\mathrm{\Theta }\left(\xi \right)$ profile within the flow region, which is spotted in both active and passive control cases, as indicated in Figure 11(a,b) with absolute error ranges ranging from 10⁻⁵ to 10⁻⁸. In Figure 11(a,b), we display the visual interpretations of how $Rd$ impact the $\mathrm{\Theta }\left(\xi \right)$ in both scenarios, while simultaneously operating an AE analysis within the range of 10⁻⁵ to 10⁻⁸. $Rd$ plays an extensive role in governing the transfer of heat energy within and between the liquid and its surroundings. The $Rd$ serves as a quantification of the impact of radiation in the situation of heat transport. A higher value $Rd$ specifies an elevated status of radiation effects, indicating to higher rates of heat transport and alterations in thermal distribution. As the $Rd$ escalates, the surface temperature also experiences an increase, then ensuing a rise in both the thickness of the thermal boundary layer and thermal resistance. Also, observing Figure 12(a,b), it becomes evident that $\varphi \left(\xi \right)$ profile experience a reduction as the value of $\gamma$ increases in case of active control while for passive control present opposite behaviour, and this is accompanied by AE ranges between 10⁻⁵ and 10⁻⁷. Similarly, the employment of higher value of $Nb$ in the system results in an expansion in $\varphi \left(\xi \right)$ profile within the flow region, which is spotted in both active and passive control cases, as indicated in Figure 13(a,b) with absolute error ranges ranging from 10⁻³ to 10⁻⁸.

Figure 9. Comparison of suggested and ref solutions and absolute errors for various values of $K^{\ast }$.

Figure 10. Comparison of suggested and ref solutions and absolute errors for various values of $M$.

Figure 11. Comparison of suggested and ref solutions and absolute errors for various values of $M$.

Figure 12. Comparison of suggested and ref solutions and absolute errors for various values of $Rd$.

Figure 13. Comparison of suggested and ref solutions and absolute errors for various values of $\gamma$.

 Figure 14(a) shows comparisons between the current study and existing literature (Ramzan et al., 2013), whereas Figure 14(b) emphasizes the numerical comparison between numerical and ANN outcomes.

Figure 14. Comparison of suggested and ref solutions and absolute errors for various values of $Nt$.

 Tables 4–7 show the significance of each variation within the suggested methodology and were individually prepared for each of the four scenarios (1–4). Figure 15(a) illustrates the contrast between our present work and the existing literature (Ramzan et al., 2013), whereas Figure 15(b) demonstrates the validation of our findings by comparing the ANN results with the numerical dataset. Both figures exhibit a high level of concordance, thereby guaranteeing the precision of our current work and preventing any potential confusion or misinterpretation.

Figure 15 (a) Assessment of $F^{\mathrm{\prime }\mathrm{\prime }}\left(0\right),G^{\mathrm{\prime }\mathrm{\prime }}\left(0\right)$ result with published work (Ramzan et al., 2013) for $\omega$, when $M=n=\mathrm{\Lambda }=\text{β}=0$, (b) Comparison of $F^{\mathrm{\prime }}\left(\xi \right)$ numerical and ANN results.

Table 4. Outcome of LMBP-NNs (Levenberg–Marquardt backpropagation neural networks) technique scenario 1.

						MSE
Case	Performance	Gradient	Mu	Epoch	Time	Train data	Validation data	Testing data
1	2.71 $\times {10}^{-10}$	2.71 $\times {10}^{-9}$	4.0 $\times {10}^2$	445	0:00:28	7.3 $\times {10}^{-11}$	3.1 $\times {10}^{-12}$	2.3 $\times {10}^{-12}$
2	7.18 $\times {10}^{-12}$	1.10$\times {10}^{-8}$	7.7$\times {10}^4$	731	0:00:35	2.3$\times {10}^{-11}$	1.7$\times {10}^{-10}$	3.8$\times {10}^{-12}$
3	5.32 $\times {10}^{-11}$	9.38 $\times {10}^{-9}$	3.8 $\times {10}^3$	513	0:00:16	1.8 $\times {10}^{-11}$	5.3 $\times {10}^{-11}$	9.5 $\times {10}^{-11}$

Table 5. Outcome of LMBP-NNs (Levenberg–Marquardt backpropagation neural networks) technique scenario II.

						MSE
Case	Performance	Gradient	Mu	Epoch	Time	Train data	Validation data	Testing data
1	2.13 $\times {10}^{-10}$	2.53 $\times {10}^{-9}$	2.9 $\times {10}^2$	336	0:00:40	2.5 $\times {10}^{-10}$	2.3 $\times {10}^{-12}$	6.1 $\times {10}^{-11}$
2	1.25 $\times {10}^{-10}$	9.02$\times {10}^{-9}$	7.2$\times {10}^3$	89	0:00:31	3.3$\times {10}^{-11}$	7.0$\times {10}^{-10}$	3.9$\times {10}^{-11}$
3	4.82 $\times {10}^{-9}$	6.88 $\times {10}^{-9}$	4.1 $\times {10}^2$	394	0:00:30	1.2 $\times {10}^{-11}$	4.6 $\times {10}^{-10}$	1.7 $\times {10}^{-11}$

Table 6. Outcome of LMBP-NNs (Levenberg–Marquardt backpropagation neural networks) technique scenario III.

						MSE
Case	Performance	Gradient	Mu	Epoch	Time	Train data	Validation data	Testing data
1	8.04 $\times {10}^{-10}$	5.23 $\times {10}^{-9}$	1.1 $\times {10}^2$	180	0:00:27	7.2 $\times {10}^{-10}$	4.7 $\times {10}^{-12}$	3.7 $\times {10}^{-12}$
2	2.93 $\times {10}^{-10}$	1.72$\times {10}^{-9}$	3.8$\times {10}^3$	338	0:00:41	3.9$\times {10}^{-10}$	7.8$\times {10}^{-12}$	4.4$\times {10}^{-11}$
3	7.26 $\times {10}^{-10}$	8.18 $\times {10}^{-9}$	3.4 $\times {10}^3$	726	0:00:34	3.5 $\times {10}^{-11}$	3.0 $\times {10}^{-11}$	1.3 $\times {10}^{-11}$

Table 7. Outcome of LMBP-NNs (Levenberg–Marquardt backpropagation neural networks) technique scenario IV.

						MSE
Case	Performance	Gradient	Mu	Epoch	Time	Train data	Validation data	Testing data
1	3.41 $\times {10}^{-10}$	8.38 $\times {10}^{-9}$	7.31 $\times {10}^2$	78	0:00:27	7.2 $\times {10}^{-10}$	4.7 $\times {10}^{-12}$	3.7 $\times {10}^{-12}$
2	5.22 $\times {10}^{-10}$	2.30$\times {10}^{-9}$	2.82$\times {10}^3$	392	0:00:41	3.9$\times {10}^{-10}$	7.8$\times {10}^{-12}$	4.4$\times {10}^{-11}$
3	1.71 $\times {10}^{-10}$	5.08 $\times {10}^{-9}$	8.74 $\times {10}^3$	309	0:00:34	3.5 $\times {10}^{-11}$	3.0 $\times {10}^{-11}$	1.3 $\times {10}^{-11}$

5. Conclusions

The present investigation aims to explore the impacts of coupling stress Casson fluid models with convectional properties under active and passive control using advanced AI-based computational methodologies. These approaches power the neural networks enhanced with the Levenberg–Marquardt Algorithm (NN-LMA). The investigation incorporates the Cattaneo–Christov heat and mass flux models and employs the Rosseland approximation to account for thermal radiation effects. The basic partial differential equations (PDEs) explaining flow patterns are transformed into ODEs through appropriate conversion manner. This system of equations are then solved using the robust bvp4c solver to produce datasets for ANN analysis. Subsequently, the BANN-LMA method is applied to the reference dataset to approximate solutions for the CSCFM models. As shown by a minimal MSE of about 10⁻¹⁰ in the training charts, the findings of this study show a high degree of agreement between the approximations and the real data. In error histograms, around 90% of the output dynamics are centred around the zero line, demonstrating a strong agreement between expected and actual values. Additionally, the regression value constantly equals 1 in each case, highlighting the precision of the results. The main objective of the current analysis is described below, focusing on how the physical characteristics of couple stress Casson fluid models change.

• The $M$ parameter reduces $F^{\mathrm{\prime }}\left(\xi \right)$ profile and results in a thinner momentum boundary layer, while enhances the fluid’s thermal behaviour.

• Increasing the $K^{\ast }$ and $\lambda$ parameter leads to an acceleration of fluid motion.

• $\mathrm{\Theta }\left(\xi \right)$ distribution can be controlled by adjusting the $Rd,Nt,Pr$ and $\gamma$ to finer values.

Both temperature and concentration distributions are more pronounced in cases of active control compared to passive control.

• As the $Nt$ increases, the mass transfer rate declines, while the $Nb$ exhibits the opposite trend.

• A higher $\gamma$ signifies increased convective effects, which enhance the temperature.

• In the future, one could employ the remarkable capabilities of artificial intelligence-based methodologies, utilizing both supervised and unsupervised neural networks, in various applications within the field of fluid mechanics such as (Nasir, Alghamdi, et al., 2023; Ramzan et al., 2013; Wang et al., 2023; K. Zhang et al., 2023). Future study may also expand on the current research to include non-Newtonian liquids and unsteady behaviour under diverse physical conditions. To ensure that the study continues to make significant contributions and remains current in its specific field, these suggested areas for extension might serve as a road map for future research directions.

Nomenclature

List of symbols

$u$, $v$ $w$	=	Components of velocities $\left(\text{m }{\text{s}}^{-1}\right)$
$B$	=	Magnetic field strength $\left(\text{Nm}{\text{A}}^{-1}\right)$
$F,G$	=	Dimensional velocity profile
$T_f,T_w$	=	Fluid and wall temperature $\left(\text{K}\right)$
$T_{\mathrm{\infty }}$	=	Free surface temperature $\left(\text{K}\right)$
$F_b$	=	Inertia coefficient
$M$	=	Magnetic parameter
$k^{\ast }$	=	Rosseland means absorption constant
$Pr$	=	Prandtl number
$Re$	=	Local Reynolds number
$C_b$	=	Drag force
$K$	=	Porous medium permeability
$D_B$	=	Brownian diffusivity
$D_T$	=	Thermophoretic diffusion coefficient
$R_d$	=	Radiation parameter
$C_f,C_w,C_{\mathrm{\infty }}$	=	Fluid, wall, free space concentration
$(C_p{)}_f$	=	Specific heat fluid $\left(\text{J}/\text{kg}\text{K}\right)$
$k_f$	=	Thermal conductivity of fluid $\left(\text{W}{\text{m}}^{-1}{\text{K}}^{-1}\right)$
$K^{\ast }$	=	Couple stress parameter
$Nt$	=	Thermophoresis parameter
$Le$	=	Lewis number
$Ec$	=	Eckert number

Greek symbols

$\mu$	=	Dynamic viscosity $\left(\text{Kg}/\text{ms}\right)$
$\upsilon$	=	Kinematic viscosity $\left({\text{m}}^2/\text{s}\right)$
$\rho$	=	Density $\left(\text{Kg}{\text{m}}^{-3}\right)$
$\mathrm{\Theta },\varphi$	=	Dimensional thermal, concentration profiles
$\xi$	=	Similarity variable
$\varphi$	=	Nanoparticle volume fraction
$\omega$	=	Ratio parameter
$\sigma$	=	Electrical conductivity $\left({\mathrm{\Omega }}^{-1}{\text{m}}^{-1}\right)$
${\sigma }^{\ast }$	=	Stefan Boltzmann constant
${\mathrm{\Psi }}_E$	=	Thermal relaxation time coefficient
${\mathrm{\Psi }}_E$	=	Concentration relaxation time coefficient
$\text{β}$	=	Casson parameter

Abbreviation

ANN	=	Artificial neural network
BANN-LMA	=	Back propagation artificial neural network with Levenberg–Marquardt Algorithm
CSCNF	=	Couple stress Casson nanofluid
MSE	=	Mean square error
AE	=	Absolute error
MLP	=	Multi-Layer perception
ARD	=	Average relative deviation
PEDs	=	Partial differential equations
MHD	=	Magneto-hydrodynamics

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors acknowledge the financial support from Khalifa University of Science, Technology and Research [grant number RC2-2018-024].