Exploring the Influence of Induced Magnetic Fields and Double-Diffusive Convection on Carreau Nanofluid Flow through Diverse Geometries: A Comparative Study Using Numerical and ANN Approaches

Abstract

This current investigation aims to explore the significance of induced magnetic fields and double-diffusive convection in the radiative flow of Carreau nanofluid through three distinct geometries. To simplify the fluid transport equations, appropriate self-similarity variables were employed, converting them into ordinary differential equations. These equations were subsequently solved using the Runge–Kutta–Fehlberg (RKF) method. Through graphical representations like graphs and tables, the study demonstrates how various dynamic factors influence the fluid’s transport characteristics. Additionally, the artificial neural network (ANN) approach is considered an alternative method to handle fluid flow issues, significantly reducing processing time. In this study, a novel intelligent numerical computing approach was adopted, implementing a Levenberg–Marquardt algorithm-based MLP feed-forward back-propagation ANN. Data collection was conducted to evaluate, validate, and guide the artificial neural network model. Throughout all the investigated geometries, both velocity and induced magnetic profiles exhibit a declining trend for higher values of the magnetic parameter. An increase in the Dufour number corresponds to a rise in the nanofluid temperature. The concentration of nanofluid increases with higher values of the Soret number. Similarly, the nanofluid velocity increases with higher velocity slip parameter values, while the fluid temperature exhibits opposite behavior, decreasing with increasing velocity slip parameter values.

1. Introduction

The circulation of blood is a fundamental component of human physiology, and comprehending its kinetics of paramount importance to numerous disciplines, such as biomedical engineering and healthcare investigation. The investigation of blood flow in bioengineering facilitates the development of medical equipment and innovations that can aid or supplant the operation of the cardiovascular system, including but not limited to artificial heart valves, stents, and pumps. Bioengineers can design devices that optimize blood flow, minimize turbulence, and reduce the risk of clotting by comprehending the fluid mechanics of blood flow. Additionally, these hemodynamic stresses can cause damage to blood vessels. The investigation of blood circulation is an essential aspect of medical research as it provides insights into various cardiovascular ailments and conditions, including but not limited to hypertension, atherosclerosis, and aneurysms. Medical professionals employ diverse methodologies, such as medical imaging and computational simulations, to scrutinize blood circulation in distinct areas of the circulatory system and evaluate the impact of different interventions or therapies. This knowledge facilitates physicians in the diagnosis and treatment of cardiovascular diseases with greater efficacy, and in the development of novel therapies that can enhance patient outcomes. The phenomenon of heat transfer holds significant importance in the context of blood circulation within the human body. The transportation of heat throughout the body is facilitated by blood, and the interaction between blood and the adjacent tissues in terms of heat exchange has an impact on various physiological mechanisms, including blood circulation [1,2,3,4,5,6,7]. Alghamdi et al. [8] examined the MHD flow of a hybrid nanofluid in a microcirculatory system with the use of the delivery procedure. The energy equation is purported to maintain a consistent temperature for the blood flow through the incorporation of the heat source/sink term. Dharmaiah et al. [9] investigated the features of heat transmission by examining the influence of the supply of blood velocity and circulation flow temperature slips via the Casson blood-flow ferromagnetic fluid throughout a sheet that was stretched through a wide range of fluid-controlling governing parameters. Jiang et al. [10] conducted research on a nonlinear fractional differential equation using the predictor-corrector compact difference system. Yang et al. [11] examine a space-time Sinc-collocation approach for solving the fourth-order nonlocal heat model emerging in viscoelasticity, a class of partial integrodifferential equations whose solution generally shows weak singularities at starting point time.

Artificial intelligence (AI), a branch of computer science, focuses on tasks with a defined goal, such as problem-solving, environmental adaption, decision-making, learning, and communication. Artificial intelligence (AI) finds application in diverse fields such as robotics, computer vision, pattern recognition, information retrieval, natural language processing, medical image computing, machine learning, image processing, data mining, knowledge representation, and other related domains. The artificial neural network (ANN) is a widely used methodology that emulates neural processes to mimic the cognitive functions of the human brain [12,13]. The significance of this factor is found to be crucial in addressing numerous intricate challenges in the field of physical engineering. Consequently, it is widely utilized with great efficacy in various mechanical systems such as combustion engines, refrigeration systems, and thermal devices. The researchers initially examined the effect of nanoparticle size, temperature, and particle concentration on thermal conductivity to estimate it using the ANN method. The utilization of artificial neural networks (ANN) is an effective method for the simulation of blood flow within the human circulatory system. This technique can facilitate the comprehension of the fundamental physiology and pathophysiology of diverse cardiovascular ailments. Artificial neural network (ANN) models have the potential to be trained on medical imaging data, including magnetic resonance imaging (MRI) or computed tomography (CT) scans. This can enable the prediction of blood flow patterns and the identification of areas within the circulatory system that may be compromised or obstructed. The aforementioned data have the potential to aid in the identification of cardiovascular ailments and strategizing of corresponding therapeutic interventions. Furthermore, artificial neural network (ANN) models have the potential to simulate the hemodynamics of blood flow within the circulatory system and investigate the impacts of different interventions or treatments. This methodology has the potential to facilitate researchers in comprehending the underlying mechanisms of diverse cardiovascular ailments and evaluating prospective therapeutic interventions. These models have been employed for the purpose of constructing predictive models for cardiovascular ailments, including hypertension, atherosclerosis, and aneurysms. Through the examination of extensive patient cohorts, researchers have the ability to instruct artificial neural network models to forecast the likelihood of disease development by utilizing diverse demographic, genetic, and environmental determinants [14,15,16,17,18,19,20]. Shilpa and Leela [21] explore the effects of local thermal nonequilibrium, induced magnetic field, and radiative heat on the magnetohydrodynamic mixed convective flow in the vertically circular permeable region. These effects are investigated using artificial neural networks (ANN) and LBM. The nanomaterial magnetohydrodynamic (MHD) flow of Ree–Eyring fluid between two spinning disks was analyzed utilizing an artificial neural network by Zhao et al. [22].

In the majority of scientific and practical applications for changing natural processes, viscoelastic non-Newtonian fluids are chosen over Newtonian fluids. The diverse properties of Newtonian materials cannot be adequately represented by a solitary constitutive relationship between sharing rate and stress (examples: water, air, alcohol and glycerol). The non-Newtonian paradigm encompasses a diverse array of constitutive relationships. Blood is classified as a non-Newtonian fluid due to its variable viscosity in response to changes in shear rate or applied force. At low shear rates, blood exhibits solid-like behavior, whereas at high shear rates, it exhibits liquid-like behavior [23,24,25,26,27,28,29]. The anomalous fluid dynamics exhibited by blood can be attributed to the existence of red blood cells, which possess the ability to undergo deformation and alter their morphology in reaction to the externally imposed shear stress. Under conditions of low shear rates, red blood cells have a tendency to aggregate and coalesce, resulting in the formation of clusters that contribute to an elevation in blood viscosity. Under conditions of high shear rates, the erythrocytes undergo alignment and directional flow, resulting in a reduction in blood viscosity. Comprehending the non-Newtonian characteristics of blood holds significant importance in numerous medical domains, including but not limited to blood rheology, hemodynamics, and blood flow simulations. The aforementioned phenomenon holds significant ramifications for the identification and management of cardiovascular ailments, including thrombosis, wherein alterations in blood viscosity can exert an impact on the genesis and endurance of blood coagulation. Comprehending the flow characteristics of biological fluids holds significance in the development of tissue-engineered structures that can emulate the performance of native tissues. Nanoparticles in an MHD Carreau fluid were subjected to a time-dependent stretching sheet with heat radiation and examined for their effects of multislip with multistratification by Faraz et al. [30]. Algehyne et al. [31] study the mixed convection flow of micropolar Carreau–Yasuda hybrid nanoliquid across a convectively heated Riga plate at a stagnation point. Shojaei et al. [32] used molecular dynamics simulation to study the impacts of atomic percentage and size of zinc nanoparticles as well as atomic porosity on the thermal behavior and mechanical performance of reinforced calcium phosphate cement. Koochaki et al. [33] employed the molecular dynamics technique and LAMMPS (www.lammps.org (accessed on 1 January 2023)) software to examine the atomic and mechanical performance of a polyethylene glycol-based hydrogel–cellulose nanocomposite as a wound-healing biostructure. Almitani et al. [34] investigated the impact of numerous surfactants on the thermal conductivity of nanofluids made of silica and deionized water. The nanofluids used in this investigation were produced using a two-step process.

Biomedical engineers, medical scientists, and physicians are interested in observing blood flow rate since it is used to diagnose cardiovascular disorders, including atherosclerosis and arrhythmia. Numerous researchers have investigated blood flow through the cardiovascular system using various non-Newtonian fluid models. Due to its shear-thinner characteristics, the Carreau fluid model may more precisely depict the rheological behavior of blood. Furthermore, magnetic properties are a noninvasive diagnostic tool that can be used to identify a variety of diseases and conditions, such as tumors, joint injuries, and injuries to the brain and spinal cord. The primary objective of the current study is to demonstrate the occurrence of double-diffusive convection in the radiative flow of blood nanofluid over three distinct geometries, namely plate, wedge, and stagnation point, under the influence of an induced magnetic field. The current literature has been extensively reviewed, revealing a lack of attempts to demonstrate the flow of forced convective blood nanofluid over three distinct geometries in the occurrence of an induced magnetic field. The current framework involves the utilization of Carreau nanofluid as a substitute for blood. The current issue is modeled using the nonhomogeneous equilibrium model. The current model has the potential to be utilized for the analysis of fluid transport characteristics in blood flow within a cardiovascular system, catheterized artery, and various hyperthermia treatments, such as cancer therapy. The self-similarity variables that were acquired are appropriate for accommodating any fluid Prandtl number. This study presents a new method for intelligent numerical computation, which involves the use of a Levenberg–Marquardt algorithm-based MLP feed-forward back-propagation ANN implementation [35,36]. The implementation of suitable self-similarity variables has enabled the transformation of equations governing fluid transport into equations of ordinary differential nature. The aforementioned equations were solved using the Runge–Kutta–Fehlberg (RKF) numerical integration technique.

2. Problem Formulation

The schematic illustration of three different geometries is shown in Figure 1. This study focuses on the Falkner–Skan unsteady forced convective flow of blood nanofluid. It is assumed that the velocity of the potential flow out from the boundary layer is given by the equation $u_{\infty }=b^{xm}$, with b standing for the constant. In the provided equations, u and v are the velocity components along the x and y directions. The wall’s temperature $\left(T_w\right)$ and concentration $\left(C_w\right)$ are elevated compared to the ambient temperatures $\left(T_{\infty }\right)$ and concentrations $\left(C_{\infty }\right)$. It is assumed that a magnetic field of uniform induction with an intensity of $H_0=H_e$ is applied to the surface in a direction perpendicular to it. Once $H_2$ reaches the surface and $H_1$ attains the value of $H_0$, the induced magnetic field’s typical component vanishes.

• This study investigates the properties of forced convective flow of Cross nanofluid, which is characterized by its laminar, steady, and incompressible nature.
• The formula for momentum neglects the body force, such as the Lorentz force.
• The rheological behavior of blood nanofluid is characterized using the Carreau nanofluid model.
• Viscose dissipation and impacts from linear radiation are taken into account in the temperature equation. The Buongiorno nanofluid model is employed to simulate the governing equations.
• The mass transfer equation incorporates a chemical reaction of a destructive nature.
• $n=\frac{1}{6}$ for plate, wedge, and stagnation point of a flat plate.
• $m=\left(\frac{1}{\left(2/{\beta }_1\right)-1}\right)$ ${\beta }_1$ = 0, 0.5, 1 represents plate, wedge and stagnation point, respectively, $n_1\in \left[0,1\right]$.

The aforementioned factors have led to the expression of the fluid transport equations in Cartesian coordinates $\left(x,y\right)$ for the current problem [37,38,39,40,41].

Continuity equation:

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

(1)

Momentum equation:

$$\begin{array}{l}u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=u_{\infty }\frac{d{u}_{\infty }}{dx}-\frac{{\mu }_{mp}{H}_e}{4\pi {\rho }_f}\frac{d{H}_e}{dx}+\frac{{\mu }_{mp}}{4\pi {\rho }_f}\left(H_1\frac{\partial H_1}{\partial x}+H_2\frac{\partial H_1}{\partial y}\right)\\ +{\upsilon }_f\frac{{\partial }^{2}u}{\partial y^2}{\left[1+{\Gamma }^2{\left(\frac{\partial u}{\partial y}\right)}^2\right]}^{\frac{n_k-1}{2}}+{\upsilon }_f\left(n_k-1\right){\Gamma }^2{\left(\frac{\partial u}{\partial y}\right)}^2\frac{{\partial }^{2}u}{\partial y^2}{\left[1+{\Gamma }^2{\left(\frac{\partial u}{\partial y}\right)}^2\right]}^{\frac{n_k-3}{2}}\end{array}$$

(2)

Induced magnetic field equation:

$$u\frac{\partial H_1}{\partial x}+v\frac{\partial H_1}{\partial y}-{\mu }_e\frac{{\partial }^2{H}_1}{\partial y^2}=H_1\frac{\partial u}{\partial x}+H_2\frac{\partial u}{\partial y}$$

(3)

Energy equation:

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}={\alpha }_f\frac{{\partial }^{2}T}{\partial y^2}+\tau \left(D_B\frac{\partial T}{\partial y}\frac{\partial C}{\partial y}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^2\right)-\frac{1}{{\left(\rho c_p\right)}_f}\frac{\partial q_r}{\partial y}+\frac{D_m{k}_T}{c_{p_f}{c}_s}\frac{{\partial }^{2}C}{\partial y^2}$$

(4)

Concentration equation:

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_B\frac{{\partial }^{2}C}{\partial y^2}+\frac{D_T}{T_{\infty }}\frac{{\partial }^{2}T}{\partial y^2}-K_r^2\left(C-C_{\infty }\right)+\frac{D_m{k}_T}{T_m}\frac{{\partial }^{2}T}{\partial y^2}$$

(5)

Subject to the boundary conditions

$$\begin{array}{l}u=L_1\frac{\partial u}{\partial y}{\left[1+{\Gamma }^2{\left(\frac{\partial u}{\partial y}\right)}^2\right]}^{\frac{n_k-1}{2}},v=0,\hspace{0.17em}\frac{\partial H_1}{\partial y}=H_2=0,\hspace{0.17em}-k_f\frac{\partial T}{\partial y}=h_f\left(T_w-T\right),C=C_w\hspace{0.17em}\hspace{0.17em}at\hspace{0.17em}y=0,\\ u=u_{\infty },\hspace{0.17em}{H}_1=H_e\to H_0,\hspace{0.17em}T\to T_{\infty },\hspace{0.17em}C\to C_{\infty }\hspace{0.17em}\hspace{0.17em}as\hspace{0.17em}\hspace{0.17em}y\to \infty .\end{array}$$

(6)

where $u_{\infty }=b{x}^m$ is the free stream velocity, b is the constant. The parameter $\lambda ={\delta }_1\sqrt{\mathrm{Re}}$, which is suitable to choose any fluid Prandtl number value. Here, ${\delta }_1=\sqrt{\frac{\mathrm{Pr}}{{\left(1+\mathrm{Pr}\right)}^{2n}}}$, $n=\frac{1}{6}$ for plate, wedge and stagnation point of a flat plate, and $\mathrm{Re}=\frac{x{u}_{\infty }}{{\upsilon }_f}$ is the Reynolds number.

The radiative heat flux is provided in Equation (4) by

$$q_r=-\frac{4{\sigma }^{*}}{3k_1^{*}}\frac{\partial T^4}{\partial y},$$

(7)

where $k_1^{*}$ is the mean absorption coefficient and ${\sigma }^{*}$ is the Stefan–Boltzmann constant. If the temperature gradients within the blood flow mass are negligibly small, then Equation (7) can be linearized by expanding $T^4$ into the Taylor’s series about $T_{\infty }$, and neglecting higher-order terms, we obtain

$$T^4\cong 4T_{\infty }^{3}T-3T_{\infty }^4$$

(8)

The appropriate variables for self-similarity are formulated in the following approach [38].

$$\begin{array}{l}\eta =\lambda \left(\frac{y}{x}\right),\hspace{0.17em}f\left(\eta \right)=\frac{\psi \left(x,y\right)}{\lambda {\alpha }^{\ast }},\hspace{0.17em}u=\frac{b{x}^m{f}^{\prime }\left(\eta \right)}{{\left(1+\mathrm{Pr}\right)}^{2n}},H_1=\frac{H_0{g}^{\prime }\left(\eta \right)x^m}{{\left(1+\mathrm{Pr}\right)}^{2n}},\\ v=-\lambda \left(\frac{{\alpha }_f}{x}\right)\left[\frac{m+1}{2}f\left(\eta \right)+\frac{m-1}{2}\eta f^{\prime }\left(\eta \right)\right],\\ H_2=-\frac{H_0\lambda }{b}\left(\frac{{\alpha }_f}{x}\right)\left[\frac{m+1}{2}g\left(\eta \right)+\frac{m-1}{2}\eta g^{\prime }\left(\eta \right)\right],\\ T=T_{\infty }+\left(T_W-T_{\infty }\right)\theta \left(\eta \right),C=C_{\infty }+\left(C_W-C_{\infty }\right)\varphi \left(\eta \right).\end{array}$$

(9)

Substituting Equation (9) into the Equations (2)–(6), we obtain

$$\begin{array}{l}\mathrm{Pr}\left(\frac{d^3}{d{\eta }^3}f(\eta )\right)\left(1+n_k\frac{\mathrm{Pr}W{e}^2}{{\left(1+\mathrm{Pr}\right)}^{6n}}{\left(\frac{d^2}{d{\eta }^2}f(\eta )\right)}^2\right){\left[1+\frac{\mathrm{Pr}W{e}^2}{{\left(1+\mathrm{Pr}\right)}^{6n}}{\left(\frac{d^2}{d{\eta }^2}f(\eta )\right)}^2\right]}^{\frac{n_k-3}{2}}\\ -{\beta }_{mp}m{\left(1+\mathrm{Pr}\right)}^{4n}+m{\left(1+\mathrm{Pr}\right)}^{4n}-m{\left(\frac{d}{d\eta }f(\eta )\right)}^2+\left(\frac{m+1}{2}\right)f(\eta )\frac{d^2}{d{\eta }^2}f(\eta )\\ -{\beta }_{mp}\left(\frac{\left(m+1\right)}{2}g(\eta )\frac{d^2}{d{\eta }^2}g(\eta )-m{\left(\frac{d}{d\eta }g(\eta )\right)}^2\right)=0\end{array}$$

(10)

$$M_{mp}\mathrm{Pr}\frac{d^3}{d{\eta }^3}g(\eta )+\left(\frac{m+1}{2}\right)\left(f(\eta )\frac{d^2}{d{\eta }^2}g(\eta )-g(\eta )\left(\frac{d^2}{d{\eta }^2}f(\eta )\right)\right)=0$$

(11)

$$\begin{array}{l}\left(1+\frac{4}{3}Rd\right)\frac{d^2}{d{\eta }^2}\theta (\eta )+\frac{\left(m+1\right)}{2}f(\eta )\frac{d}{d\eta }\theta (\eta )\\ +\mathrm{Pr}\left(N_T{\left(\frac{d}{d\eta }\theta (\eta )\right)}^2+N_B\frac{d}{d\eta }\theta (\eta )\frac{d}{d\eta }\varphi (\eta )\right)+\mathrm{Pr}{D}_r\frac{d^2}{d{\eta }^2}\varphi (\eta )=0\end{array}$$

(12)

$$\frac{\mathrm{Pr}}{Sc}\frac{d^2}{d{\eta }^2}\varphi (\eta )+\frac{(m+1)}{2}f(\eta )\frac{d}{d\eta }\varphi (\eta )+\frac{\mathrm{Pr}}{Sc}\frac{N_T}{N_B}\frac{d^2}{d{\eta }^2}\theta (\eta )-\gamma {\left(1+\mathrm{Pr}\right)}^{2n}\varphi (\eta )+\mathrm{Pr}{S}_r\frac{d^2}{d{\eta }^2}\theta (\eta )=0$$

(13)

Subject to the boundary conditions

$$\begin{array}{l}\frac{d}{d\eta }f(\eta )=\frac{{\alpha }_s\sqrt{\mathrm{Pr}}}{{\left(1+\mathrm{Pr}\right)}^n}\frac{d^2}{d{\eta }^2}f(\eta ){\left[1+\frac{\mathrm{Pr}W{e}^2}{{\left(1+\mathrm{Pr}\right)}^{6n}}{\left(\frac{d^2}{d{\eta }^2}f(\eta )\right)}^2\right]}^{\frac{n_k-1}{2}},f(\eta )=0,\hspace{0.17em}\\ \frac{d^2}{d{\eta }^2}g(\eta )=g(\eta )=0,\hspace{0.17em}\frac{d}{d\eta }\theta (\eta )=-\frac{B_i{\left(1+\mathrm{Pr}\right)}^n}{\sqrt{\mathrm{Pr}}}\left(1-\theta (\eta )\right),\varphi (\eta )=1\hspace{0.17em}\hspace{0.17em}at\hspace{0.17em}\eta =0\\ \frac{d}{d\eta }f(\eta )=\frac{d}{d\eta }g(\eta )={\left(1+\mathrm{Pr}\right)}^{2n},\hspace{0.17em}\theta (\eta )\to 0,\hspace{0.17em}\varphi (\eta )\to 0\hspace{0.17em}\hspace{0.17em}as\hspace{0.17em}\hspace{0.17em}\eta \to \infty \end{array}$$

(14)

where $m$ is the Hartree pressure gradient parameter, $\mathrm{Pr}=\frac{{\upsilon }_f}{{\alpha }_f}$ is the Prandtl number, $We=\sqrt{\frac{{\Gamma }^2{u}_{\infty }^3}{x{\upsilon }_f}}$ is the Weissenberg number, $M_{mp}=\frac{{\mu }_e}{{\upsilon }_f}$ is the magnetic Prandtl number, ${\beta }_{mp}=\frac{{\mu }_{mp}}{4\pi {\rho }_f}{\left(\frac{H_0}{b}\right)}^2$ is the magnetic parameter, $N_T=\frac{\tau D_T\left(T_w-T_{\infty }\right)}{T_{\infty }{\upsilon }_f}$ is the thermophoresis parameter, $N_B=\frac{\tau D_B\left(C_w-C_{\infty }\right)}{{\upsilon }_f}$ is the Brownian motion parameter, $Rd=\frac{4{\sigma }^{*}{T}_{\infty }^3}{k_1{}^{*}{k}_f}$ is the radiation parameter, $S_r=\frac{D_m{k}_T\left(T_w-T_{\infty }\right)}{{\upsilon }_f\left(C_w-C_{\infty }\right)T_m}$ is the Soret number, $D_r=\frac{D_m{k}_T\left(C_w-C_{\infty }\right)}{{\upsilon }_f\left(T_w-T_{\infty }\right)c_{p_f}{c}_s}$ is the Dufour number, $Sc=\frac{{\upsilon }_f}{D_B}$ is the Schmidt number, $\gamma =\frac{x{K}_r^2}{u_{\infty }}$ is the chemical reaction parameter, $B_i=\frac{h_f}{k_f}\sqrt{\frac{x{\upsilon }_f}{u_{\infty }}}$ is the Biot number, and ${\alpha }_s=L_1\sqrt{\frac{u_{\infty }}{x{\upsilon }_f}}$ is the velocity slip parameter.

The values for the skin friction factor, heat transfer rate, and mass transfer rate are provided as follows:

$$\begin{array}{l}{C}_f^{*}=\frac{2\sqrt{\mathrm{Pr}}}{{\left(1+\mathrm{Pr}\right)}^{3n}}{\left(1+\frac{\mathrm{Pr}W{e}^2}{{\left(1+\mathrm{Pr}\right)}^{6n}}{\left(f^{″}\left(0\right)\right)}^2\right)}^{\frac{n_k-1}{2}}{f}^{″}\left(0\right),\\ N{u}^{\ast }=-\left[1+\frac{4}{3}Rd\right]{\theta }^{\prime }\left(0\right),\\ S{h}^{*}=-{\varphi }^{\prime }\left(0\right).\end{array}$$

(15)

where $C_f^{\ast }=C_f{\mathrm{Re}}^{1/2},\hspace{0.17em}N{u}^{\ast }=Nu{\mathrm{Re}}^{1/2}{\delta }_1{}^{-1},\hspace{0.17em}S{h}^{\ast }=Sh{\mathrm{Re}}^{1/2}{\delta }_1{}^{-1},\hspace{0.17em}N{n}^{\ast }=Nn{\mathrm{Re}}^{1/2}{\delta }_1{}^{-1}$.

3. ANN Modeling

The artificial neural network is a contemporary computer system approach that is based on the concept of the human brain functioning as a network of interconnected neural cells. This phenomenon has been identified as a replication of the evolutionary process of neural networks observed in the human brain. This model exhibits comparable performance to the human brain with respect to optimization, clustering, learning, classification, prediction, and generalization [12].

The subsequent phrases outline the principal benefits of utilizing the artificial neural network methodology.

• The artificial neural network (ANN) has demonstrated remarkable effectiveness and efficacy even when deployed on a limited hardware infrastructure.
• The intricate process of class-distributed mapping is surprisingly simplified by the use of artificial neural networks.
• The input vector is responsible for determining the appropriate outcomes within the training set.
• The weights that represent outcomes are acquired through iterative training.

The artificial neural network (ANN) utilized in this study is a computational approach inspired by the human brain’s network of interconnected neurons. It is designed to handle complex tasks such as problem-solving, learning, and prediction. The ANN consists of multiple layers, including an input layer that receives data, hidden layers that process information, and an output layer that generates predictions. The ANN model employed in this research is a multilayer perceptron (MLP) feed-forward network, which is a widely used architecture. It comprises nodes arranged in layers, with connections between nodes in adjacent layers. The back-propagation algorithm is utilized for training, which is applied to adjust the weights of connections iteratively to minimize prediction errors. A key parameter is the number of hidden layers and nodes, which is determined through iterative optimization. This ANN model is trained on data, utilizing 70% for training, 15% for validation, and 15% for testing. The model’s ability to predict fluid flow characteristics, such as skin friction and heat transfer rates, is demonstrated through comparison with numerical simulations. The ANN’s ability to capture intricate relationships within the data enables it to produce accurate predictions, offering an efficient alternative to traditional computational methods. The development of a training algorithm and the establishment of interneuronal connections yield diverse neural network structures. The stratification of neural networks is frequently attributed to the close interplay among individual neurons. The artificial neural network (ANN) methodology comprises three unique strata, namely the input, hidden, and output layers. The layers of the system receive external information, undergo processing, and subsequently transmit the output through the artificial neural network. The input layer transmits information to the hidden layer neurons in an unmodified state without undergoing any processing by the input layer’s processing components. It is imperative to acknowledge that the weights, connection lines, and interconnecting neurons facilitate the transmission of information. A database is maintained by the system to facilitate training of artificial neural networks (ANNs), wherein input values and weights are stored. The construction of an artificial neural network is guided by the utilization of data, which takes into account various factors, such as determining the optimal number of layers and hidden neurons.

The multilayer perceptron architecture-based feed-forward neural network (FFNN) has gained significant popularity and interest as an artificial neural network (ANN) model and is currently the most commonly employed. Alternative techniques for training feed-forward neural networks exhibit inferior efficiency compared to the back-propagation method. The back-propagation algorithm is capable of modifying the weights of individual neurons during the computation of the network’s output error. This weight adjustment is uniformly implemented across all neurons to reduce the output error.

The subsequent expression represents the net input of the jth hidden neuron, as depicted in Figure 2.

The ith node input layer is denoted as $x_i$, the jth node hidden layer is indicated as $a_j$, and the linking weight between $x_i$ and $a_j$ is represented as $W{1}_{ji}$.

The output’s jth hidden node is represented as follows:

$$z_j\left(x\right)=\frac{1}{1+e^{-y_j\left(x\right)}},$$

The output layer’s kth node is indicated as follows:

$$o_k\left(x\right)={\displaystyle \sum _{j=1}^{m}W{2}_{kj}{z}_j+b_k}$$

$W{2}_{kj}$ is the connecting weight between the kth node of the output layer and the jth node of the hidden layer, where $b_k$ is the biasing term at the kth node of the output layer.

The present study involves the quantification of skin friction and heat transfer rates for representative samples of artificial neural network (ANN) output nodes, as depicted in Figure 3. The trained network’s characteristics, including its layer counts, are listed in

Table 1. Neural networking detail.

Input Layer	Hidden Layers	Output Layer
$4({\alpha }_s$$,We$$,{\beta }_{mp}$$,\mathrm{and}{M}_{mp}$)	3	1 (skin friction factor)
$11({\alpha }_s$$,We$$,{\beta }_{mp}$$,M_{mp}$$,Rd$$,N_T$$,N_B$$,D_r$$,S_r$$,B_i$ and $\gamma$)	5	2 (heat and mass transfer rate)

. The parameters $We,\hspace{0.17em}{\beta }_{mp}$ and ${\alpha }_s$ and $B_i,\hspace{0.17em}{N}_T,\hspace{0.17em}{N}_B,\hspace{0.17em}{D}_r$, and $S_r$ are estimated for the input node samples. The process of trial and error is employed to ascertain the optimal number of nodes in the hidden layer of a neural network, taking into account the number of training epochs necessary for the network, preventing the over- or under-setting of input parameters, and guaranteeing the successful completion of the learning process. After conducting multiple iterations, it was determined that the convergence criteria utilized involved the incorporation of a single hidden layer consisting of five neurons. This approach was implemented to mitigate the discrepancy between the projected values of $C_f^{*}$, $N{u}^{\ast }$, and $S{h}^{*}$. In the present study, 24 datasets are obtained for ${\alpha }_s$, $We$, ${\beta }_{mp}$, and $M_{mp}$, and 48 datasets are obtained for ${\alpha }_s$, $We$, ${\beta }_{mp}$, $M_{mp}$, $Rd$, $N_T$, $N_B$, $D_r$, $S_r$, $\gamma$, and $B_i$. It is important to note that the appropriate neural network can be chosen by minimizing the mean square error between the network model’s target data and the sample data. The training process concludes upon reaching a point where the error stabilizes (see Figure 4), indicating that the trained network has achieved the desired level of accuracy. Figure 5 presents an error histogram showcasing errors detected during the training phase of the ANN. These values are small when considering the error values along the x-axis of this histogram. Upon examining the data histogram, which illustrates error values across three distinct data sets, it becomes apparent that the errors are heavily concentrated very close to the yellow line representing the zero-error threshold. These findings, derived from an analysis of the error histogram, conclusively demonstrate the successful completion of the ANN model’s training phase, characterized by an impressively low error rate. The results pertaining to the skin friction coefficient and heat transfer rate in the training, validation, and test sets of the ANN model are presented in Figure 6 and Figure 7. The provided elements equip artificial neural network models with the necessary components to replicate intricate relationships between input and output variables.

The outcomes of the artificial neural network (ANN) model exhibit a remarkable concurrence with the figures derived from computational analysis.

Table 2. Effects of $We,\hspace{0.17em}{\beta }_{mp}$, and ${\alpha }_s$ on local skin friction coefficient $C_f^{*}$, dimensionless local rate of heat transfer $N{u}^{\ast }$, and dimensionless local rate of mass transfer $S{h}^{*}$ over the wedge plate, wedge, and stagnation point.

	Parameter Values	Physical Quantities	Plate	Wedge	Stagnation Point
$We$	0.5	$C_f^{*}$	0.61098	1.15894	1.62372
		$N{u}^{\ast }$	0.17579	0.19120	0.20069
		$S{h}^{*}$	0.13759	0.13997	0.14865
	1.5	$C_f^{*}$	0.64079	1.24752	1.76215
		$N{u}^{\ast }$	0.17602	0.19160	0.20115
		$S{h}^{*}$	0.13738	0.13956	0.14819
	2.5	$C_f^{*}$	0.68280	1.34075	1.88226
		$N{u}^{\ast }$	0.17648	0.19216	0.20162
		$S{h}^{*}$	0.13712	0.13924	0.14797
${\beta }_{mp}$	0.1	$C_f^{*}$	0.60769	1.14176	1.58952
		$N{u}^{\ast }$	0.17581	0.19116	0.20062
		$S{h}^{*}$	0.13763	0.14006	0.14880
	0.3	$C_f^{*}$	0.53777	1.03182	1.46035
		$N{u}^{\ast }$	0.17341	0.18966	0.19966
		$S{h}^{*}$	0.13769	0.13949	0.14766
	0.5	$C_f^{*}$	0.46320	0.89631	1.29484
		$N{u}^{\ast }$	0.17038	0.18750	0.19825
		$S{h}^{*}$	0.13787	0.13882	0.14612
${\alpha }_s$	0.1	$C_f^{*}$	0.62774	1.37457	2.16400
		$N{u}^{\ast }$	0.16060	0.17898	0.19102
		$S{h}^{*}$	0.14033	0.14040	0.14600
	0.4	$C_f^{*}$	0.61465	1.19645	1.71000
		$N{u}^{\ast }$	0.17304	0.18928	0.19932
		$S{h}^{*}$	0.13803	0.13991	0.14809
	0.7	$C_f^{*}$	0.59115	1.04187	1.38777
		$N{u}^{\ast }$	0.18010	0.19385	0.20237
		$S{h}^{*}$	0.13714	0.14048	0.15002

,

Table 3. Effects of $B_i,\hspace{0.17em}{N}_T,\hspace{0.17em}{N}_B,\hspace{0.17em}{D}_r$, and $S_r$ on dimensionless local rate of heat transfer $N{u}^{\ast }$ and dimensionless local rate of mass transfer $S{h}^{*}$ over the wedge plate, wedge, and stagnation point.

	Parameter Values	Physical Quantities	Plate	Wedge	Stagnation Point
$B_i$	0.1	$N{u}^{\ast }$	0.04254	0.04352	0.04406
		$S{h}^{*}$	0.15246	0.15893	0.17031
	0.4	$N{u}^{\ast }$	0.14697	0.15769	0.16414
		$S{h}^{*}$	0.14085	0.14435	0.15382
	0.7	$N{u}^{\ast }$	0.22673	0.25245	0.26896
		$S{h}^{*}$	0.13191	0.13219	0.13939
$N_T$	0.01	$N{u}^{\ast }$	0.16741	0.18669	0.19806
		$S{h}^{*}$	0.10187	0.08966	0.08950
	0.02	$N{u}^{\ast }$	0.13749	0.17012	0.18823
		$S{h}^{*}$	0.10891	0.06094	0.04049
	0.03	$N{u}^{\ast }$	0.08634	0.14125	0.17092
		$S{h}^{*}$	0.21073	0.08891	0.02709
$N_B$	0.1	$N{u}^{\ast }$	0.11288	0.15630	0.18007
		$S{h}^{*}$	0.15873	0.15577	0.16153
	0.2	$N{u}^{\ast }$	0.03178	0.09207	0.13861
		$S{h}^{*}$	0.17854	0.17381	0.17441
	0.3	$N{u}^{\ast }$	0.00284	0.03414	0.08648
		$S{h}^{*}$	0.18389	0.18913	0.18987
$D_r$	0.01	$N{u}^{\ast }$	0.16430	0.18535	0.19741
		$S{h}^{*}$	0.14148	0.14208	0.14992
	0.02	$N{u}^{\ast }$	0.14719	0.17720	0.19314
		$S{h}^{*}$	0.14749	0.14504	0.15149
	0.03	$N{u}^{\ast }$	0.11976	0.16514	0.18724
		$S{h}^{*}$	0.15770	0.14970	0.15381
$S_r$	0.1	$N{u}^{\ast }$	0.17528	0.19078	0.20035
		$S{h}^{*}$	0.14923	0.15465	0.16529
	0.4	$N{u}^{\ast }$	0.17567	0.19106	0.20055
		$S{h}^{*}$	0.14055	0.14372	0.15294
	0.7	$N{u}^{\ast }$	0.17607	0.19135	0.20075
		$S{h}^{*}$	0.13174	0.13270	0.14051

,

Table 4. Skin friction factor values for different ${\alpha }_s,\hspace{0.17em}We,{\beta }_{mp}$ and $M_{mp}$.

${\mathit{\alpha }}_{\mathit{s}}$	$\mathit{W}\mathit{e}$	${\mathit{\beta }}_{\mathit{m}\mathit{p}}$	${\mathit{M}}_{\mathit{m}\mathit{p}}$	${\mathit{C}}_{\mathit{f}}^{*}$
				NM	ANN	Error
0	0.1	0.1	0.1	2.341742	2.341054	6.87 × 10−4
0.2	0.1	0.1	0.1	1.997203	1.997393	1.90 × 10−4
0.4	0.1	0.1	0.1	1.710001	1.709799	2.02 × 10−4
0.6	0.1	0.1	0.1	1.482665	1.482392	2.72 × 10−4
0.8	0.1	0.1	0.1	1.303260	1.302654	6.06 × 10−4
0.5	0.2	0.1	0.1	1.594459	1.597219	2.76 × 10−3
0.5	0.4	0.1	0.1	1.612127	1.615347	3.22 × 10−3
0.5	0.6	0.1	0.1	1.636474	1.636499	2.52 × 10−5
0.5	0.8	0.1	0.1	1.663976	1.662062	1.91 × 10−3
0.5	1	0.1	0.1	1.692508	1.694016	1.51 × 10−3
0.5	0.1	0.2	0.1	1.528476	1.519159	9.32 × 10−3
0.5	0.1	0.4	0.1	1.383355	1.377509	5.85 × 10−3
0.5	0.1	0.6	0.1	1.190723	1.199874	9.15 × 10−3
0.5	0.1	0.8	0.1	0.901431	0.898421	3.01 × 10−3
0.5	0.1	1	0.1	0.292491	0.294073	1.58 × 10−3
0.5	0.1	0.1	0.2	1.589078	1.589140	6.21 × 10−5
0.5	0.1	0.1	0.4	1.588526	1.589259	7.32 × 10−4
0.5	0.1	0.1	0.6	1.588166	1.589272	1.11 × 10−3
0.5	0.1	0.1	0.8	1.587900	1.589409	1.51 × 10−3
0.5	0.1	0.1	1	1.587689	1.588979	1.29 × 10−3

and

Table 5. Heat transfer rate and mass transfer rate for different ${\alpha }_s,\hspace{0.17em}We,{\beta }_{mp},\hspace{0.17em}{M}_{mp},\hspace{0.17em}Rd,\hspace{0.17em}{N}_T,\hspace{0.17em}{N}_B,\hspace{0.17em}{D}_r,S_r, \gamma \mathrm{and} B_i$.

${\mathit{\alpha }}_{\mathit{s}}$	$\mathit{W}\mathit{e}$	${\mathit{\beta }}_{\mathit{m}\mathit{p}}$	${\mathit{M}}_{\mathit{m}\mathit{p}}$	$\mathit{R}\mathit{d}$	${\mathit{N}}_{\mathit{T}}$	${\mathit{N}}_{\mathit{B}}$	${\mathit{D}}_{\mathit{r}}$	${\mathit{S}}_{\mathit{r}}$	$\mathit{\gamma }$	${\mathit{B}}_{\mathit{i}}$	$\mathit{N}{\mathit{u}}^{\ast }$			$\mathit{S}{\mathit{h}}^{*}$
											NM	ANN	Error	NM	ANN	Error
0	0.1	0.1	0.1	1	0.001	0.01	0.001	0.5	1	0.5	0.1842	0.1891	4.90 × 10−3	0.1460	0.1450	9.62 × 10−4
0.2	0.1	0.1	0.1	1	0.001	0.01	0.001	0.5	1	0.5	0.1950	0.1950	6.98 × 10−6	0.1466	0.1466	2.84 × 10−5
0.4	0.1	0.1	0.1	1	0.001	0.01	0.001	0.5	1	0.5	0.1993	0.1992	8.00 × 10−5	0.1481	0.1481	2.37 × 10−5
0.6	0.1	0.1	0.1	1	0.001	0.01	0.001	0.5	1	0.5	0.2016	0.2015	6.08 × 10−5	0.1494	0.1496	1.18 × 10−4
0.5	0.2	0.1	0.1	1	0.001	0.01	0.001	0.5	1	0.5	0.2006	0.2007	6.93 × 10−5	0.1488	0.1487	7.70 × 10−5
0.5	0.4	0.1	0.1	1	0.001	0.01	0.001	0.5	1	0.5	0.2007	0.2006	5.71 × 10−5	0.1487	0.1486	1.17 × 10−4
0.5	0.6	0.1	0.1	1	0.001	0.01	0.001	0.5	1	0.5	0.2007	0.2006	1.54 × 10−4	0.1486	0.1486	2.39 × 10−5
0.5	0.8	0.1	0.1	1	0.001	0.01	0.001	0.5	1	0.5	0.2008	0.2008	4.59 × 10−5	0.1485	0.1486	1.55 × 10−4
0.5	0.1	0.2	0.1	1	0.001	0.01	0.001	0.5	1	0.5	0.2002	0.2002	5.13 × 10−5	0.1483	0.1481	1.86 × 10−4
0.5	0.1	0.4	0.1	1	0.001	0.01	0.001	0.5	1	0.5	0.1990	0.1989	1.37 × 10−4	0.1470	0.1466	3.95 × 10−4
0.5	0.1	0.6	0.1	1	0.001	0.01	0.001	0.5	1	0.5	0.1972	0.1967	5.75 × 10−4	0.1451	0.1450	1.09 × 10−4
0.5	0.1	0.8	0.1	1	0.001	0.01	0.001	0.5	1	0.5	0.1936	0.1932	4.11 × 10−4	0.1422	0.1435	1.25 × 10−3
0.5	0.1	0.1	0.2	1	0.001	0.01	0.001	0.5	1	0.5	0.2006	0.2006	3.22 × 10−6	0.1488	0.1488	1.56 × 10−5
0.5	0.1	0.1	0.4	1	0.001	0.01	0.001	0.5	1	0.5	0.2006	0.2005	1.23 × 10−4	0.1488	0.1487	4.70 × 10−5
0.5	0.1	0.1	0.6	1	0.001	0.01	0.001	0.5	1	0.5	0.2006	0.2004	1.93 × 10−4	0.1488	0.1488	2.24 × 10−5
0.5	0.1	0.1	0.8	1	0.001	0.01	0.001	0.5	1	0.5	0.2006	0.2005	5.83 × 10−5	0.1488	0.1489	8.30 × 10−5
0.5	0.1	0.1	0.1	0.1	0.001	0.01	0.001	0.5	1	0.5	0.0207	0.0207	2.97 × 10−5	0.1463	0.1461	1.15 × 10−4
0.5	0.1	0.1	0.1	0.2	0.001	0.01	0.001	0.5	1	0.5	0.0412	0.0404	7.99 × 10−4	0.1466	0.1468	1.46 × 10−4
0.5	0.1	0.1	0.1	0.3	0.001	0.01	0.001	0.5	1	0.5	0.0616	0.0617	1.40 × 10−4	0.1469	0.1474	4.23 × 10−4
0.5	0.1	0.1	0.1	0.4	0.001	0.01	0.001	0.5	1	0.5	0.0818	0.0836	1.86 × 10−3	0.1473	0.1480	7.13 × 10−4
0.5	0.1	0.1	0.1	1	0.01	0.01	0.001	0.5	1	0.5	0.1981	0.1980	2.63 × 10−5	0.0895	0.0896	5.67 × 10−5
0.5	0.1	0.1	0.1	1	0.02	0.01	0.001	0.5	1	0.5	0.1882	0.1882	5.24 × 10−6	0.0405	0.0404	8.69 × 10−5
0.5	0.1	0.1	0.1	1	0.03	0.01	0.001	0.5	1	0.5	0.1709	0.1693	1.64 × 10−3	0.0271	0.0445	1.74 × 10−2
0.5	0.1	0.1	0.1	1	0.04	0.01	0.001	0.5	1	0.5	0.1463	0.1463	3.01 × 10−5	0.0674	0.0672	2.13 × 10−4
0.5	0.1	0.1	0.1	1	0.001	0.10	0.001	0.5	1	0.5	0.1801	0.1813	1.25 × 10−3	0.1615	0.1603	1.26 × 10−3
0.5	0.1	0.1	0.1	1	0.001	0.15	0.001	0.5	1	0.5	0.1618	0.1617	5.48 × 10−5	0.1673	0.1674	3.81 × 10−5
0.5	0.1	0.1	0.1	1	0.001	0.20	0.001	0.5	1	0.5	0.1386	0.1372	1.41 × 10−3	0.1744	0.1749	4.53 × 10−4
0.5	0.1	0.1	0.1	1	0.001	0.25	0.001	0.5	1	0.5	0.1126	0.1127	8.95 × 10−5	0.1822	0.1822	3.52 × 10−5
0.5	0.1	0.1	0.1	1	0.001	0.01	0.01	0.5	1	0.5	0.1974	0.1972	1.59 × 10−4	0.1499	0.1499	4.22 × 10−5
0.5	0.1	0.1	0.1	1	0.001	0.01	0.015	0.5	1	0.5	0.1954	0.1952	2.36 × 10−4	0.1507	0.1510	3.74 × 10−4
0.5	0.1	0.1	0.1	1	0.001	0.01	0.02	0.5	1	0.5	0.1931	0.1929	2.09 × 10−4	0.1515	0.1520	5.18 × 10−4
0.5	0.1	0.1	0.1	1	0.001	0.01	0.025	0.5	1	0.5	0.1905	0.1906	5.21 × 10−5	0.1525	0.1523	2.20 × 10−4
0.5	0.1	0.1	0.1	1	0.001	0.01	0.001	0.1	1	0.5	0.2003	0.2002	1.18 × 10−4	0.1653	0.1653	3.98 × 10−5
0.5	0.1	0.1	0.1	1	0.001	0.01	0.001	0.15	1	0.5	0.2004	0.2003	5.56 × 10−5	0.1632	0.1633	4.70 × 10−5
0.5	0.1	0.1	0.1	1	0.001	0.01	0.001	0.2	1	0.5	0.2004	0.2004	2.99 × 10−6	0.1612	0.1613	1.11 × 10−4
0.5	0.1	0.1	0.1	1	0.001	0.01	0.001	0.25	1	0.5	0.2005	0.2005	5.38 × 10−5	0.1591	0.1593	1.52 × 10−4
0.5	0.1	0.1	0.1	1	0.001	0.01	0.001	0.5	0	0.5	0.2017	0.2017	1.01 × 10−5	0.0723	0.0727	4.06 × 10−4
0.5	0.1	0.1	0.1	1	0.001	0.01	0.001	0.5	0.2	0.5	0.2015	0.2014	4.18 × 10−5	0.0904	0.0898	5.45 × 10−4
0.5	0.1	0.1	0.1	1	0.001	0.01	0.001	0.5	0.4	0.5	0.2012	0.2013	2.11 × 10−5	0.1067	0.1064	3.08 × 10−4
0.5	0.1	0.1	0.1	1	0.001	0.01	0.001	0.5	0.6	0.5	0.2010	0.2011	9.63 × 10−5	0.1218	0.1220	2.45 × 10−4
0.5	0.1	0.1	0.1	1	0.001	0.01	0.001	0.5	1	0.2	0.0860	0.0860	1.65 × 10−5	0.1646	0.1645	3.64 × 10−5
0.5	0.1	0.1	0.1	1	0.001	0.01	0.001	0.5	1	0.4	0.1641	0.1643	1.30 × 10−4	0.1538	0.1539	1.01 × 10−4
0.5	0.1	0.1	0.1	1	0.001	0.01	0.001	0.5	1	0.6	0.2355	0.2358	2.56 × 10−4	0.1440	0.1441	7.48 × 10−5
0.5	0.1	0.1	0.1	1	0.001	0.01	0.001	0.5	1	0.8	0.3010	0.3010	6.59 × 10−5	0.1350	0.1350	5.53 × 10−5

present the skin friction factor, heat transfer rate, and mass transfer rate magnetic parameter $\left({\beta }_{mp}=0.1,\hspace{0.17em}0.2,\hspace{0.17em}0.3\right)$, Weissenberg number $\left(We=0.5,\hspace{0.17em}2.0,\hspace{0.17em}3.5\right)$, velocity slip parameter $\left({\alpha }_s=0.1,\hspace{0.17em}0.3,\hspace{0.17em}0.5\right)$, magnetic Prandtl number $\left(M_{mp}=0.1,\hspace{0.17em}0.3,\hspace{0.17em}0.5\right)$, thermal radiation parameter $\left(Rd=1.0,\hspace{0.17em}2.0,\hspace{0.17em}3.0\right)$, Dufour number $\left(D_r=0.001,\hspace{0.17em}0.005,\hspace{0.17em}0.010\right)$, Biot number $\left(Bi=0.1,\hspace{0.17em}0.3,\hspace{0.17em}0.5\right)$, thermophoresis parameter $\left(N_T=0.001,\hspace{0.17em}0.005,\hspace{0.17em}0.010\right)$, Brownian motion parameter $\left(N_B=0.1,\hspace{0.17em}0.2,\hspace{0.17em}0.3\right)$, Soret number $\left(S_r=0.1,\hspace{0.17em}0.5,\hspace{0.17em}1.0\right)$, chemical reaction parameter $\left(\gamma =0.0,\hspace{0.17em}1.0,\hspace{0.17em}2.0\right)$, and Schmidt number $\left(Sc=0.5,\hspace{0.17em}1.0,\hspace{0.17em}1.5\right)$ values for a range of parameters. In addition to the quantitative results, the findings of the artificial neural network model are demonstrated to be favorable. Thus far, the findings of this investigation have indicated that the artificial neural network (ANN) is capable of accurately forecasting both skin friction and heat transfer rate.

4. Results and Discussion

This section is intended to demonstrate double-diffusive convection in radiative flow of Carreau nanofluid over three geometries (plate, wedge, and stagnation point) with induced magnetic field. In this section, the physical importance of velocities $\left(f^{\prime }\left(\eta \right)\right)$, induced magnetic field $\left(g^{\prime }\left(\eta \right)\right)$, temperature $\left(\theta \left(\eta \right)\right)$, concentration $\left(\varphi \left(\eta \right)\right)$, skin friction $\left(C_f^{\ast }\right)$, Nusselt number $\left(N{u}^{\ast }\right)$, and mass transfer rate $\left(S{h}^{\ast }\right)$ are visualized and thoroughly discussed. Under specific boundary conditions, the dimensional version of the equations for flow, heat transport, and concentration is solved using the Runge–Kutta–Fehlberg for plate $\left(\beta =0\right)$, wedge $\left(\beta =0.5\right)$, and stagnation point $\left(\beta =1.0\right)$ are shown as solid, dashed, and dotted lines in the comparison to the thermal radiative flow of ferromagnetic physical features over the three distinct geometries of the images, respectively. Comparing the results shows a significant degree of agreement, as shown in

Table 6. Comparison of the present result $\left(N{u}^{\ast }\right)$ with the results of Lin and Lin [42] in the absences of $We,{\beta }_{mp},B_i,N_T,N_B,D_r,S_r,M_{mp},n_k,We,Rd$, $S_r$, and ${\alpha }_s$.

$\mathbf{Pr}$	Lin and Lin [42] Results			Present Results
	${\mathit{\beta }}_1=0$	${\mathit{\beta }}_1=0.5$	${\mathit{\beta }}_1=1.0$	${\mathit{\beta }}_1=0$	${\mathit{\beta }}_1=0.5$	${\mathit{\beta }}_1=1.0$
1	0.372722	0.493968	0.640326	0.3727218	0.4939669	0.6403256
10	0.343388	0.477039	0.631365	0.3433875	0.4770383	0.6313644
100	0.339208	0.482208	0.644454	0.3392574	0.4822070	0.6444532
1000	0.338766	0.486599	0.653023	0.3387666	0.4865983	0.6530225

. This indicates that the accuracy of the numerical simulation’s outcomes.

Figure 8 and Figure 9 depict how the magnetic parameter $\left({\beta }_{mp}=0.1,\hspace{0.17em}0.2,\hspace{0.17em}0.3\right)$ affected the velocities $\left(f^{\prime }\left(\eta \right)\right)$ and induced magnetic field $\left(g^{\prime }\left(\eta \right)\right)$ profiles for ${\beta }_1$ = 0, 0.5, 1 s, respectively. It has been determined that the velocity profile $f^{\prime }\left(\eta \right)$ decreases for higher values of ${\beta }_{mp}$. A similar nature is observed in the induced magnetic profile $\left(g^{\prime }\left(\eta \right)\right)$. The magnetic field deflects particles near the walls less than those in the center because the force acting on them is weaker. As a result, the magnetic field in the vicinity of the walls weakens while remaining strong at the wall’s center. As a result, the magnetic field intensity in the velocity profile and induced magnetic field are decreased.

Figure 10 and Figure 11 describe how the velocities $\left(f^{\prime }\left(\eta \right)\right)$ and induced magnetic field $\left(g^{\prime }\left(\eta \right)\right)$ profiles are influenced for the variations in Weissenberg number $\left(We=0.5,\hspace{0.17em}2.0,\hspace{0.17em}3.5\right)$ blood nanofluid within the various geometries ${\beta }_1$ = 0, 0.5, 1. Note that the velocity profile diminishes for both $\left(f^{\prime }\left(\eta \right)\right)$ and $\left(g^{\prime }\left(\eta \right)\right)$ and rises for $\left(We\right)$. The fluid’s elasticity predominates as the Weissenberg number rises, and the blood-based fluid flow shows more prominent viscoelastic effects. For instance, the fluid may deform and stretch in reaction to the flow due to the elasticity, creating elastic stresses that hinder the flow. When compared to a fluid that is only viscous, this may result in the velocity profile being more wide and flat. The fluid’s relaxation time, which is a component in the Weissenberg number definition, serves as a defining characteristic of this relaxation process. As a result, the Weissenberg number decreases and the fluid’s elasticity becomes less prominent as the fluid moves downgradient, resulting in a narrower velocity profile that resembles the profile for a purely viscous fluid.

The effect of the velocity slip parameter $\left({\alpha }_s=0.1,\hspace{0.17em}0.3,\hspace{0.17em}0.5\right)$ on velocities $\left(f^{\prime }\left(\eta \right)\right)$, induced magnetic field $\left(g^{\prime }\left(\eta \right)\right)$ and temperature $\left(\theta \left(\eta \right)\right)$ profiles changes are shown in Figure 12, Figure 13 and Figure 14. These graphs indicate that increasing ${\alpha }_s$ values lead to an increase in $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$ for the various geometries ${\beta }_1$ = 0, 0.5, and 1, but it is reduced for the temperature profile $\left(\theta \left(\eta \right)\right)$. The blood fluid nanoparticles can interact with the solid surface to form a thin layer of adsorbed particles, which can change the surface’s boundary conditions. The velocity slip parameter’s value reflects the possibility of a slip velocity between the fluid and the surface. Due to the enhanced contacts between the nanoparticles and the surface, the slip velocity may rise as the fluid’s nanoparticle concentration rises. The velocity slip parameter’s value may rise as a result of this. In general, when the blood nanofluid temperature rises, the fluid’s viscosity decreases and the blood cells and nanoparticles move through the fluid at a faster rate. The velocity slip parameter might rise as a result of the larger velocity differential between the fluid and the solid surface caused by the increasing velocity.

Figure 15 shows the effect of different magnetic Prandtl numbers $\left(M_{mp}=0.1,\hspace{0.17em}0.3,\hspace{0.17em}0.5\right)$ on the induced magnetic field $\left(g^{\prime }\left(\eta \right)\right)$ profile. It is acknowledged that the $g^{\prime }\left(\eta \right)$ decreases when the magnetic Prandtl number $M_{mp}$ increases. A blood-based nanofluid encounters the Lorentz force when an external magnetic field is introduced, which can cause the fluid to flow and produce heat transfer. The magnetic Prandtl number illustrates how important magnetic diffusion is in this process in relation to fluid viscosity. Additionally, the viscosity and thermal conductivity of the nanofluid may alter due to the nanoparticles, which may have an effect on the magnetic Prandtl number as a whole. Overall, the interaction between the magnetic nanoparticles, magnetic field, and fluid characteristics is the physical cause of the drop in magnetic Prandtl number in blood-based nanofluids under generated magnetic field profiles.

Figure 16 illustrates how the thermal radiation parameter $\left(Rd=1.0,\hspace{0.17em}2.0,\hspace{0.17em}3.0\right)$ influence of temperature $\left(\theta \left(\eta \right)\right)$ changes. According to this graph, the temperature $\left(\theta \left(\eta \right)\right)$ rises with increasing $Rd$ values. The interaction of the nanoparticles with thermal radiation, which can improve heat transfer and change the optical characteristics of the nanofluid, is the physical cause for the rise in radiation parameter in the blood-based nanofluid temperature profile.

The fluctuations in $\theta \left(\eta \right)$ are shown in Figure 17 for different values of the Dufour number $\left(D_r=0.001,\hspace{0.17em}0.005,\hspace{0.17em}0.010\right)$. It is detected that the $\theta \left(\eta \right)$ profiles increase by enhancing the Dufour number $\left(D_r\right)$ values. When the temperature of the Dufour number rises, it indicates that thermal diffusion is occurring at a slower rate than the rate at which mass is transported through diffusion. This may occur because when the mixture’s chemical composition varies with temperature, so do the components’ diffusivities. This may lead to a rise in the Dufour number.

Figure 18 shows the temperature $\left(\theta \left(\eta \right)\right)$ profile with different Biot number $\left(Bi=0.1,\hspace{0.17em}0.3,\hspace{0.17em}0.5\right)$ values applied to the various geometries ${\beta }_1$ = 0, 0.5, 1. The temperature $\left(\theta \left(\eta \right)\right)$ profile is increased when the Biot number $\left(Bi\right)$ is increased. The Biot number is described as the relationship between a body’s internal and exterior thermal resistances. In a temperature profile, an increasing Biot number indicates that the body’s internal thermal resistance is rising compared to its exterior thermal resistance.

Figure 19 and Figure 20 depict how the thermophoresis parameter $\left(N_T=0.001,\hspace{0.17em}0.005,\hspace{0.17em}0.010\right)$ affected the temperature $\left(\theta \left(\eta \right)\right)$ and concentration $\left(\varphi \left(\eta \right)\right)$ profiles for ${\beta }_1$ = 0, 0.5, 1 s, respectively. Figure 19 describes how the $\theta \left(\eta \right)$ influenced the three geometries to the variations in $\left(Nt\right)$. It is observed that $\theta \left(\eta \right)$ intensification boosts $\left(Nt\right)$. This is due to the action of thermophoresis, which draws large thermally conducting nanoparticles deeper into the fluid and exposes a stronger thermal boundary layer. The differences in a concentration profile $\left(\varphi \left(\eta \right)\right)$ for several thermophoresis values $\left(Nt\right)$ are shown schematically in Figure 20. The $\theta \left(\eta \right)$ increases for superior values of $\left(Nt\right)$. The wall slope of concentration profiles increases as the fluid concentration rises, nonetheless only partly.

The Brownian motion parameter $\left(N_B=0.1,\hspace{0.17em}0.2,\hspace{0.17em}0.3\right)$ effect on the temperature profile $\left(\varphi \left(\eta \right)\right)$ and concentration profile $\left(\varphi \left(\eta \right)\right)$ in the case of ${\beta }_1$ = 0, 0.5, 1 was investigated with results shown in Figure 21 and Figure 22. The effect of $\theta \left(\eta \right)$ on the changes in the Brownian motion parameter $\left(N_B\right)$ is shown in Figure 21. It can be seen that $\theta \left(\eta \right)$ intensifications increase $\left(N_B\right)$. Blood nanofluid flow collides with the basic liquid particles when there is an increase in the Brownian motion parameter. This results in an increase in temperature as well as an increase in the kinetic energy. The differences of a concentration profile $\left(\varphi \left(\eta \right)\right)$ for several values of Brownian motion $\left(N_B\right)$ are shown schematically in Figure 22. The blood nanofluid of the concentration profile $\left(\varphi \left(\eta \right)\right)$ rises for superior values of $\left(N_B\right)$. Physically, the random movement of the fluid particles improves with increasing Brownian motion, which increases heat generation and raises the fluid’s temperature.

Figure 23 shows the concentration profile $\left(\varphi \left(\eta \right)\right)$ with various Soret number $\left(S_r=0.1,\hspace{0.17em}0.5,\hspace{0.17em}1.0\right)$ values applied to the various geometries ${\beta }_1$ = 0, 0.5, 1. The concentration profile $\left(\varphi \left(\eta \right)\right)$ is increased when the Soret number $\left(S_r\right)$ is increased. The relative rates of mass transport caused by temperature and concentration gradients in a fluid mixture are described by the dimensionless Soret number. The rate of mass transfer due to heat gradients increases relative to the rate of mass transport due to concentration gradients as the Soret number for a concentration profile in blood nanofluid flow increases.

Figure 24 illustrates how the chemical reaction parameter $\left(\gamma =0.0,\hspace{0.17em}1.0,\hspace{0.17em}2.0\right)$ influences in thblood nanofluid flow with response to changes in the concentration profile $\left(\varphi \left(\eta \right)\right)$. According to this graph, the concentration profile $\left(\varphi \left(\eta \right)\right)$ decays with increasing values of $\gamma$. The blood nanofluid flow concentration profiles show a drop in the chemical reaction parameter when compared to diffusion or convection, which indicates that the pace of chemical reactions is slowing.

Figure 25 illustrates how the Schmidt number $\left(Sc=0.5,\hspace{0.17em}1.0,\hspace{0.17em}1.5\right)$ influences blood nanofluid flow with response to concentration profile $\left(\varphi \left(\eta \right)\right)$ changes. According to this graph, the concentration profile $\left(\varphi \left(\eta \right)\right)$ decays with increasing values of $Sc$.

5. Conclusions

The purpose of this study is to illustrate double-diffusive convection for the fluid transport properties of blood-based Carreau nanofluid flow over three distinct geometries with an induced magnetic field. This type of theoretical investigation is extensively used in modern aspects of biological and chemical engineering, such as designing pumping devices for medical diagnostics, mixing samples at certain temperatures, and generating microscale flows using stimulus-responsive working fluids. The effect of important parameters like Weissenberg number, magnetic parameter, thermophoresis, Brownian motion, thermal radiation, Biot number, Soret number, and Dufour number are analyzed. The results are shown through two-dimensional graphs and tables. The following significant findings emerged from this investigation:

• The artificial neural network model does not require linearization, is fast convergent, and has a reduced processing cost.
• The velocity $f^{\prime }\left(\eta \right)$ and induced magnetic profiles decrease for higher values of ${\beta }_{mp}$ magnetic parameter in all the geometries.
• Velocity profile diminishes for both $\left(f^{\prime }\left(\eta \right)\right)$ and $\left(g^{\prime }\left(\eta \right)\right)$ given higher values of the Weissenberg number $\left(We\right)$.
• Radiation, thermophoresis, and temperature ratio parameter increase corresponding to a rise in blood nanofluid temperature.
• The concentration profile $\left(\varphi \left(\eta \right)\right)$ decays with increasing values in the chemical reaction parameter $\left(\gamma \right).$ The temperature $\left(\theta \left(\eta \right)\right)$ profile is increased when the Biot number $\left(Bi\right)$ is increased.
• Among the three geometries, the fluid flow over the plate has the largest heat transfer rate.