Machine learning simulation for radiation effects on MoS₂-Al₂O₃ hybrid nanoliquid with local thermal non-equilibrium conditions

Graphical abstract

Abstract

The current study examines the Soret-Dufour influence on a MoS₂-Al₂O₃ hybrid nanomaterial with LTNECs, using a novel intelligent numerical computing paradigm. The current study uses a simple mathematical model to investigate the features of heat transmission in the absence of LTECs. It can help forecast and optimize medication delivery since temperature and concentration gradients can better direct nanoparticles in blood. The model also aids in understanding heat and mass transmission in stenosed arteries, which is critical for designing medical remedies like hyperthermia therapy and preventing arterial blockages. It also sheds light on biomedical device design, such as stents or targeted nanofluid-based cooling systems, by accurately simulating blood flow under complex temperature and concentration scenarios. The ODEs are then numerically solved using MATLAB’s built-in bvp4c solver. Comprehensive error analysis, histogram evaluations, regression metrics, and mean squared error (MSE) figures of merit were routinely employed to address fluid model variants, allowing us to assess the efficacy of the newly given ANN-BLMA methodology. As the interphase heat transfer parameter increases, the solid phase thermal profile falls while the liquid phase thermal profile rises.

Introduction

An artificial neural network has become essential elements in the development of cooling systems over recent times. In the thermal industry, the operational characteristics of heating equipment are often controlled by artificial connections, assess the efficiency of radiator systems and accelerate the boiling and investigation procedures. As a result, scientists have employed ANNs (artificial neural networks) to analysis thermal transfer data, keep an eye on the heating procedure and compute the heat transmission coefficient. One crucial step in the central heating or cooling of a particular entity is the transmission of thermal energy from one place to each other. To guarantee the efficient and reliable operation of a device or system at peak performance conditions, the excessive thermal energy generated within it must be reduced and maintained within acceptable temperature limits. Fluid refrigerants have historically been utilized across numerous industries, including technology and automotive, to lessen the impact of heat on machinery, like processors. Khan et al. [1] used artificial intelligence techniques to study the performance characteristics of HNF (hybrid nanofluids) based on carbon nanotubes for wastewater treatment applications. Khan et al. [2] examined the MHD Maxwell HNF model within advanced deep learning-based supervised backpropagated recurrent neural networks. Abbas et al. [3] used an ANNs model to inspect the temperature aspects of oxytactic bacteria in bioconvective slip flow of THNF with activation energy effects. Sait et al. [4] employed ANNs and ML (machine learning) techniques based on linear regression to examine the result of thermal radiation on MHD flow of NNF (non-Newtonian fluids). Abdelfattah et al. [5] used semi-analytical and intelligent computational approaches to study the thermal rheology of organic nanofluids within thermophoretic diffusion. Khan et al. [6] investigated how intelligent predictive networks affected radiative magnetohydrodynamic hybrid nanofluid flow's thermal efficiency while taking heat generation and absorption effects into account. Hadidi et al. [7] employed a BRNN (Bayesian-regularized neural network) structure to investigate the thermal transmission features of the CMC-water HNF under LTNECs. Alzahrani et al. [8] studied neural network-based predictions of MHD natural convection in THNF contained within a curved cavity. Liaqat et al. [9] investigated a Bayesian regularization optimization approach for HNF velocity applying the CCHFM (Cattaneo–Christov heat flux model). Alharbi and Usman [10] used a non-Fourier heat conduction model to investigate how, an inclined MF (magnetic field) improved the properties of velocity and temperature transmission of THNF. Ben Hamida et al. [11] examined the thermal characteristics of HNF containing gyrotactic microorganisms and bioconvective effects employing an ANNs framework. Badawi et al. [12] utilized ANNs techniques to study NNF flow along various viscosity and magnetohydrodynamic impacts within wire coating. A Bayesian-regularized algorithm was used by Liaqat et al. [13] to scrutinize slip properties in solarized Boger NF (nanofluid) flow when activation energy effects were present. Levenberg–Marquardt backpropagation ANNs were used by Ali et al. [14] to scrutinize magnetohydrodynamic NF flow across a vertical surface. Liaqat et al. [15] employed an intelligent Levenberg–Marquardt method to compare the XM and HCM for Boger trihybrid nanofluid. In order to inspect the effects of MHD enhanced velocity in porous channels, Batool et al. [16] used ANNs techniques. Aoudia et al. [17] investigated how artificial neural network techniques could increase heat and heat transfer in HNF comprising gyrotactic microbes in conditions of LTNE. Shah et al. [18] examined a bioconvective nanofluid model's behavior employing ANNs backpropagation. The influence of SB (Stefan blowing) on chemically reactive BNF (Boger nanofluid) velocity with thermophoretic diffusion using an ANN was analyzing by Shaaban et al. [19]. Aoudia et al. [20] used an ANN study of LTNE properties to examine the use of Nimonic NF in industrial systems. Shaaban et al. [21] scrutinized the Boger HNF thermal analysis for industrial heat transmission. Kumar et al. [22] examined the deep learning of heat transmission in NNF flow via convective cone. Khan et al. [23] investigate the ANN investigation for entropy optimized flow of Jeffrey NF invoking Cattaneo-Christov theory. Akbar et al. [24] investigated the ANN-assisted modeling of cilia-driven seminal fluid transfer via the ductuli efferentes. Hayat et al. [25] investigate the NN (neural network)-based investigation for entropy rheological material. Akbar et al. [26] explored the ANN-based prediction of thermal improvement in PTSC for solar-powered aircraft wings. Iqbal et al. [27] inspected Bayesian regularization for the peristaltic propulsion of channel walls. Akbar et al. [28] explored the NF flow driven by metachronal beating of artificial cilia. Iqbal et al. [29] explored the thermal expansion and energy minimization using NF flow in PTSC for solar-powered aircraft.

The existence of nanoparticles smaller than 100 nm has led to the popularity of nanofluids in a variety of engineering and industrial uses, which significantly advance thermal conductivity and heat transmission performance. Heat exchangers, solar collectors, reactor technologies, pharmaceutical processes, refrigeration and chiller systems, absorption technologies, coolant applications, and aggregate manufacturing are just a few of the many industries that make extensive use of these fluids. Nanofluids have also been used in new fields such as safe cooling methods, cancer treatment, heart surgery, and brain tumour therapy. Apart from nanofluids, liquid velocity through permeable media is important in a number of engineering domains, like as industrial, petroleum, and geothermal engineering. Malik et al. [30] looked into thermal transmission and HNF velocity in inclined porous cylinders: Radiation and MHD significance are numerically analyzed using a coupled ANNs. The magnetothermal behavior of tangent hyperbolic HNF flow has been investigated by Atha et al. [31] employing artificial neural network modelling. In order to highlight its applicability to bio-nanomedicine, Subramanyam et al. [32] discovered pulsating HNF velocity in the existence of viscous dissipation and entropy generation effects. The influence of entropy generation as well as magnetohydrodynamics on HNF flow in a staggered cavity were observed by Turabi et al. [33]. The HNF flow over a variable porous region between two stretchable rotating discs was observed by Ali et al. [34]. A computational study of magnetized three-dimensional spinning HNF flow over a sheet with temperature radiation was carried out by Anjum et al. [35]. The liquid-like material that flows through an animal's body is called blood. According to medical theory, blood viscosity is not static and can adjust hemoglobin ratio, thermal, body pressure, and vessel/artery size since the liquid used in this research is thought to be the purest blood. In this study, the large blood vessel is taken into account. Using hybrid and ternary solid nanoparticles, Kumar et al. [36] investigated response optimization and spectral quasi-linearization on MHD flow via stenosed arteries. Faqihi et al. [37] deliberate the STG flow of a viscoelastic THNF across a Riga plate with elastic deformation. For biomedical applications, Kumar et al. [38] inspected the prediction of heat transmission in an adaptive neuro-fuzzy implication system is used in Casson THNF thin film flow. Akbar et al. [39] explored the thermodynamic analysis of nanofluid. Razzaq et al. [40] explored the Double-diffusive convective in rheological NMs with Cattaneo-Christov fluxes for thermal energy storage schemes. Akbar et al. [41] explored the thermal transportation in peristaltic flow of NFs through porous media NNs. Khan et al. [42] explored the improved solutal and thermal fluxes through convective flow of Reiner-Rivlin material. Akbar et al. [43] explored the thermodynamic analysis of peristaltic propulsion of MoS2 nanomaterials in HNF.

Particles are transported via the thermos-diffusion mechanism employing a number of variables that are calculated and combined with a temperature gradient. The SDE (Soret-Dufour effect) occur in the flow regime when a large density difference continues. In liquid motion, instantaneous heat and mass conduction complicate the relationship between driving potentials and fluxes. This manifestation results from a persistent difference between one species' concentration and that in a chemical process of other species. The SDE can help transfer both heat and mass in transitional atomic weight gases using binary systems. Because of its many uses in geothermal energy systems, energy storage units, nuclear waste management, catalytic reactors, thermal insulation, and drying processes, the study of thermal and solutal transport in permeable media has generated a great deal of theoretical and practical interest. Because of its many uses in geothermal energy systems, energy storage units, nuclear waste management, catalytic reactors, thermal insulation, and drying processes, the study of permeable media has generated a great deal of theoretical and practical interest. The properties of slip-on MHD spinning nanoliquid flows, as well as Soret-Dufour diffusion were studied by Sivanandam and Alqurashi [44]. The impact of MHD orientation on Dufour and Soret properties in inclined crenelated triangular cavities filled with NNF was deliberate by Shah et al. [45]. Mhamdi et al. [46] studied MHD convection in the existence of SDE using a mesoscopic lattice Boltzmann numerical approach. Sarma and Sarma [47] examined the mutual influences of thermophoresis, MHD flow, Dufour diffusion, and Soret diffusion on a stretching plate. Salahuddin et al. [48] studied the solutal behavior of SPF (stagnation-point flow) in MHD tangent hyperbolic fluids, including SDE. The flow characteristics of an magnetohydrodynamic Casson fluid with radiation and viscous dissipation effects during Dufour diffusion across a surface in the occurrence of AE (activation energy) were observed by Reddy et al. [49]. Dharmaiah et al. [50] studied the mass transmission and entropy production of spinning surfaces with dissipative Eyring-Powell flows.

Because of its widespread use in biomedical engineering, particularly thermal therapy techniques, the result of radiation on blood flow has emerged as an important subject of research. Because of its capacity to directly warm blood capillaries in damaged tissues, infrared radiation is commonly used as a heat treatment approach for various parts of the human body. Infrared heat therapy has been shown to be useful in the conduct of myalgia (muscle pain), fibromyalgia, muscle spasms, and contracture diseases caused by persistent muscle shortening. Furthermore, this therapeutic method is frequently used to treat bursitis, which is characterized by inflammation of the fluid-filled sac found between tendons and bones or between tendons and skin.

This approach aims to control therapy options for artery disease using numerical models. The usefulness of nano-drug delivery systems drives the research. The innovative characteristic of this study is the examination of the outcome of Dufour and Soret on HNF in an artery with stenosis using the LTNE model (Fig. 1(a)). LTNE conditions in stenotic arteries are crucial for biomedical engineering and medical research. These situations are caused by fluctuations in blood and artery wall temperatures, resulting in non-uniform thermal profiles. LTNE can impact heat transfer through the arterial walls as well as blood flow in stenosed arteries. The LTNECs can improve the precision and effectiveness of thermal therapy procedures by allowing for accurate temperature regulation while minimizing damage to surrounding healthy tissues. Furthermore, LTNE analysis helps to develop better diagnostic approaches for monitoring temperature distribution in stenosed arteries, which improves the diagnosis and treatment of cardiovascular disease. This is especially significant in therapies that involve regulated heating, such as clot dissolution and cancer treatment using hyperthermia. Based on the existing literature, the present work is unique and was not been previously examined.

Fig. 1a.

Stenotic artery.

The objectives of the test are as given:

• ❖This work uses LTNECs to examine the effect of nanoparticles volume friction on the mass transfer as well as heat rate of a blood-based HNF within a stenotic artery.
• ❖The study will look at how MHD affects a blood-based HNF in a stenotic artery with LTNECs.
• ❖To scrutinize the properties of Soret and Dufour on the solutal distributions of blood hybrid nanofluid.
• ❖The goal is to use a modified Hamilton-Crosser model to determine the most effective HNF (hybrid nanofluid) for thermal and mass transport.

Mathematical formulation

Analyze the SDE on a blood-based HNF with steady flow through an artery under LTNE conditions. The model currently in use has selected blood as the fundamental fluid. $\left(\frac{L_0}{2}\right)$ is the location of the stenosed artery, the axes of coordinates $x$and $r$. Fig. 1(b) depicts the physical configuration of the suggested model, which includes the subsequent governing parameters and effects? The $R\left(x\right)-$radius of the artery, maximum height of stenosis, the $\left(2R_0\right)-$diameter of the unimpeded zone of an artery with stenosis are all represented by $\left({\alpha }^{\prime }\right)$in a flow problem and displays the stenosed area's outline [[51], [52], [53], [54], [55], [56], [57]]: The following shows the object's flow molds:

• ❖The consequence of heat radiation and heat generation are examined.
• ❖The result of Dufour and Soret on blood based HNF using LTNECs.
• ❖The intermission of NPs, the continuous flow of a blood-based HNF (hybrid nanofluid) is scrutinized.

Fig. 1b.

Flow description.

The momentum, mass, energy equations are defined as follows [[51], [52], [53], [54], [55], [56], [57]]:

$$\begin{array}{cc}R\left(x\right)\hfill & =R_0-\frac{\alpha }{2}\left(1+\text{cos}\left(\frac{4\pi x}{L_0}\right)\right),-\frac{L_0}{4}<x<\frac{L_0}{4}\\ \hfill & =R_0.\end{array}\frac{\partial \left(rv\right)}{\partial r}+\frac{\partial \left(ru\right)}{\partial x}=0,$$ (1)

$$\left(\frac{\partial u}{\partial x}u+\frac{\partial u}{\partial r}v\right){\rho }_{hnf}={\mu }_{hnf}\frac{\partial }{r\partial r}\left(r\frac{\partial u}{\partial r}\right)-\frac{{\mu }_{hnf}}{k^{*}}u-{\sigma }_{hnf}B{\left(x\right)}^{2}u,$$ (2)

$$\left(v\frac{\partial T}{\partial r}+u\frac{\partial T}{\partial x}\right){\left(\rho C_p\right)}_{hnf}=\left(k_{hnf}+\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}}\right)\frac{\partial }{r\partial r}\left(r\frac{\partial T}{\partial r}\right)+\left(r\frac{\partial \mathrm{C}}{\partial r}\right)\frac{\partial }{r\partial r}\left(\frac{D_{\mathrm{m}}{k}_{\mathrm{T}}}{C_{\mathrm{s}}}\right)+h_{fs}\left(-\mathrm{T}+T_p\right)$$ (3)

$$0=\left(\frac{\partial }{r\partial r}\left(r\frac{\partial T_p}{\partial r}\right)\right)\frac{k_p}{{\left(\rho c_p\right)}_p}+\frac{h_{fs}}{{\left(\rho c_p\right)}_p}\left(-T_p+\mathrm{T}\right),$$ (4)

$$\left(u\frac{\partial \mathrm{C}}{\partial x}+v\frac{\partial \mathrm{C}}{\partial r}\right)=D_B\frac{\partial }{r\partial r}\left(r\frac{\partial \mathrm{C}}{\partial r}\right)+\frac{D_{\mathrm{m}}{k}_{\mathrm{T}}}{T_{\mathrm{m}}}\frac{\partial }{r\partial r}\left(r\frac{\partial T}{\partial r}\right).$$ (5)

The BCs are [43-41]:

$$v=0,T=T_{\mathrm{w}},u=u_w,T_p=t_{\mathrm{w}},C=C_{\mathrm{w}},\text{at}r=R,$$ (6)

$$T\to T_{\infty },T_p\to t_{\infty },C\to C_{\infty },u\to 0,\text{at}r\to \infty .$$ (8)

The similarity transformations [57]:

$$\mathrm{u}=\frac{x{u}_0}{L_0}{f}^{\prime }\left(\eta \right),v=\frac{R}{r}\sqrt{\frac{{\upsilon }_f{u}_0}{L_0}}f\left(\eta \right),$$ (9)

$$\eta =\frac{r^2-R^2}{2R}\sqrt{\frac{u_0}{{\upsilon }_f{L}_0}},\Theta \left(\eta \right)=\frac{T_p-T_{\infty }}{T_w-T_{\infty }},$$ (10)

$$\varphi \left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},u=r^{-1}{\psi }_r,v=-r^{-1}{\psi }_x.$$ (11)

ODEs

$$E_1\left(1+2\delta \right)f^{″\prime }+E_{1}2\delta f^{″}+E_2\left(f^{″}\mathrm{f}-f{{}^{\prime }}^2\right)-E_{3}M{f}^{\prime }-E_{1}K{f}^{\prime }=0,$$ (12)

$$\left(E_4+Rd\right)(\left(1+2\delta ){\theta }^{″}+2\delta {\theta }^{\prime }\right)+Pr{E}_5\left(f{\theta }^{\prime }-f^{\prime }\theta \right)+PrH\left(-\theta +\Theta \right)+PrDu(2\delta {\theta }^{\prime }+\left(1+2\delta ){\theta }^{″}\right)=0,$$ (13)

$$\left(1+2\delta ){\Theta }^{″}+2\delta {\Theta }^{\prime }\right)+\text{Pr}\gamma H\left(\theta -\Theta \right)=0,$$ (14)

$$(1+2\delta ){\varphi }^{″}+2\delta {\varphi }^{\prime }+Sc\left(F{\varphi }^{\prime }-F^{\prime }\varphi \right)+ScSr(\left(1+2\delta ){\theta }^{″}+2\delta {\theta }^{\prime }\right)=0$$ (15)

BCs

$$\Theta \left(0\right)=1,f^{\prime }\left(0\right)=\lambda ,\varphi =1,\theta \left(0\right)=1,f\left(0\right)=0,$$ (16)

$$\theta \left(\infty \right)\to 0,\Theta \left(\infty \right)\to 0,f^{\prime }\left(\infty \right)\to 0,\varphi \left(\infty \right)\to 0.$$ (17)

Where Magnetic parameter$\mathrm{M}=\frac{B_0^2{L}_0{\sigma }_f}{{\rho }_f},$ Schmidt number $\text{Sc}=\frac{{\upsilon }_f}{D_B}$, Flow parameter$\delta =\sqrt{\frac{{\upsilon }_f{L}_0}{L_0{R}^2}}$, $\text{Du}=\frac{D_m{K}_T\left(C_w-C_{\infty }\right)}{C_p{C}_s\left(T_w-T_{\infty }\right)}$Dufour number,$\text{Sr}=\frac{D_m{K}_T\left(T_w-T_{\infty }\right)}{T_{\infty }\left(C_w-C_{\infty }\right)}$Soret number. The values of based liquid and NPs (nanoparticles) are displayed in Table 1.

$$E_1=\frac{{\mu }_{\text{hnf}}}{{\mu }_{\mathrm{f}}},E_2=\frac{{\rho }_{\text{hnf}}}{{\rho }_{\mathrm{f}}},E_3=\frac{{\sigma }_{\text{hnf}}}{{\sigma }_f},E_4=\frac{k_{\text{hnf}}}{k_f},E_5=\frac{{\left(\rho C_p\right)}_{\text{hnf}}}{{\left(\rho C_p\right)}_{\mathrm{f}}}.$$

Table 1.

Levenberg-marquardt optimizer results.

case	Time	Epoch	Gradient	Performance	Training	Validation	Testing
1	04	227	9.97 × E–08	2.65 × E–10	2.31 × E–10	3.17 × E–10	5.10 × E–10
2	16	681	9.80 × E–08	2.22 × E–10	2.25 × E–10	2.51 × E–10	1.45 × E–10
3	18	624	9.87 × E–08	7.56 × E–11	6.74 × E–11	1.12 × E–10	9.68 × E–11

The rate of heat transfer rate $Nu,$ mass transmission $Sh$, and skin friction $Cf$, are well-defined as given:

$$Nn=\frac{x{q}_{\mathrm{n}}}{k_p\left(t_w-t_{\infty }\right)},Sh=\frac{x{q}_{\mathrm{m}}}{D_B\left(C_w-C_{\infty }\right)},$$ (18)

$$Cf=\frac{{\tau }_w}{\frac{1}{2}{\rho }_f{U}_w^2},Nu=\frac{x{q}_{\mathrm{w}}}{k_f\left(T_w-T_{\infty }\right)},$$ (19)

$${\tau }_w={\mu }_{hnf}{\frac{\partial u}{\partial r}|}_{r=0},q_{\mathrm{w}}=-{k_{hnf}+\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}}\frac{\partial T}{\partial r}|}_{r=0},$$ (20)

$$q_{\mathrm{m}}=-{D_B\frac{\partial C}{\partial r}|}_{r=0},q_n=-{k_p\frac{\partial T_p}{\partial r}|}_{r=0},$$ (21)

$$Cf=E_1{f}^{″}\left(0\right),R{e}^{\frac{1}{2}}Nu=-\left(E_4+\text{Rd}\right){\theta }^{{}^{\prime }}\left(0\right),$$ (22)

$$R{e}^{\frac{1}{2}}Nn=-{\Theta }^{{}^{\prime }}\left(0\right),ShR{e}^{\frac{1}{2}}=-{\varphi }^{{}^{\prime }}\left(0\right).$$ (23)

Artificial neural network

The proposed ANN architecture consists of one input neuron, one HL (hidden layer) consisting of fifteen neurons, and one output neuron. While the HL employs the tangent sigmoid (tansig) AF, the output layer uses a linear (purelin) activation function to offer accurate prediction of the continuous output variables. In the fields of science, engineering, and mathematics, machine learning algorithms (MLAs) have become potent computational tools for solving challenging issues. By creating connections between inputs and intended outputs, supervised learning algorithms (SLAs) allow neural networks (NNs) to learn from labeled datasets. In order to diminish the difference among target and predicted values and increase prediction accuracy, the network iteratively modifies its weights using error back-propagation throughout the training phase. The performance and results of the suggested m ANN-LMA are succinctly summarized in this section (Table 1).

To address the suggested influence of the fluid model in example 1, an ANN optimized using LM is established. The effects of the model are displayed in Fig. 2. In this instance, Fig. 2(a) illustrate the model's MSE value for resolving the Dufour and Soret effects. At epoch 227, the ANN network's best testing performance value is 5.10 × E^–10. The ST (state transition) plot utilized to report the effects of the proposed model is shown in Fig. 2(b). The EH (error histogram) plot derived from ANN models is revealed in Fig. 2(c). Our created approach's prediction accuracy is validated by the minimal values of the error histogram and mean square error. The regression examination is revealed in Fig. 2(d), where the correlation coefficient R=1 indicates that there is an association among the actual and anticipated values, demonstrating that the ANN model correctly characterizes the velocity field's properties. Plotting ANN prediction versus target data, Fig. 2(e) illustrates function fit. High fitting precision is established by the close match between the expected and actual numbers. The model consequences for case 2 that were gained using the ANN-LM method are displayed in Fig. 3. In this instance, the MSE value for the testing, training, and validation data is revealed in Fig. 3(a). The training data error begins to drop at the start of each epoch and continues to decrease as the epoch progresses. At epochs 681, the ANN-LM network's best testing performance value is 1.45 × E^–10. The gradient, mean, and validation results at epoch 681 are presented in Fig. 3(b). The EH, which demonstrations the dispersion of error in the predictions, is displayed in Fig. 3(c). The ANN's extremely accurate predictions are demonstrated by the errors' clustering towards zero. Fig. 3(d) demonstrate the RA (regression analysis) plot from the ANN models that were used to address the effects. The function fit in Fig. 3(e) demonstrates the model's accuracy in simulating thermal activity because the target and anticipated thermal data almost perfectly overlap. The model consequences for case 3 that were acquired using the ANN-LM method are displayed in Fig. 4. The MSE value for the testing data in this instance is displayed in 4(a). The exercise data error starts to decline at the beginning of each epoch and keeps deteriorating as the epoch goes on. The ANN-LM network's best testing performance value at epochs 624 is 9.68 × E^–11. The training states are shown in 4(b), which shows the evolution of gradient values, learning rate, and validation checks throughout iterations. Stable optimization is explained by the gradient's stabilization. The error histogram is shown in Fig. 4(c), with slight departures from zero ranging from positive-to-positive temperature forecasts. The regression examination is revealed in Fig. 4(d), where R=1 value designates that the ANN has been successful in achieving nearly ideal correspondence between anticipated and observed thermal outputs, supporting its efficacy in thermal field modeling. The FC (fitness curve) plot for determining the proposed model is shown in 4(e). The small error found indicates that the predictable data most closely resembles the reference data.

Fig. 2.

Case 1: (a) MSE performance; (b) state-transition analysis; (d) regression analysis; (c) error histogram; and (e) curve-fitting performance.

Fig. 3.

Case 2: MSE performance (a), state-transition analysis (b), error histogram (c), regression analysis (d), and curve-fitting performance (e).

Fig. 4.

Case 3: MSE performance (a), state-transition analysis (b), regression analysis (d), error histogram (c), and curve-fitting performance (e).

Fig. 5(a) displays the result of $Du$on $\theta \left(\eta \right)$. Fig. 5(b) shows the absolute error plot variation in $Du$. The heat flow brought on by a solutal gradient is referred to as the Dufour influence. When the Dufour result is present, the thermal profile functions well; otherwise, it behaves poorly. The TBL (thermal boundary layer) thickness increases dramatically when the Dufour number rapidly rises, indicating a stronger boundary layer flow. Fig. 6(a) displays the result of $Sr$on $\varphi \left(\eta \right)$. Fig. 6(b) demonstrations the absolute error plot variation in $Sr$. As shown in Fig. 6(a), the concentration profiles get better as $Sr$ values rise. The consequence of $H$ on $\Theta$and $\theta$is shown in Figs. 7(a) and 8(a). Augmented molar mass diffusivity brought on by growing$Sr$ contributes to an increase in $\varphi$. The absolute error plot variation in $H$ is shown in Figs. 7(b) and 8(b). As we rise the value of $H$, the thermal field of the solid phase declines and the thermal distribution of the fluid phase rises. Fast heat transmission among phases is the problem in this instance. The graphic designates that the system is trying to alter the LTE phase since the $H$ has been moved to a higher position. Taking into account $H$ values in the middle range significantly alters how the two phases appear. According to some, the flow in this area may be affected by the LTNE. The essential adjustments are made to the system to achieve thermal equilibrium. The thermal of the fluid phase gradually drops that of the solid phase, even if there may be an initial temperature difference. The system ultimately reaches an LTE state as heat transfer in both phases gets closer to parity and $H$ rises. Fig.s 9(a) and 10(a) illustrate how δ affects the thermal profile and velocity of the liquid phase. Figs. 9(b) and 9(b) illustrate how the absolute error plot in $H$ varies. The fluid phase thermal profile falls and the velocity profile rises as δ increases. Fig. 10(a) illustrates how $\Phi$ affects $Nu$. As heat transfer rates rise, so do the volume friction values of nanoparticles. Fig. 10(b) illustrates the significance of $Du$ to $Nu$. Fig. 10(c) illustrates the significance of $Sr$ to $Sh$. When the Soret number upsurges, so does the mass transmission rate. The basic assets of HNF are show in Table 2. Table 3 demonstration the features of based liquid and NPs. The results by likening them to previously study in which the additional mechanisms were set to zero are show in Table 4. The reliability and accuracy of the study are supported by a correlation among the present results and those published in earlier publications (Figs. 11a, 11b, 11c).

Fig. 5a.

Effect of $Du$ on $\theta$.

Fig. 5b.

Absolute error plot variation in$Du$.

Fig. 6a.

Properties of $Sr$ on $\varphi$.

Fig. 6b.

Absolute error plot variation in$Sr$.

Fig. 7a.

Influence of $H$ on $\theta$.

Fig. 8a.

Properties of $H$ on $\Theta$.

Fig. 7b.

Absolute error plot variation in$H$.

Fig. 8b.

Absolute error plot variation in$H$.

Fig. 9a.

Influence of $\delta$ on $\theta$.

Fig. 10a.

Effect of $\delta$ on $\theta$.

Fig. 9b.

AE plot variation in$\delta$.

Fig. 10b.

AE plot variation in$\delta$.

Table 2.

The properties of HNF [51,52].

Properties	HNF
${\mu }_{hnf}$	$E_1=\frac{{\mu }_{hnf}}{{\mu }_f}=\frac{1}{{\left(-{\Phi }_2+1\right)}^{2.5}{\left(-{\Phi }_1+1\right)}^{2.5}}$
${\rho }_{hnf}$	$E_2=\frac{{\rho }_{hnf}}{{\rho }_f}=\left[{\Phi }_2\frac{{\rho }_{s_2}}{{\rho }_f}+\left(1-{\Phi }_2\right)\left\{{\Phi }_1\frac{{\rho }_{s_1}}{{\rho }_f}+\left(1-{\Phi }_1\right)\right\}\right]$
${\sigma }_{hnf}$	$E_3=\frac{{\sigma }_{hnf}}{{\sigma }_f}=\left(1+\frac{3\left(\frac{{\sigma }_1{\Phi }_1+{\sigma }_2{\Phi }_2}{{\sigma }_f}-\left({\Phi }_1+{\Phi }_2\right)\right)}{\left(\frac{{\sigma }_1{\Phi }_1+{\sigma }_2{\Phi }_2}{{\sigma }_f}+2\right)-\left(\frac{{\sigma }_2{\Phi }_2+{\sigma }_1{\Phi }_1}{{\sigma }_f}-{\Phi }_1{\Phi }_2\right)}\right)$
${\left(\rho c_p\right)}_{hnf}$	$E_5=\frac{{\left(\rho c_p\right)}_{hnf}}{{\left(\rho c_p\right)}_f}=\left[\left(-{\Phi }_2+1\right)\left\{\left(-{\Phi }_1+1\right)+{\Phi }_1\frac{{\left(\rho c_p\right)}_{s_1}}{{\left(\rho c_p\right)}_f}\right\}+{\Phi }_2\frac{{\left(\rho c_p\right)}_{s_2}}{{\left(\rho c_p\right)}_f}\right]$

Table 3.

The basic properties of NPs' and base liquid's [58,59].

Properties constituents	$c_p$	$k$	$\sigma$	$\rho$
Blood	3617	0.52	0.667	1050
$\mathit{A}{\mathit{l}}_2{\mathit{O}}_3$	765	40	3.69 ×107	3970
$\mathit{MoS}\_2$	397.7	904.7	2.09×106	5060

Table 4.

Nusselt number and Skin friction coefficient valuation data.

	Published results (Waqas et al. [57])		Current results
$\mathit{\delta }$	Skin friction	Skin friction
0.10	0.939968	0.939989
0.12	0.924794	0.924771
0.14	0.911311	0.911323

Fig. 11a.

Outcome of $\Phi$ on $\theta$.

Fig. 11b.

Properties of $\text{Du}$ on $\text{Nu}$.

Fig. 11c.

Outcome of $\text{Sr}$ on $Sh$.

Conclusions

There are significant biomedical and technological ramifications to the machine learning-based simulation of SDE on mediated blood HNF flow in a stenosed artery using the LTNE model. It can aid in predicting and optimizing drug delivery since nanoparticles in blood can be steered more efficiently by temperature and concentration gradients. The model also helps to comprehend heat and mass transmission in stenosed arteries, which is important for developing medical therapies such as hyperthermia therapy and preventing arterial obstructions. It also offers insights into biomedical device design, such as stents or targeted nanofluid-based cooling systems, by correctly simulating blood flow under complex temperature and concentration conditions. The study's robustness is further increased by using an ANN (artificial neural network) to learn and replicate the complex nonlinear correlations obtained from the numerical solutions. The trained network significantly reduces processing time as a predictive surrogate model without compromising accuracy. The subsequent are the study's key results:

• ❖To accurately assess and forecast numerical outputs using a Levenberg-Marquardt neural network.
• ❖Increasing the interphase heat transmission parameter results in a higher heat profile for the liquid phase as well as a lower thermal profile for the solid phase.
• ❖The thermal profile decreased as well as the blood HNF velocity profile increased as the velocity factors value increased.
• ❖Thermal, concentration, mass transmission, and rate of temperature all rises as the Soret and Dufour numbers rise.
• ❖Model correctness is assessed using regression analysis, absolute error, comparison solution, and error distribution measures.
• ❖As the interphase thermal transmission factors increases in value, the rate of thermal transmission in the dense phase decreases.

Future work: Future applications of the suggested artificial neural network-mediated blood tetra hybrid nanofluid model include the optimization of drug delivery systems based on nanoparticles in stenosed arteries, personalized treatment planning, and advanced cardiovascular disease diagnosis. The development of targeted therapies, treatments for hyperthermia, and the design of biomedical nanodevices may benefit from the more precise prediction of heat and mass transport phenomena made possible by the integration of thermophoretic particle deposition and the Yamada–Ota thermal conductivity model. Additionally, the ANN framework can support patient-specific simulations, digital twin technologies, and real-time hemodynamic analysis to enhance clinical decision-making and cardiovascular healthcare outcomes.

Nomenclature

$q_w$Heat flux $T_w$Temperature at the surface $Sc$Schmidt number $H$Inter-phase heat transfer factor $C_m$Concentration at the surface $K$Porosity parameter $\delta$Flow parameter $h_{fs}$Heat transmission ${\rho }_f$Fluid density$\left(kg{m}^{-3}\right)$ $\left(x,r\right)$Cartesian coordinates $k_f$Thermal conductivity$\left(W{m}^{-1}{k}^{-1}\right)$ $Nu$Nusselt number ${\sigma }_f$Electrical Conductivity $\mathrm{Pr}$Prandtl number ${\mu }_f$Dynamic viscosity $\left(kg{m}^{-1}{s}^{-1}\right)$ $B_0$Uniform magnetic field$\left(kg{s}^{-2}{A}^{-1}\right)$ $\left(u,v\right)$Velocity fields of fluid$\left(m{s}^{-1}\right)$ coefficient for the fluid–solid

$C_p$Specific heat $\left(Jk{g}^{-1}{k}^{-1}\right)$ $Sh$Sherwood number $C$Concentration $T$Fluid temperature $\left(k\right)$ $M$Magnetic parameter ${\upsilon }_f$Kinematic viscosity $\left(m^2{s}^{-1}\right)$ $Ec$Eckert number $T_p$Fluid temperature for solid phase $T_0$Constants SDE-Soret-Dufour effect NNF-Non-Newtonian fluids HNF-Hybrid nanofluids LTNECs-Local thermal non-equilibrium conditions NF-Nanofluid CCHFM-Cattaneo–Christov heat flux model TEHNF-Ternary-hybrid nanofluid NPs-Nanoparticles

Authors' contributions

Mohamad Amin Khadimalah and Joffin J Ponnore: Supervision, Methodology, Conceptualization. Elimem Ali and Farkhod Alisherov: Methodology, Formal analysis.^, Saba Liaqat and Mohamed Abbas El-Naggar: Writing – review & editing.

Declaration of competing interest

The authors have no relevant financial or non-financial interests to disclose.

Data availability

The authors do not have permission to share data.