A semi analytical simulation and Bayesian Regularization technique for thermophoretic particle deposition in radiative nanofluid flow with Brownian motion and Stefan blowing impacts

Abstract

By using the computational power of Bayesian Regularization Optimizer Supervised Neural Networks (BRO-SNNs), this semi-analytical model, which investigates thermophoretic particle deposition in a Boger nanofluid flow over an inclined sheet using Brownian motion and Stefan blowing, has important applications in advanced heat management and materials processing. It provides a key framework for enhancing heat exchanger designs, demonstrating how controlled nanoparticle deposition can improve heat transfer or avoid surface fouling. The findings are immediately applicable to coating and thin-film manufacturing processes, allowing for fine control over deposit uniformity and thickness by manipulating thermophoresis and Stefan blowing. Furthermore, the model can be used in environmental engineering to estimate particle deposition in scrubbers and filtration systems, as well as in aircraft to manage dust or ice formation on critical components. The viscoelastic character of Boger fluids makes it especially useful for creating next-generation smart cooling systems based on polymer-based nanofluids. The HAM solves the non-dimensional non-linear governing ODEs (ordinary differential equations) that are derived following appropriate similarity transformations. The velocity field upsurges as increase the values of nanoparticles solvent friction while thermal profile decline.

Introduction

Boger fluids also known as constant-viscosity elastic liquids, are ideal for both theoretical and experimental studies due to their unique rheological properties. Because of their very low polymer content, Boger fluids do not show shear-thinning behaviour like the majority of viscoelastic fluids, which keeps the viscosity nearly constant even when deformed. These fluids are created by combining a highly viscous Newtonian solvent with a tiny amount of polymer; common examples include corn syrup solutions and polyacrylamide mixtures. Their composition allows for the separation of elastic and shear effects during analysis. The flow properties of fluids with comparable viscosities that are Newtonian and BF (Boger fluid), researchers can gain a better understanding of how elasticity affects flow resistance and transport phenomena. In order to comprehend non-Newtonian performance, this information is essential. The BF are distinct rheological properties make it useful in an extensive range of engineering. In fluid rheology and mechanics research, complex liquids must be used as model fluids in order to comprehend their behaviour. The Boger fluid is used by biomedical researchers to optimise the creation of pharmaceuticals and medical equipment delivery systems because it has a viscosity that is comparable to biological fluids. They are also used in consumer goods manufacturing, where exact control over resistance is essential, such as in cosmetics and food. The BF are used in the gas and oil sector to boost drilling efficacy. Additionally, they are employed in engineering projects to design checking systems that offer security and stability. In order to prevent the polymer accompaniments’ shear thinning properties, the most commonly used solvents in Boger fluids are very sticky. In order to investigate mass transmission and MHD flow in a Boger THNF over a porous disk, combined an ANN technique with the CCHFM was analyze by Rehman et al. [1]. The deep neural networks had been employed by Jalili et al. [2] to forecast the temperature and flow distributions of magnetised Boger micropolar THNF. Jangid et. al [3] inspected the mathematical examination of Cattaneo-Christov heat flux model on Carreau THNF flow across a sheet. Kumar et al. [4] scrutinized the impacts of chemical reactions on the flow behavior of BF among two parallel disks using an ANN framework. Okasha et al. [5] analyze the LTNE model and a modified HCM to analyze the elastic deformation features of HNF (hybrid nanofluids). Baskaran et al. [6] used the Cattaneo-Christov model to perform a computational heat transmission study of THNF flow across a disk.

The Buongiorno’s nanofluid model is used in this NF (nanofluid) velocity model. The base liquid for this model contains NPs (nanoparticles) ranging in size from 10 to 100 nm. Under these conditions, the solid phase is made up of suspended nanoparticles, while the continuous phase is liquid. The resultant nanofluid, which is made up of NPs suspended in a base fluid, possesses basic features like density, viscosity, specific heat capacity and temperature conductivity. When NPs are present, the overall characteristics of the nanofluid are altered in comparison to the base liquid alone, even in the dense phase. Yet, the nanoparticles themselves are scattered throughout the base liquid in the dense phase and do not change its characteristics. Instead, thermophoresis impacts and Brownian motion result from the random motions of nanoparticles, and these have a big impact on mass and heat transport. The result of magnetic fields on HNF (hybrid nanofluid) flow in an annulus under two different heat sources was investigated by Thriveni et al. [7] using response surface methodology and an enhanced Buongiorno model. Norzawary et al. [8] used a modified Buongiorno model that included Arrhenius activation energy effects to analyze the flow of liquid over a surface. Oluwaseun and Agbaje [9] numerically investigated hydromagnetic flow and thermal transportation in a HNF over a revolving disc, taking into account slip boundary conditions and Hall current influences within a porous medium. The pulsatile MHD flow of a micropolar NF (nanofluid) in a vertical permeable surface was scrutinized by Rajkumar et al. [10] using the Buongiorno formulation, taking radiative heat transfer, thermophoresis, and Brownian diffusion into consideration. Nath et. al. [11] investigated MHD transport of micropolar and Casson hybrid nanofluids through a porous channel under nonlinear thermal radiation using a modified Buongiorno model.

Aerosol particles transition from regions with high to low temperatures of a gas through a physical phenomenon known as thermophoresis. This phenomenon has drawn a lot of attention as a means of gathering submicrometer and nanometre particles. In fluid dynamics with highly temperature-dependent viscosities, the relationship between the momentum and energy equations makes viscous heating crucial. Because the higher temperature of a semiconductor wafer facilitates the formation of the transition structure required for atomic jumps, impurity ions may migrate from the cold to the hot side. Diffusion flux can occur both up and down the temperature gradient, depending on the materials used. Thermophoretic force is used in commercial precipitators for the same reasons as electrostatic precipitators. In order to create optical fibres, it is employed in the vapour deposition process. It might be helpful in fouling as a transport mechanism. Because thermophoresis allows for the identification of aptamer interaction by contrasting the target molecule’s motion when bound and unbound, it has also been demonstrated to have the potential to support drug discovery. This technique is known as microscale thermophoresis. Thermophoresis is a method used in field flow fractionation to separate different polymer particles. Deposition of contaminating particles on product surfaces is particularly significant in assessing product quality in the electronic industry. Using microorganisms and an internal heat source, Iqbal et al. [12] investigated thermophoretic particle deposition in bioconvective nanofluid flow. Prasannakumara and Gowda [13] examined how radiative fluid flow behaved in terms of mass and heat transmission when thermophoretic deposition and a uniform horizontal magnetic field were both present. The TPD (Thermophoretic particle deposition) in NF flow across a disc was investigated by Ragupathi et al. [14] while taking non-Fourier heat flux factors into consideration. In an inclined porous annulus, Shilpa et al. [15] examined the thermo-solutal convection of an aqueous FeO₄ nanofluid while taking electrophoretic and thermophoretic forces into account. Khan et al. [16] performed a numerical investigation of wall-jet flow using spherical graphene oxide nanoparticles in a NNF (non-Newtonian fluid) with a non-uniform heat source that included thermophoretic deposition. Abbas et al. [17] inspected Marangoni convection under the effects of thermophoresis in a dusty hybrid nanofluid containing gyrotactic microbes.

The Stefan-blowing impact is caused by species diffusion, which creates bulk fluid motion. Stefan blowing affects fluid bulk motion, and species transmission varies with the flow field. As a result, modelling the flow requires a combination of momentum and NPVF. Evaporation is considered a natural phenomenon in soil science and process industries, based on the Stefan blowing principle. The literature states that when there is a mass flux from the surface to the environments during paper drying, SB (Stefan blowing) takes place. The SB is fundamentally dissimilar from conventional mass injection or transpiration-induced blowing. In transpiration-driven processes, injection or suction takes place through a permeable surface, whereas Stefan blowing originates from an impermeable boundary due to phase change effects. With consideration for thermal radiation, activation energy, and electric field effects, the influence of SB on unsteady magnetohydrodynamics NF flow across a sheet was scrutinize by Haider et al. [18]. Geetha et al. [19] inspected the mass and heat transmission behaviour of magnetohydrodynamics NF flow across a stretching surface embedded in a permeable medium during SB. Ali et al. [20] inspected the impacts of heat transmission on flow across a spinning disk in a permeable medium while controlling for Stefan blowing. The electro-osmotic flow of a tangent hyperbolic NF across a permeable spinning disc, accounting for SB, thermal radiation, and temperature-dependent thermal conductivity was analyse by Senbagaraja and De [21]. Furthermore, Thenmozhi et al. [22] scrutinised the result of SB on the flow and thermal transportation of a NNF across a non-uniform sheet. The combined properties of double stratification and SB on convective HNF flow, including entropy generation and CBCs (convective boundary conditions) was examining by Joyce et al. [23]. Jangid et. al [24] and Loganathan et. al [25] analyse mass and heat transmission in Williamson and Oldroyd-B nanofluids across a sheet and second-order slip conditions. Heat transmission in Casson fluid motion decreased by a flat plate with Ohmic heating was modelled and simulated by Dey et al. [26]. Choudhury et al. [27] examined hydromagnetic oscillatory slip flow of a visco-elastic fluid over a channel. The properties of viscous dissipation and slip on viscous fluid flow with different fluid specifications were combined by Kalita et al. [28]. Mixed convection characteristics on unsteady SPF (Stagnation Point Flow) of a NF with heat generation were scrutinized by Islam et al. [29].

By providing real-time pressure, velocity and thermal predictions for intricate fluid flow applications, ML (machine learning) techniques—such as neural networks and regression models—offer a high computational power substitute for traditional CFD (computational fluid dynamics) simulations. Thus, ML is perfect for improving heat exchangers and energy-saving fluid systems as it reduces computational cost while maintaining correctness. Additionally, compared to traditional turbulence models, ML-based models may advance turbulence predictions by learning from high-fidelity simulations like DNS (Direct Numerical Simulation). ML advances fluid flow examination in uses ranging from industrial cooling systems to aerodynamic efficacy by identifying crucial factors that maximize efficacy and minimize energy losses. Fluid dynamics research and engineering uses will change as a result of this integrating ML with CFD, which reduces computing cost and advances accurateness. Aoudia et al. [30] investigated the use of ANN (artificial neural networks) to enhance heat and mass transportation in HNF with gyrotactic microbes and LTNE phenomena. The thermal behavior paradigm in tangent hyperbolic HNF flow was inspected by Athar et al. [31] using ANN. For the Boger THNF, Liaqat et al. [32] investigated the intelligent Levenberg–Marquardt approach in comparison to XM and HCM. Hadidi et al. [33] inspected the Bayesian-regularized neural network analysis of heat transmission HNF with LTNE influence. The design-oriented imitation of a HNF based on CMC was considered by Abbas et al. [34]. The ANN investigation of the basic characteristics of MHD flow in a cylinder with radiation was inspected by Bairagi et al. [35]. The computational assessment of bioconvective gyrotactic microorganisms wedge flow of Carreau fluid with thermal radiations was inspected by Zeeshan et al. [36] using ANN.

The aforementioned literature reviews highlight an important study gap in the research of the result of SB on thermophoretic particle deposition in unsteady MHD. The Boger nanofluid velocity across a stretched, inclined surface within thermal radiation and velocity slip when a fluctuating, inclined magnetic field is present. To statistically close this gap, the following main objectives guided the current study’s conduct:

• Provide a model that illustrates a viscous, incompressible, radiating, electrically conducting nanofluid flow across an angled stretched sheet within a permeable media with a variable MF (magnetic field).

• For the solution of the fluid flow model in the form of non-linear ODEs, a novel solution technique of BRO-SNNs is suggested.

• Once the governing equations have undergone the appropriate similarity transformations, a system of non-linear ODEs can be solved semi-analytically.

• Analyse how thermophoresis and Brownian motion affect the Boger fluid’s thermal and solutal boundary layer flow.

• Analyze how Stefan blowing affects thermophoresis in a Boger tunnel.

• Analyze and physically interpret the model’s dominant parameters as they affect the fluid and boundary properties.

Mathematical formulation of the problem

Imagine the Stefan blowing on thermophoresis (referenced by Kanwal et. al. [30]) in two-dimensional MHD Boger NF flow over an inclined stretching under the following prescribing assumptions:

• The nanofluid is incompressible, conducting electrically and radiating

• The velocity is over an inclined unsteady sheet at $y=0$, stretching with velocity of $u_w$ $(x,\mathit{t})$ in a permeable media

• The flow is unsteady due to the unsteadiness of the sheet.

• Both the concentration at the surface, $C_w$ $(x,\mathit{t})$, and the surface heat, $T_w$ $(x,\mathit{t})$, are thought to be time dependent and vary along the stretching sheet.

• The stretching sheet’s inclination is exposed to an inclined variable MF.

The magnetic field is defined as $=B_0{x}^{\left(m-1\right)/2}{(1-{\beta }_{2}t)}^{-1/2}$. Where $B_{0}0$ the magnetic field strength, m is a positive integer and ${\beta }_2$ is a constant. Using$m$ $=1$, reduces the variable magnetic field to be strictly a function of time, as $B\left(t\right)=B_0{(1-{\beta }_{2}t)}^{-1/2}$. The momentum, continuity, and energy equations that follow [1–7,37,38] are therefore provided as follows: The schematic diagram (see Fig. 1(a) illustrates the physical scenario described under these conditions.

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$ 1

$$\frac{\partial u}{\partial t}+v\frac{\partial u}{\partial y}+u\frac{\partial u}{\partial x}=v(\frac{1+\beta }{1+{\beta }_1})\frac{{\partial }^{2}u}{\partial y^2}-\frac{\sigma B^2\left(t\right)}{\rho }\left(\mathrm{si}{\mathrm{n}}^2\gamma \right)u-\frac{\nu }{k}u+g{\beta }_C\left(C-C_{\infty }\right)\cos \gamma +g{\beta }_T\left(T-T_{\infty }\right)\cos \gamma$$ 2

$$\frac{\partial T}{\partial t}+v\frac{\partial T}{\partial y}+u\frac{\partial T}{\partial x}=\left(\alpha +\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}}\right)\frac{{\partial }^{2}T}{\partial y^2}+\tau \left(D_B\frac{\partial C}{\partial y}\frac{\partial T}{\partial y}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^2\right)+\frac{Q_0}{{\rho C}_p}\left(T-T_{\infty }\right),$$ 3

$$\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_B\frac{{\partial }^{2}C}{\partial y^2}-\frac{\partial }{\partial \mathrm{y}}(\frac{\partial \mathrm{T}}{\partial y}\left(C-C_{\infty }\right)k_t\frac{\nu }{T_{\mathit{ref}}})$$ 4

$$u=u_w\left(x,t\right)+\gamma \left(\frac{1+\beta }{1+{\beta }_1}\right)\frac{\partial u}{\partial y}\mathit{;v}=\frac{D_m}{1-C_w}\frac{\partial C}{\partial y}$$ 5

$$T=T_w,C=C_w,\mathit{at\; y}=0$$ 6

$$T\to T_{\infty },u\to 0,C\to C_{\infty },\mathit{as\; y}\to \infty$$ 7

Fig. 1.

a Flow description. b, c and d ℏ-curves for HAM solution

The BCs are

Electrical conductivity is denoted by $\sigma$, kinematic viscosity by $v$, and components of velocity in the x and y axes by ($u$,$v$). The Brownian and thermophoretic diffusion coefficients are signified by the parameters $D_B$ and $D_T$. Furthermore, it is determined that the NPs effective thermal capacity ratio to the base liquid is $\tau =(\rho C{)}_P/(\rho C{)}_f$. The fluid density is represented by $\rho$, the applied variable MF strength is indicated by $B$, and the thermal diffusivity is represented by $\alpha =\frac{K}{{\rho C}_P}$, In this case, $C_P$ stands for specific heat and $K$ for thermal conductivity. The final symbols are $T$ for the fluid’s temperature and $C$ for its concentration.

According to [16], the velocity of the sheet can be calculated as follows:

$$u_w\left(x,t\right)=\frac{\mathit{bx}}{1-{\beta }_{2}t}$$ 8

Valid for $t\beta 1,$ unless $\beta =0$

Similarity transformation variables can be used to transform the governing PDEs into a system of nonlinear ODEs. We introduce the subsequent transformations of similarity:

$$\eta ={\left[\frac{b}{v}\right]}^{-1/2}{\left(1-{\beta }_{2}t\right)}^{(-1/2)}y$$ 11

$$\psi ={\left(\mathit{vb}\right)}^{1/2}{\left(1-{\beta }_{2}t\right)}^{(-1/2)}\mathit{xF}\left(\eta \right)$$ 12

$$T=T_{\infty }+T_0(\frac{b{x}^2}{2v}){\left(1-{\beta }_{2}t\right)}^{-\frac{3}{2}}\theta \left(\eta \right)$$ 13

$$C=C_{\infty }+C_0(\frac{b{x}^2}{2v}){\left(1-{\beta }_{2}t\right)}^{-\frac{3}{2}}\varphi \left(\eta \right)$$ 14

where $\psi$ is the function of streams that complies with Eq. (1) for continuity. The velocity’s components are then provided by:

$$u=\frac{\partial \psi }{\partial y}=u_w{F}^{\prime }\left(\eta \right)=\frac{\mathit{bx}}{\left(1-\beta t\right)}{F}^{\prime }\left(\eta \right)$$ 15

$$u=-\frac{\partial \psi }{\partial x}=-{\left(\frac{\mathit{vb}}{\left(1-\beta t\right)}\right)}^{\frac{1}{2}}F\left(\eta \right)$$ 16

ODEs

$$(\frac{1+\beta }{1+{\beta }_1})F^{\prime \prime \prime }+F{F}^{\prime \prime }-F^{\prime 2}-\left(M{\sin }^2\gamma +K\right)F^{\prime }+\left({\lambda }_1\cos \gamma \right)\theta +\left({\lambda }_2\cos \gamma \right)\varphi =0$$ 17

$$\left(\frac{4}{3}\mathit{Rd}+1\right){\theta }^{\prime \prime }+\mathrm{Pr}\left(-2F^{\prime }\theta +F{\theta }^{\prime }-\frac{A}{2}\left(3\theta +\eta \theta \prime \right)\right)+\mathit{PrNb}{\varphi }^{\prime }{\theta }^{\prime }+\mathit{PrNt}{\left({\theta }^{\prime }\right)}^2+Q_T\theta =0$$ 18

$${\varphi }^{\prime \prime }+\mathit{Sc}\left(F{\varphi }^{\prime }-2F^{\prime }\varphi -\frac{A}{2}\left(3\varphi +{\eta \varphi }^{\prime }\right)\right)-\mathrm{\tau }\mathrm{Sc}\left(\varphi {\mathrm{\theta }}^{\prime \prime }+{\mathrm{\theta }}^{\prime }{\varphi }^{\prime }\right)=0$$ 19

BCs:

$$F(\eta )=\frac{\mathit{Sb}}{\mathit{Sc}}\varphi \prime (0),F\prime (\eta )=1+\delta f\left(\frac{1+\beta }{1+{\beta }_1}\right)F\prime \prime (0),\theta (\eta )=1,\varphi (\eta )=1,\mathit{at}\eta =0$$ 20

$$\theta (\eta )\to 0,\varphi (\eta )\to 0,F\prime (\eta )\to 0,\mathit{as}\eta \to \infty$$ 21

Where $G{r}_T=(g{\beta }_T\left(T_w-T_{\infty }\right)x^3/v^2$ is the thermal Grashof number, $A={\beta }_2/b$ is the unsteadiness parameter, $\mathit{Rd}=\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}k}$ is the radiation factor, ${\lambda }_2=G{r}_c/{\mathrm{Re}}_x^2,G{r}_c=(g{\beta }_c\left(C_w-C_{\infty }\right)x^3/v^2$ is the concentration Grashof number, $M=\sigma /\left(b\rho \right)B_0$ is the magnetic factor, $r=v/\alpha$ = is the Prandtl number, $K=(v/(\mathit{bk}))\left(1-{\beta }_2\right)$ is the porosity parameter, ${\lambda }_1=G{r}_T/{\mathrm{Re}}_x^2,\mathit{Sc}=v/D_B$ is the Schmidt number, $\mathit{Sb}=\frac{C_w-C_{\infty }}{1-C_w}$ Stefan blowing parameter.

The Sherwood number ($\mathit{Sh}$), Nusselt number ($\mathit{Nu}$) and skin friction coefficient ($\mathit{Cf}$), are computed using:

$$\mathit{Cf}=\frac{2\mu (\partial u/\partial y{)}_{y=0}}{{\rho U}_w^2}=F^{\prime \prime }\left(0\right)R{e}_x^{-(1/2)}$$ 22

$$\mathit{Nu}=\frac{-x(k+\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}})\frac{\partial T}{\partial y}}{K\left(T_w-T_{\infty }\right)}=-{\theta }^{\prime }\left(0\right)\left(\frac{4}{3}\mathit{Rd}+1\right)R{e}_x^{\frac{1}{2}}$$ 23

$$\mathit{Sh}=\frac{x(\partial C/\partial y{)}_{y=0}}{\left(C_w-C_{\infty }\right)}=-R{e}_x^{\frac{1}{2}}{\varphi }^{\prime }\left(0\right)$$ 24

where the local Reynold’s number, $\mathrm{Re}=(U_{w}x)/v$ depends on the stretching sheet’s velocity.

Homotopy analysis method

The use of the HAM has several notable advantages (Homotopy Analysis Method) when dealing with extremely nonlinear situations (chu et al. [39]). One of its main advantages is the ability to customize auxiliary parameters like the convergence-control parameter ℏ and the solution series base function. Moreover, HAM does not depend on any physical parameters, no matter how big or small, allowing it to be used to a broader spectrum of nonlinear equations without relying on perturbation assumptions. It also avoids the discretization inherent in numerical approaches, resulting in analytical approximations in series form. The method’s built-in convergence guarantee via the ℏ-curve enables systematic confirmation of the solution’s accuracy. These characteristics combine to make HAM a versatile and reliable analytical tool for complicated nonlinear processes in science and engineering.

Convergence exploration

Homotopy is a superior technique among the many ways. HAM is extremely beneficial for nonlinear problems arising from physical situations in a wide range of engineering and science sectors. Finding convergent series solutions is critical for the implication of a Homotopy scheme. The auxiliary parameters $h_1,h_2$ and $h_3$ are used to calculate the target convergent region. The ranges of these factors show that they contribute significantly to obtaining a series solution in convergent form. The applicable ranges for the relevant auxiliary factors are$-1.4\le h_1\le 0.4,-1.7\le h_2\le 1.6$ and $-0.6\le h_3\le 0.5$. The optimal values of $h_1,=-0.5,h_2=-1$ and $h_3=-0.1$ for range of $\eta (0\le \eta \le \infty )$. Figure 1(b, c and d) show $\hslash$-curves for HAM solution.

Artificial neural network technique

The MLAs (Machine learning algorithms) have been developed by numerous academic disciplines, such as engineering, science and mathematics, to tackle challenging real-world problems. The Supervised Learning Algorithm (SLA) mimics human learning through tasks or examples. To aid in this process, a training dataset—also known as a teaching signal—is provided to the neural network (NN). By back-propagating the error between the expected and actual outputs during training, the model can modify its internal weights to enhance performance and guarantee that the neural network learns the desired mapping. An overview of the results for the proposed Artificial Neural Network is given in this section.

The computational efficacy of the Bayesian Regularization Optimizer Supervised Neural Networks (BRO-SNNs) are demonstrated by the training curves’ smooth and consistent convergence over all epochs. The distribution of prediction errors following training can be better understood by looking at the error histogram in Fig. 2(a-c). With the majority of errors clustered close to zero, it shows how well the expected outputs match the goal values. This shows that in every case analyzed, there is an agreement among the actual and expected outcomes. Fitness assessment charts in Fig. 3(a-c) display a thorough contrast among the reference data and the calculated solutions. The simulation’s excellent capability to effectively capture and reproduce the primary behavior of the fluid dynamic system is established by the close contract among the two curves. This agreement supports the recommended neural network model’s structural soundness and prediction capability. The model’s better performance over the training period is shown by the graphical results and the steadily decreasing mean squared error (MSE) in Table 1. This steady decline in MSE shows that lower error levels are associated with the BRO-SNNs model’s enhanced generalization ability and increased prediction accuracy. Additionally, the steady and well-behaved convergence patterns demonstrate how robust the training procedure is, guaranteeing dependable and ideal learning. Table 1 demonstrates the strong correlation between reduced MSE values and improved precision and prediction quality. Further evidence of the model’s convergence and stability can be found in Fig. 4(a-c), which shows how the gradient and Mu factors changed during training. The gradient gradually declining until it reaches a minimal value indicates convergence to an ideal solution. The $\mathit{Mu}$ factor simultaneously energetically changes to regulate the learning process in order to ensure stable and effective optimization. Lastly, regression analysis shows how training and testing dataset’s expected and target outputs relate to one another (Fig. 5(a-c)). Strong prediction correctness and model consistency across several data subsets are designated by the close alignment of data points along the diagonal reference line 6(a-c). Overall, these findings confirm the BRO-SNNs framework’s excellent predictive ability, stability, and dependability.Fig. 6a b

Fig. 2.

ANN error histograms for each of the three scenarios

Fig. 3.

Artificial neural network’s fitness analysis

Table 1.

Artificial neural network technique’s numerical results

Scenarios	Epochs	Performance	Training	Testing
I	352	5.31×E-10	6.43×E-11	3.21×E-09
II	489	2.64×E-11	2.88×E-11	1.26×E-11
III	646	2. 68×E-11	2.75×E-11	2.30×E-11

Fig. 4.

ANN training outcomes for each of the three scenarios

Fig. 5.

ANN state transition study for each of the three scenarios

Fig. 6.

ANN regression analysis in three scenarios

Discussion of results

This semi-analytical model has significant uses in materials processing and advanced thermal management. The study investigates thermophoresis in a Boger NF with Brownian motion and Stefan blowing. It offers a crucial framework for heat exchanger design optimization, where controlled nanoparticle deposition can either improve heat transmission or stop surface fouling. By manipulating thermophoresis and Stefan blowing, the findings are immediately relevant to coating and thin-film manufacturing processes, allowing for fine control over deposit uniformity and thickness. Additionally, by forecasting particle deposition in scrubbers and filtering systems, the model helps environmental engineering. In aerospace, it helps control dust or ice buildup on delicate components. It is especially useful for creating next-generation smart cooling systems that use polymer-based nanofluid because of the viscoelastic character of Boger fluid. The result of β on thermal and velocity fields is portrayed in Figs. 7(a) and 7(b). Changing the solvent fraction $\beta$ enhance the velocity distribution while reducing the heat distribution. Boger nanofluids have exceptional elasticity at constant viscosity due to their high velocity rate at low Reynolds numbers. Figure 8(a&b) and 9(a) show the effects of Sb on thermal profiles and velocity concentration. Energy injection into a liquid to boost its velocity is called “Stephan blowing.” Because of the extra energy source, the fluid accelerates. Consequently, the velocity of the Boger nanofluid increases. The $\mathit{Nt}$ and $\mathit{Nb}$ show effects on the thermal profile are shown in Figs. 9(b) and 10(a). The higher the $\mathit{Nb}$ values, the lower the thermal profile. As $\mathit{Nt}$ values increase, the thermal profile also increases. The thermal profile’s $Q_T$ results are displayed in Fig. 10(b). As the thermal generation factors values rises, the thermal profile rises as well. An example of how τ affects the concentration field is revealed in Fig. 11(a). As the thermophoretic factor are upsurges, the concentration field gets smaller. Figures 11(b), 12(a&b), and 13(a&b) demonstrate the impacts of $\delta f$ on the profiles of temperature, velocity, mass transfer rate, heat transmission rate, and solutal. The thermal and solutal field rises as $\delta$ increases. The liquid velocity near to the sheet is not equal to the stretched sheet velocity, though, as the velocity profile change. This indicates that only a portion of the sheet’s pulling is transmitted to the liquid velocity when there is increasing sliding. Mass and thermal transmission rates rise as the velocity slip parameter values rises. In Fig. 14(a&b), the effect of $\mathit{Sb}$ on $\mathit{Nu}$ and $\mathit{Sh}$ is shown. The rate of mass and heat transmission rises with values of the Stefan blowing parameter. Figure 15(a) shows how Nu reacts to $Q_T$. As the heat generation parameter’s values rise, so does the heat transmission rate. Figure 15(b) show effect of τ on Sh is shown. As thermophoretic parameter values increases, the mass transmission rate falls. Figure 16(a) illustrates the respective effects of $\mathit{Sb}$ and $\delta$ on $\mathit{Cf}$. Higher values of $\delta$ and $\mathit{Sb}$ indicate higher skin friction. Figure 16(b) illustrates how $\mathit{Nt}$ and $\mathit{Nb}$ affect $\mathit{Sh}$. The mass transmission rate rises as $\mathit{Nt}$ and $\mathit{Nb}$ values rise. The $\mathit{Nt}$ and $\mathit{Nb}$ ‘s impact on Nu is shown in Fig. 16(c). As the values of $\mathit{Nb}$ and $\mathit{Nt}$ rises, so does the rate of heat transmission. Table 2 display the sensitivity analysis of auxiliary parameters. Table 3, which has the additional constraints set to zero, shows how the results of this investigation relate to those of previous issued investigations. The updated data exhibits excellent agreement with earlier findings.

Fig. 7.

(a) Outcome of $\beta =\mathrm{0.2,0.4,0.6,0.8}$ on $F\prime \left(\eta \right)$ (b) Impact of $\beta =\mathrm{0.2,0.4,0.6,0.8}$ on $\theta \left(\eta \right)$

Fig. 8.

(a) Outcome of $\mathrm{Sb}=\mathrm{0.1,0.2,0.3,0.4}$ on $F^{\prime \left(\eta \right)}$ (b) Outcome of $\mathit{Sb}=\mathrm{0.1,0.2,0.3,0.4}$ on $\theta \left(\eta \right)$

Fig. 9.

(a) Outcome of $\mathrm{Sb}=\mathrm{0.1,0.2,0.3,0.4}$ on $\varphi \left(\eta \right)$ (b) Impact of $\mathrm{Nb}=\mathrm{0.4,0.5,0.6,0.7}$ on $\theta \left(\eta \right)$

Fig. 10.

(a) Outcome of $\mathrm{Nt}=\mathrm{0.1,0.3,0.5,0.7}$ on $\theta \left(\eta \right)$ $\theta \left(\eta \right)$ (b) Impact of $Q_T=\mathrm{0.3,0.6,0.9,1.2}$ on $\theta \left(\eta \right)$

Fig. 11.

(a) Outcome of $\tau =\mathrm{0.5,0.7,0.9,1.1}$ on $\varphi \left(\eta \right)$ (b) Results of $\delta f=0.7,\mathrm{0.8,0.9,1.0}$ on $\varphi \left(\eta \right)$

Fig. 12.

(a) Results of $\delta f=\mathrm{0.7,0.8,0.9,1.0}$ on $F^{\prime \left(\eta \right)}$ (b) Impact of $\delta f=\mathrm{0.7,0.8,0.9,1.0}$ on $\theta \left(\eta \right)$

Fig. 13.

(a) Impact of $\delta f=\mathrm{0.7,0.8,0.9,1.0}$ on $\mathit{Nu}$ (b) Impact of $\delta f=\mathrm{0.7,0.8,0.9,1.0}$ on $\mathit{Sh}$

Fig. 14.

(a) Effect of $\mathit{Sb}=\mathrm{0.2,0.4,0.6,0.8}$ on $\mathit{Nu}$ (b) Effect of $\mathit{Sb}=\mathrm{0.2,0.4,0.6,0.8}$ on $\mathit{Sh}$

Fig. 15.

(a) Effect of $Q_T=\mathrm{0.1,0.2,0.3,0.4}$ on $\mathit{Nu}$ (b) Effect of $\tau =\mathrm{0.3,0.4,0.5,0.6}$ on $\mathit{Sh}$

Fig. 16.

(a) Conclusion of $\mathit{Sb}$ and $\delta$ on $\mathit{Cf}$ (b) Outcome of $\mathit{Nb}$ and $\mathit{Nt}$ on $\mathit{Sh}$ (c) Influence of $\mathit{Nt}$ and $\mathit{Nb}$ on

Table 2.

Sensitivity Analysis of Auxiliary Parameters ($h,$-Curve Validation)

Parameter	Range	Optimal Value	Effect on Convergence	Error Estimate
$h_1(\mathrm{for}\mathit{F}\prime )$	$-1.4\le h_1\le 0.4$	$-0.5$	Stable convergence region	${10}^{-8}$
$h_2(\mathrm{for}$$\theta )$	$-1.7\le h_2\le 1.6$	$-1.0$	Smooth decay in residuals	${10}^{-9}$
$h_3$$(\mathrm{for}$$\varphi )$	$-0.6\le h_3\le 0.5$	$-0.1$	Rapid convergence	${10}^{-9}$

Table 3.

Correlation of the present and previous outcomes for convective heat transfer and mass transfer

Rd	Nt	Nb	M	$-{\theta }^{\prime }\left(0\right)\left(\frac{4}{3}\mathit{Rd}+1\right)R{e}_x^{\frac{1}{2}}$
				Algehyne et al. [40]	Present	Algehyne et al. [40]	Present
0.1 0.2 0.3	0.1	0.1	0.1	0.542132 0.678534 0.743212	0.542121 0.678527 0.743217	0.856542 0.822676 0.820743	0.856547 0.822664 0.820737
	0.2 0.3			0.780238 0.832984	0.780241 0.83300	0.79978 0.724832	0.799741 0.724829
		0.2 0.3		0.953278 0.97547	0.953295 0.975461	0.889533 0.902375	0.889533 0.902371
			0.2 0.3	0.562143 0.573424	0.562157 0.573439	0.853056 0.825734	0.853053 0.825734

Conclusion

This study examined the Stefan Blowing effect on thermophoresis in a magnetohydrodynamics Boger nanofluid across a permeable inclined sheet with heat radiation and velocity slip, using a semi-analytical method. Based on the analysis of pertinent fluid parameters and characteristics, which are displayed in graphs and tables, the subsequent conclusions can be made:

• The reliability, stability, and convergence of the proposed BRO-SNN algorithm were evaluated using a fitness metric based on MSE, histogram and regression analysis for every fluid model scenario.

• While velocity slip has a negative impact on the velocity profile, it has a positive and important influence on the thermal and solutal profiles.

• The thermal profile upsurges with increasing values of the heat generation factor.

• The velocity and thermal fields rise and fall in proportion to the solvent friction parameter values.

• The temperature field declines as Nb rises, but the concentration field upsurges as $\mathit{Nt}$ increases.

• As the thermophoretic parameter values rise, the 2D and 3D concentration profiles fall.

• The heat profile rises in tandem with the heat generation parameter’s values.

Limitation of the current work

The current study has numerous notable restrictions. First, the model relies entirely on semi-analytical and computational methods (HAM and BRO-SNNs). Second, the governing equations omit several physically significant phenomena, including Joule heating, viscous dissipation, and nanoparticle coagulation, all of which can influence thermophoretic deposition. Finally, the study is limited to 2D flow over a flat inclined sheet, whereas real-world uses in aircraft surfaces, heat exchangers, and filtration systems involve complex three-dimensional geometries.

Abbreviation

ANN

Artificial Neural Network

BCs

Boundary Conditions

BF

Boger Fluid

BHNF

Boger Hybrid Nanofluids

BRO-SNNs

Bayesian Regularization Optimizer Supervised Neural Networks

CCHFM

Cattaneo-Christov Heat Flux Model

HAM

Homotopy Analysis Method

HCM

Hamilton-Crosser Model

HNF

Hybrid Nanofluid

MLAs

Machine Learning Algorithms

MSE

Mean Squared Error

NF

Nanofluid

NN

Neural Network

NPs

Nanoparticles

ODEs

Ordinary Differential Equations

PDEs

Partial Differential Equations

SLA

Supervised Learning Algorithm

A

Unsteadiness parameter

B

Applied variable magnetic field strength

$B_0$

Magnetic field strength constant

b

Constant for stretching velocity

$C_p$

Specific heat

$C_f$

Skin friction coefficient

C

Fluid concentration

$C_w$

Surface concentration

$C_{\infty }$

Ambient concentration

$D_T$

Thermophoretic diffusion coefficient

$D_B$

Brownian diffusion coefficient

$g$

Gravitational acceleration

F(η)

Dimensionless stream function

${\mathit{Gr}}_T$

Thermal Grashof number

${\mathit{Gr}}_C$

Concentration Grashof number

k

Thermal conductivity

K

Porosity parameter

m

Positive integer (magnetic field variation)

M

Magnetic parameter

Nt

Thermophoresis parameter

Nb

Brownian motion parameter

Pr

Prandtl number

Nu

Nusselt number

Rd

Radiation parameter

$Q_T$

Heat generation parameter

Sb

Stefan blowing parameter

R_ex

Local Reynolds number

Sh

Sherwood number

Sc

Schmidt number

$T_w$

Surface temperature

T

Fluid temperature

t

Time

$T_{\infty }$

Ambient temperature

$u_{\mathrm{w}}$

Stretching sheet velocity

u

Velocity component in x-direction

v

Velocity component in y-direction

x, y

Cartesian coordinates

Author contributions

Muhammad Azhar Iqbal: Supervision, Methodology, Conceptualization. M. Saif Aldien and Abaker A. Hassaballa: Methodology, Formal analysis, Asma A. Alhashmi and Ansar Abbas: Methodology, Data curation. Gulnar Hamidova Abdulhamid: Writing – review & editing.

Funding

The authors would like to acknowledge the Deanship of Graduate Studies and Scientific Research, Taif University for funding this work.

Data availability

All data used in this manuscript have been presented within the article.

Declarations

Competing interests

The authors declare no competing interests.