Levenberg–Marquardt Training Technique Analysis of Thermally Radiative and Chemically Reactive Stagnation Point Flow of Non-Newtonian Fluid with Temperature Dependent Thermal Conductivity

Abstract

We have examined the magnetized stagnation point flow of non-Newtonian fluid towards an inclined cylindrical surface. The mixed convection, thermal radiation, viscous dissipation, heat generation, first-order chemical reaction, and temperature-dependent thermal conductivity are the physical effects being carried for better novelty. Mathematical equations are constructed for four different flow regimes. The shooting method is used to evaluate the heat transfer coefficient at the cylindrical surface with and without heat generation/thermal radiation effects. For better examination, we have constructed artificial neural networking models with the aid of the Levenberg–Marquardt training technique and Purelin and Tan-Sig transfer functions. The Nusselt number strength is greater for fluctuations in the Casson fluid parameter, Prandtl number, heat generation, curvature, and Eckert number when thermal radiations are present.

1. Introduction

A non-Newtonian fluid is one whose shear stress does not always maintain a linear connection with the rate of shear strain. Non-Newtonian fluids are classified into several categories based on their unique rheological characteristics, and they are employed as additives, media, and protective materials in a variety of industries. Such flows have applications in mechanical processing, dampening equipment, individual protection equipment, and in increasing fluid rheological properties. Owing to the importance of non-Newtonian fluid flows, various non-Newtonian fluid models have been proposed and the Casson fluid model is one of them. For several materials, it has been claimed that the Casson fluid model fits rheological data more accurately than conventional viscoplastic models. Casson fluid can also be used to treat human blood [1]. Human red blood cells can form aggregates, sometimes referred to as rouleaux, which are shaped like chains and are caused by the presence of numerous components such as fibrinogen, protein, and globulin in an aqueous base plasma. A yield stress that is comparable to the constant yield stress in Casson’s fluid exists if the rouleaux exhibit plastic solid behavior. Further, it has been discovered that the flow curves of the pigment suspensions in the lithographic varnishes used to prepare silicon suspensions and printing inks are accurately described by Casson’s constitutive equation. The shear stress–shear rate connection proposed by Casson provides a satisfactory description of the characteristics of numerous polymers. Casson fluid possesses shear-thinning aspects that are considered to have a yield stress below which no flow occurs, zero viscosity at an infinite rate of shear, and an infinite viscosity at zero rates of shear. Owing to such importance, various attempts [2,3,4,5,6,7,8,9,10,11] on Casson fluid models in different configurations are reported by researchers, such as the traditional Fluid Particle Model (FPM), created by Espaol, which was created by Paszyski and Schaefer [12] to represent linear Newton fluid flow. They introduced an FPM modification that enables the simulation of flows for a class of fluids whose constitutive equations are nonlinear. The regionally varying viscosity was taken into account when adjusting the inter-particle interaction parameters. The technique was used to imitate blood flow in arteries with a focus on capturing the propagation of the pulse wave. The non-linear Casson law was used to model the blood. For the carotid artery, numerical results contrast favorably with experimental data. Additionally discussed were the specifics of a parallel implementation of the algorithm. An incompressible Casson chemically reactive fluid towards a porous surface with viscous dissipation with suction was studied by Animasaun [13] to determine the impacts of thermophoresis, Dufour, temperature-dependent thermal conductivity, and viscosity. The pressure drop and flow rate as the fluid passed a porous media were thought to have a non-linear relationship. The governing equations were converted to coupled ordinary differential equations using similarity transformations, and the numerical solutions for the velocity, temperature, and concentration profiles were obtained by shooting, Runge–Kutta–Gill, and quadratic interpolation (Muller’s scheme). Various values of the Casson parameter, Prandtl number, thermophoretic parameter, temperature-dependent viscosity, temperature-dependent thermal conductivity, magnetic parameter, Darcy and Forchheimer parameters were used to study the behavior of dimensionless velocity, temperature, and concentration within the boundary layer. Except in a few instances, the flow regulating parameters were shown to have a significant impact on the final flow profiles. For some cases, the local skin friction, Nusselt number, and Sherwood number were also provided. Theoretical research was conducted by Reddy [14] on the continuous two-dimensional MHD convective boundary layer flow of a Casson fluid across an exponentially inclined porous stretched surface when thermal radiation and chemical reaction are present. It was presumed that wall temperature, velocity and concentration would all vary in accordance with a particular exponential form. We considered thermal, velocity and singular slips, chemical reaction, thermal radiation, and blowing/suction. The suggested model takes into account buoyant flows that are both helping and hindering. Through transformations, the flow equations were transformed into coupled ordinary differential equations which were then numerically solved using the shooting technique and a fourth-order Runge–Kutta scheme. The numerical solutions were shown in tabular form and visually presented for the relevant parameters of the dimensionless temperature, velocity, concentration, heat transfer coefficient, skin friction coefficient, and Sherwood number. Ali et al. [15] investigated the impact of magnetized Casson fluid toward the horizontal cylinder. The pressure gradient’s oscillations were what caused the flow. The fractional partial differential equations were solved by Laplace and finite Hankel transforms. Graphical representations of the effects of various parameters on fluid flow and magnetic particles were given. The analysis demonstrates that, in comparison to the conventional model, the model with fractional order derivatives brings about notable modifications. According to the study, an applied magnetic field lowers the speed of magnetic and blood particles. Nandkeolyar [16] offered magnetized Casson fluid toward the stretched surface. The extremely nonlinear time-dependent partial differential equations make up the fluid flow model. The order of equations was dropped by the use of suitable transformations. The spectral quasilinearization method was used for the solution. The effects of flow parameters, including the Hall current parameter, Casson liquid parameter, unsteadiness parameter, magnetic parameter, and radiative parameter, were investigated in depth. Additionally, the behavior of developing engineering-relevant parameters such as the skin friction and Nusselt were examined. The characteristics of Marangoni fluid flow by way of the disk were studied by Mahanthesh et al. [17]. Magnetohydrodynamics were integrated into the flow analysis. Additionally, the effects of solar radiation, viscous dissipation, and Joule heating were used. On the surface of the disk, the temperature and solute field varies quadratically. Von Kármán transformations were used to produce the ODEs. The RK-Fehlberg-based shooting scheme was used to obtain a numerical solution for the resulting problem under examination. Graphical drawings were used to explore the implications of relevant flow factors involved. Goud et al. [18] examined the impact of a vertically fluctuating porous surface subject to magnetized Casson fluid. By adding similarity variables, dimensionally non-linear coupled differential equations become dimensionally reduced and solved by the use of the Galerkin element method to obtain velocity and temperature. Graphs were used to illustrate the effects of flow parameters such as permeability, Casson fluid, phase angle, magnetic number, and Prandtl number. Through tables, Nusselt number and skin friction were also taught. Additionally, raising the heat source parameter causes a rise in temperature and velocity. Mahdy et al. [19]’s numerical research highlighted the Casson nanofluid about a spinning sphere with convective endpoint conditions. For the issue at hand, a two-phase nanofluid flow model was used. The sphere rotates with an angular velocity that changes continuously over time, just like the free stream. By using the proper dimensionless values, the highly nonlinear partial differential structures that represent the case study were transformed into nonlinear ordinary differential structures. By using MATLAB and the fourth-order Runge–Kutta–Fehlberg technique with a firing scheme, numerical solutions for ordinary differential structures were obtained. Using several charts and tables, specific aspects of such controlled physical characteristics were highlighted and explored. Results from a comparison with the earlier literature were also obtained, and it was discovered that there was an excellent agreement with those results. Bilal et al. [20] investigated the thermal properties of a non-linearly changing viscoelastic fluid towards an inclined isothermal Riga surface. The mathematical formulation was made possible by imposing momentum and energy conservation laws in the form of PDEs. In the presence of radiative heat flux, thermal characteristics were highlighted. For the transformation of PDEs into ODEs, similarity variables were capitalized. The Laplace transform technique was used to obtain an analytical solution for the obtained differential setup. The impact of flow-related variables on associated distributions was depicted graphically. Variations in engineering quantities such as temperature, wall shear stress, and mass fluctuation at the surface were also measured. The accelerating parameter, chemical reaction parameter, radiation and Hartmann number were iterated to improve skin friction, while the heat absorption parameter decreased friction. Furthermore, an increase in chemical reaction and heat absorption characteristics results in a decrease in momentum distribution. Hussain et al. [21] concentrated on the usage of Casson nanofluid in the flow of a porous solar collector on an endless surface. Stretched surface induction altered nanofluid flow. Nonlinear ordinary differential equations (ODEs) were created and improved by reducing boundary conditions to a suitable similarity transformation. The Keller box technique was used to obtain the solution of a set of ODEs. The findings were analyzed and expanded upon. For a higher magnetic parameter the Nusselt number is decreased, while the opposite is the case for the skin friction coefficient.

The purpose of the research is to offer a comparative analysis of heat transfer normal to surface in magnetized Casson fluid flow by the use of an artificial neural networking model. To be more specific, we considered the stagnation point flow of Casson fluid over an inclined stretched cylindrical surface. The flow field includes the following effects, namely first-order chemical reaction, viscous dissipation, thermal radiation, heat generation, and variable thermal conductivity. The whole flow regime is controlled by using mathematical equations and is solved by the shooting method. Key attention is paid to the estimation of the Nusselt number at the surface by using artificial neural networking. For better analysis, four different thermal flow fields are considered and debated numerically. The present evaluation of the heat transfer rate will be helpful for researchers and engineers when operating and designing heat exchangers. The article is designed as follows: A motivational literature survey on Casson fluid flow in various configurations is offered in Section 1, while the mathematical formulation is concluded in Section 2. The adopted numerical scheme is reported in Section 3. The comparative analysis on the numerical data of the Nusselt number is given in Section 4, and the construction of ANN models is debated in Section 5. The conclusion of the present research is summarized in Section 6. It is believed that the present findings on the heat transfer coefficient in a Casson fluid flow regime may help to extend the idea for the examination of time-independent shear rate and shear stress aspects of molten chocolates, yogurt, blood, and many other foodstuffs and biological materials.

2. Mathematical Formulation

The heat transfer characteristics in Casson fluid by way of an inclined cylindrical surface are well-thought-out. The magnetized flow field is carried with the stagnation point flow assumption. The energy equation is considered by incorporating viscous dissipation, variable thermal conductivity, thermal radiations, and heat generation effects. The concentration equation is carried along with the first-order chemical reaction effect. The present flow narrating key concluded mathematical equations are as follows:

$$\frac{\partial (\tilde{R}\tilde{U})}{\partial \tilde{X}}+\frac{\partial (\tilde{R}\tilde{V})}{\partial \tilde{R}}=0 ,$$

(1)

$$\begin{array}{l}\tilde{U}\frac{\partial \tilde{U}}{\partial \tilde{X}}+\tilde{V}\frac{\partial \tilde{U}}{\partial \tilde{R}}=\nu \left(1+\frac{1}{\beta }\right)\left(\frac{1}{\tilde{R}}\frac{\partial \tilde{U}}{\partial \tilde{R}}+\frac{{\partial }^2\tilde{U}}{\partial {\tilde{R}}^2}\right)+{\mathrm{g}}_0{\beta }_T(\tilde{T}-{\tilde{T}}_{\infty })\cos (\alpha )\\ +{\mathrm{g}}_0{\beta }_C(\tilde{C}-{\tilde{C}}_{\infty })\cos (\alpha )+{\tilde{U}}_e\frac{\partial {\tilde{U}}_e}{\partial \tilde{X}}-\frac{\sigma B_0^2}{\rho }(\tilde{U}-{\tilde{U}}_e) ,\end{array}$$

(2)

$$\rho c_{\rho }\left(\tilde{U}\frac{\partial \tilde{T}}{\partial \tilde{X}}+\tilde{V}\frac{\partial \tilde{T}}{\partial \tilde{R}}\right)=\frac{1}{\tilde{R}}\frac{\partial }{\partial \tilde{R}}\left(\kappa \frac{\partial \tilde{T}}{\partial \tilde{R}}\right)-\frac{1}{\tilde{R}}\frac{\partial }{\partial \tilde{R}}\left(\tilde{R}\overline{q}\right)+\overline{\mu }\left(1+\frac{1}{\beta }\right){\left(\frac{\partial \tilde{U}}{\partial \tilde{R}}\right)}^2+Q_0\left(\tilde{T}-{\tilde{T}}_{\infty }\right) .$$

(3)

$$\tilde{U}\frac{\partial \tilde{C}}{\partial \tilde{X}}+\tilde{V}\frac{\partial \tilde{C}}{\partial \tilde{R}}=D_m\frac{{\partial }^2\tilde{C}}{\partial {\tilde{R}}^2}-k_c(\tilde{C}-{\tilde{C}}_{\infty }).$$

(4)

In Equations (1)–(4), ${\beta }_C$ denotes the solutal expansion coefficient, ${\beta }_T$ is the thermal expansion coefficient, $\tilde{C}$ is concentration, $D_m$ mass diffusivity, $B_0$ is uniform magnetic field, and $k_c$ is the rate of chemical reaction. The radioactive heat flux is defined as:

$$\overline{q}=-\frac{\partial \tilde{T}}{\partial \tilde{R}} \frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}} .$$

(5)

The relation for variable thermal conductivity is written as:

$$\kappa \left(\tilde{T}\right)={\kappa }_{\infty }\left(1+\epsilon \frac{\tilde{T}-{\tilde{T}}_{\infty }}{\Delta T}\right) ,$$

(6)

and

$$\Delta T={\tilde{T}}_w-{\tilde{T}}_{\infty }.$$

(7)

The boundary conditions are elaborated as:

$$\tilde{U}(\tilde{X},\tilde{R})={\tilde{U}}_w=a\tilde{X}, \tilde{V}(\tilde{X},\tilde{R})=0, \tilde{C}={\tilde{C}}_w, \tilde{T}={\tilde{T}}_w, \mathrm{at} \tilde{R}=R_1 ,$$

(8)

$$\tilde{U}={\tilde{U}}_e=d\tilde{X}, \tilde{C}\to {\tilde{C}}_{\infty }, \tilde{T}\to {\tilde{T}}_{\infty }, \mathrm{as} \tilde{R}\to \infty .$$

(9)

For order reduction we used

$$\begin{gathered} \tilde{U}=\tilde{X}\frac{U_0}{L}{F^{\prime }}_C(\eta ), \tilde{V}=-\frac{R_1}{\tilde{R}}\sqrt{\frac{\nu U_0}{L}}{F}_C(\eta ), \\ {\theta }_C(\eta )=\frac{\tilde{T}-{\tilde{T}}_{\infty }}{{\tilde{T}}_w-{\tilde{T}}_{\infty }}, {\varphi }_C(\eta )=\frac{\tilde{C}-{\tilde{C}}_{\infty }}{{\tilde{C}}_w-{\tilde{C}}_{\infty }}, \eta =\frac{{\tilde{r}}^2-{\tilde{R}}_1{}^2}{2{\tilde{R}}_1}\sqrt{\frac{U_0}{\nu L}} . \end{gathered}$$

(10)

Incorporating Equation (10) in Equations (2)–(4) results in

$$\begin{array}{l}\left(1+1/\beta \right)\left({F_C}^{‴}(1+2\gamma \eta )+2\gamma {F_C}^{″}\right)-{F_C}^{\prime 2}+F_C{F_C}^{″}+G_T{\theta }_C\cos \left(\alpha \right)+G_C{\varphi }_C\cos \left(\alpha \right)\\ \\ -M^2({F_C}^{\prime }-A)+A^2=0 ,\end{array}$$

(11)

$$\begin{array}{l}\left(1+\frac{4}{3}R\right)\left({{\theta }_C}^{″}(1+2\eta \gamma )+2\gamma {{\theta }_C}^{\prime }\right)+\epsilon \left((+{{\theta }_C}^{\prime 2}+{\theta }_C{{\theta }_C}^{″})(1+2\eta \gamma )+2\gamma {\theta }_C{{\theta }_C}^{\prime }\right)\\ \\ +\mathrm{Pr}E\left(1+2\eta \gamma \right)\left(1+\frac{1}{\beta }\right){F_C}^{″2}+\mathrm{Pr}H{\theta }_C+\mathrm{Pr}{F}_C{{\theta }_C}^{\prime }=0 ,\end{array}$$

(12)

$${{\varphi }_C}^{″}(1+2\eta \gamma )+2\gamma {{\varphi }_C}^{\prime }+Scf{{\varphi }_C}^{\prime }-ScRc{\varphi }_C=0 ,$$

(13)

and boundary conditions become:

$$F_C=0, {F_C}^{\prime }=1,{\theta }_C=1,{\varphi }_C=1, \mathrm{at} \eta =0$$

(14)

$${F_C}^{\prime }=A, {\theta }_C=0, {\varphi }_C=0, \mathrm{as} \eta \to \infty .$$

(15)

The flow parameters are:

$$\begin{gathered} \beta =\frac{\overline{\mu }\sqrt{2{\pi }_c}}{{\tau }_r}, R=\frac{4{\sigma }^{*}{\tilde{T}}_{\infty }^3}{\kappa k^{*}}, \gamma =\sqrt{\frac{\nu L}{c^2{U}_0}}, A=\frac{d}{a}, \\ {G}_T=\frac{g_0{\beta }_T({\tilde{T}}_w-{\tilde{T}}_{\infty })L^2}{U_0\tilde{x}}, G_C=\frac{g_0{\beta }_C({\tilde{C}}_w-{\tilde{C}}_{\infty })L^2}{U_0\tilde{x}}, M=\sqrt{\frac{\sigma B_0^{2}L}{\rho U_0}}, Pr=\frac{\overline{\mu }{c}_p}{\kappa }, \\ E=\frac{U_0^2{(\tilde{x}/L)}^2}{c_p({\tilde{T}}_w-{\tilde{T}}_{\infty })}, Sc=\frac{\nu }{D_m}, Rc=\frac{k_{c}L}{U_0},H=\frac{L{Q}_0}{U_0\rho c_p} . \end{gathered}$$

(16)

Here, $\gamma$, A, $\beta$, $G_T, G_C,$ Sc, Pr, E, H, Rc, M and R are the curvature, velocities ratio, Casson fluid parameters, temperature Grashof, concentration Grashof, Schmidt, Prandtl, Eckert number, heat generation, chemical reaction, magnetic, and radiation parameters, respectively. Having the thermophysical aspects of the flow field, we considered the Nusselt number at the surface. The Nusselt number is expressed as:

$$\begin{array}{l}N{u}_{\tilde{x}}=\frac{\tilde{x}{q}_w}{\kappa ({\tilde{T}}_w-{\tilde{T}}_{\infty })},\\ q_w=-\kappa \left(1+\frac{16{\sigma }^{*}{\tilde{T}}_{\infty }^3}{3k^{*}\kappa }\right){\left(\frac{\partial \tilde{T}}{\partial \tilde{r}}\right)}_{\tilde{r}=c},\\ \frac{N{u}_{\tilde{x}}}{\sqrt{{\mathrm{Re}}_{\tilde{x}}}}=-\left(1+\frac{4}{3}R\right){{\theta }_C}^{\prime }(0)\end{array}\} .$$

(17)

We have considered four unlike flow fields. The solutions for the existence of heat generation and thermal radiation effects are obtained by using Equations (11)–(15). For the absence of thermal radiations effect, Equation (12) reduces to

$$\begin{array}{l}{{\theta }_C}^{″}(1+2\eta \gamma )+2\gamma {{\theta }_C}^{\prime }+\epsilon \left(({{\theta }_C}^{\prime 2}+{\theta }_C{{\theta }_C}^{″})(1+2\eta \gamma )+2\gamma {\theta }_C{{\theta }_C}^{\prime }\right)\\ \\ +\mathrm{Pr}E\left(1+2\eta \gamma \right)\left(1+\frac{1}{\beta }\right){F_C}^{″2}+\mathrm{Pr}H{\theta }_C+\mathrm{Pr}{F}_C{{\theta }_C}^{\prime }=0 ,\end{array}$$

(18)

while the relation of Nusselt number is modified as:

$$\begin{array}{l}N{u}_{\tilde{x}}=\frac{\tilde{x}{q}_w}{\kappa ({\tilde{T}}_w-{\tilde{T}}_{\infty })},\\ q_w=-\kappa {\left(\frac{\partial \tilde{T}}{\partial \tilde{r}}\right)}_{\tilde{r}=c},\\ \frac{N{u}_{\tilde{x}}}{\sqrt{{\mathrm{Re}}_{\tilde{x}}}}=-{\theta }^{\prime }(0)\end{array}\} .$$

(19)

Further, for the heat transfer aspects in Casson fluid flow towards the stretching cylinder in the absence of heat generation effect, we incorporate H = 0 into Equation (12). The ultimate energy equation becomes:

$$\begin{array}{l}\left(1+\frac{4}{3}R\right)\left({{\theta }_C}^{″}(1+2\eta \gamma )+2\gamma {{\theta }_C}^{\prime }\right)+\epsilon \left(({\theta }_C{{\theta }_C}^{″}+{{\theta }_C}^{\prime 2})(1+2\eta \gamma )+2\gamma {\theta }_C{{\theta }_C}^{\prime }\right)\\ \\ +\mathrm{Pr}E\left(1+2\eta \gamma \right)\left(1+\frac{1}{\beta }\right){F_C}^{″2}+\mathrm{Pr}{F}_C{{\theta }_C}^{\prime }=0 .\end{array}$$

(20)

We noted that, for the case of the absence of the heat generation effect, Equation (20) is jointly solved with Equations (11) and (13). During this examination, the boundary conditions Equations (14) and (15) remain the same.

3. Numerical Method

Due to its coupled and nonlinear nature, the system of equations Equations (11)–(15), which describes the flow of magnetized chemically reactive Casson fluid towards a cylindrical surface, along with other physical phenomena, cannot be solved analytically. As a result, a shooting method utilizing the RK-4 is employed. For numerical simulation, we have to obtain the initial value problem. Therefore, let us suppose

$$\begin{array}{l}{Y}_1=F_C(\eta ), Y_2={F_C}^{\prime }(\eta ), Y_3={F_C}^{″}(\eta ), Y_4={\theta }_C(\eta ), \\ \\ Y_5={{\theta }_C}^{\prime }(\eta ),Y_6={\varphi }_C(\eta ), Y_7={{\varphi }_C}^{\prime }(\eta ),\end{array}$$

(21)

Incorporating Equation (21) into Equations (11)–(13) results in

$${Y_{{}_1}}^{\prime }=Y_2,$$

(22)

$${Y_2}^{\prime }=Y_3,$$

(23)

$${Y_3}^{\prime }=\frac{1}{\left(1+\frac{1}{\beta }\right)\left(1+2\eta \gamma \right)}\left[\begin{array}{l}-2\gamma Y_3\left(1+\frac{1}{\beta }\right)+Y_2^2-Y_1{Y}_3-G_T{Y}_4\cos \alpha -G_C{Y}_6\cos \alpha \\ -M^2(Y_2-A)-A^2\end{array}\right],$$

(24)

$${Y_4}^{\prime }=Y_5,$$

(25)

$${Y_5}^{\prime }=-\frac{1}{\left(1+\frac{4}{3}R\right)\left(1+2\eta \gamma \right)+\epsilon (1+2\eta \gamma )Y_4}\left[\begin{array}{l}(1+\frac{4}{3}R)(2\gamma Y_5)+\epsilon ((1+2\eta \gamma )Y_5^2+2\gamma Y_4{Y}_5)+\\ \mathrm{Pr}{Y}_1{Y}_5+\mathrm{Pr}E(1+2\eta \gamma )(1+\frac{1}{\beta })Y_3^2+\mathrm{Pr}H{Y}_4\end{array}\right],$$

(26)

$${Y_6}^{\prime }=Y_7,$$

(27)

$${Y_7}^{\prime }=\frac{ScRc{Y}_6-Sc{Y}_1{Y}_6-2\gamma Y_7}{(1+2\eta \gamma )}.$$

(28)

Therefore, the reduced initial conditions are

$$\begin{array}{l}{Y}_1(0)=0,\\ Y_2(0)=1,\\ Y_3(0)={\eta }_1(\mathrm{unknown}),\\ Y_4(0)=1,\\ Y_5(0)={\eta }_2(\mathrm{unknown}),\\ Y_6(0)=1,\\ Y_7(0)={\eta }_{{}_3}(\mathrm{unknown}).\end{array}$$

(29)

It is important to note that unknowns ${\eta }_1,{\eta }_2 \mathrm{and}{\eta }_3$ are initial guesses and they were improved by using the Newton method. The system of Equations (22)–(29) is solved by the RK scheme until the solutions reach zero with the efficiency 10⁻⁵. The simulations are carried until the following boundary conditions hold:

$$\begin{array}{l}{Y}_2(\infty )\to A, \\ Y_4(\infty )\to 0 , \\ Y_6(\infty )\to 0.\end{array}$$

(30)

The results for the magnetized chemically reactive Casson fluid flow by means of stretched inclined surface are simulated using MATLAB 9.6 R2019a software. The range of parameters, namely $R,\beta ,E \mathrm{and} H$, is selected subject to the convergence of the numerical scheme.

4. Numerical Results and Discussion

We have considered Casson fluid flow over a stretching cylinder along with various pertinent physical effects. Four different flow fields are considered. In the first case (Model 1), the heat transfer features are considered in the absence of thermal radiations. In the second case (Model 2), we considered heat transfer aspects along with thermal radiations. In the third case (Model 3), the flow regime is considered in the nonexistence of the heat generation effect, while in the fourth case (Model 4) the flow regime is considered along with the heat generation effect. The Nusselt number is calculated at an inclined cylindrical surface by using the right-hand side of the last term in Equation (17).

Table 1. Impact of variable thermal conductivity on Nusselt number with and without heat generation.

$\mathit{\epsilon }$	$-(1+\frac{4}{3}\mathit{R}){\mathit{\theta }}^{\prime }(0)$
	H = 0	H = 0, ANN Values	H = 0.2	H = 0.2, ANN Values
0.2	7.3889	7.550384	7.3982	7.450857
0.3	6.7858	6.773642	6.7955	6.850917
0.4	6.2197	6.08312	6.2298	6.281561
0.5	5.6793	5.505007	5.6900	5.724124
0.6	5.1535	5.063317	5.1653	5.168366
0.7	4.6317	4.727323	4.6445	4.612874
0.8	4.1020	4.11615	4.1161	4.06405
0.9	3.5498	3.610177	3.5658	3.534021
1.0	2.9554	2.981183	2.9740	3.037938
2.0	2.8925	2.820152	2.8483	2.850564

,

Table 2. Impact of Eckert number on Nusselt number with and without heat generation.

E	$-(1+\frac{4}{3}\mathit{R}){\mathit{\theta }}^{\prime }(0)$
	H = 0	H = 0, ANN Values	H = 0.2	H = 0.2, ANN Values
0.2	8.8527	8.600711	8.8610	8.678086
0.3	9.0238	8.807419	9.0316	9.042452
0.4	9.1951	9.016825	9.2025	9.282963
0.5	9.3664	9.227915	9.3734	9.456052
0.6	9.5379	9.439715	9.5445	9.595065
0.7	9.7096	9.651195	9.7157	9.718458
0.8	9.8812	9.861192	9.8871	9.836141
0.9	10.053	10.06833	10.058	9.953408
1.0	10.225	10.27096	10.230	10.07319
2.0	11.953	11.69185	11.954	11.90829

,

Table 3. Impact of Prandtl number on Nusselt number with and without heat generation.

Pr	$-(1+\frac{4}{3}\mathit{R}){\mathit{\theta }}^{\prime }(0)$
	H = 0	H = 0, ANN Values	H = 0.2	H = 0.2, ANN Values
0.2	8.6112	8.469963	8.6170	8.340907
0.3	8.6819	8.550283	8.6905	8.539814
0.4	8.7519	8.633519	8.7633	8.70777
0.5	8.8214	8.717695	8.8352	8.84951
0.6	8.8901	8.801995	8.9063	8.969425
0.7	8.9581	8.885986	8.9769	9.071427
0.8	9.0256	8.969338	9.0468	9.158906
0.9	9.0926	9.051724	9.1159	9.234733
1.0	9.1589	9.132774	9.1845	9.301304
2.0	9.7955	9.789138	9.8396	9.773815

,

Table 4. Impact of thermal radiation parameter on Nusselt number with and without heat generation.

R	$-(1+\frac{4}{3}\mathit{R}){\mathit{\theta }}^{\prime }(0)$
	H = 0	H = 0, ANN Values	H = 0.2	H = 0.2, ANN Values
0.2	7.8596	8.045086	7.8676	8.016374
0.3	8.6819	8.787988	8.6905	8.968767
0.4	9.5032	9.266173	9.5125	9.546804
0.5	10.324	10.1253	10.336	10.26306
0.6	11.914	11.57866	11.155	11.12373
0.7	11.963	12.10208	11.974	12.03928
0.8	12.782	12.88408	12.794	12.8184
0.9	13.598	13.56171	13.612	13.62378
1.0	14.415	14.13501	14.427	14.73928
2.0	22.538	22.48489	22.562	22.54377

,

Table 5. Impact of curvature fluid parameter on Nusselt number with and without heat generation.

$\mathit{\gamma }$	$-(1+\frac{4}{3}\mathit{R}){\mathit{\theta }}^{\prime }(0)$
	H = 0	H = 0, ANN Values	H = 0.2	H = 0.2, ANN Values
0.2	1.1786	1.210011	8.6125	1.257366
0.3	1.5576	1.508415	8.5866	1.657654
0.4	1.9527	1.89567	8.5736	1.926305
0.5	2.3574	2.287011	8.5659	2.303666
0.6	2.7681	2.825819	8.5607	2.743256
0.7	3.1829	3.259274	8.5569	3.139169
0.8	3.6004	3.710605	8.5543	3.54816
0.9	4.0198	4.121886	8.5521	4.08963
1.0	4.4406	4.314493	8.5504	4.541768
2.0	8.6819	8.676551	8.5426	8.672115

,

Table 6. Impact of Casson fluid parameter on Nusselt number with and without heat generation.

$\mathit{\beta }$	$-(1+\frac{4}{3}\mathit{R}){\mathit{\theta }}^{\prime }(0)$
	H = 0	H = 0, ANN Values	H = 0.2	H = 0.2, ANN Values
0.2	8.6037	8.360935	8.6125	8.326413
0.3	8.5776	8.327485	8.5866	8.483554
0.4	8.5647	8.298506	8.5736	8.59004
0.5	8.5569	8.278881	8.5659	8.657284
0.6	8.5517	8.283509	8.5607	8.694983
0.7	8.5482	8.344537	8.5569	8.71096
0.8	8.5453	8.484703	8.5543	8.711303
0.9	8.5432	8.639533	8.5521	8.70063
1.0	8.5414	8.722334	8.5504	8.682384
2.0	8.5337	8.465836	8.5426	8.406681

,

Table 7. Impact of variable thermal conductivity on Nusselt number for non-radiative and radiative flow fields.

$\mathit{\epsilon }$	$-(1+\frac{4}{3}\mathit{R}){\mathit{\theta }}^{\prime }(0)$
	R = 0	R = 0, ANN Values	R = 0.2	R = 0.2, ANN Values
0.2	5.5160	5.419962	3.3695	3.442163
0.3	5.1033	5.067823	3.1065	3.141072
0.4	4.7308	4.784497	2.8619	2.896393
0.5	4.3901	4.471492	2.6307	2.585507
0.6	4.0736	4.141892	2.4083	2.363242
0.7	3.7748	3.821497	2.1901	2.167554
0.8	3.4875	3.426805	1.9716	1.894237
0.9	3.2056	3.165933	1.7475	1.727995
1.0	1.3924	1.369884	1.5113	1.56238
2.0	1.0839	1.090832	0.7082	0.688535

,

Table 8. Impact of Eckert number on Nusselt number for non-radiative and radiative flow fields.

E	$-(1+\frac{4}{3}\mathit{R}){\mathit{\theta }}^{\prime }(0)$
	R = 0	R = 0, ANN Values	R = 0.2	R = 0.2, ANN Values
0.2	6.7414	6.622941	8.5229	8.295121
0.3	7.0344	6.988948	8.8985	9.070103
0.4	7.3272	7.372972	9.2741	9.34344
0.5	7.6199	7.672083	9.6494	9.580389
0.6	7.9126	7.928536	10.024	9.93963
0.7	8.2051	8.181332	10.399	10.3908
0.8	8.4976	8.457681	10.775	10.83179
0.9	8.7900	8.767011	11.151	11.18968
1.0	9.0822	9.099016	11.525	11.45648
2.0	12.000	11.97278	15.271	15.26398

,

Table 9. Impact of Prandtl number on Nusselt number for non-radiative and radiative flow fields.

Pr	$-(1+\frac{4}{3}\mathit{R}){\mathit{\theta }}^{\prime }(0)$
	R = 0	R = 0, ANN Values	R = 0.2	R = 0.2, ANN Values
0.2	6.1663	6.106121	7.8043	7.523102
0.3	6.2241	6.279857	7.8744	7.927426
0.4	6.2812	6.354533	7.9436	7.95101
0.5	6.3376	6.364053	7.8982	8.016983
0.6	6.3933	6.355947	8.0800	8.016515
0.7	6.4484	6.371431	8.1473	8.067419
0.8	6.5029	6.432032	8.2139	8.206568
0.9	6.5568	6.537401	8.2799	8.32734
1.0	6.6102	6.671025	8.3453	8.418052
2.0	7.1180	7.133448	8.9711	8.969116

,

Table 10. Effect of heat generation parameter on Nusselt number for radiative and non-radiative flow fields.

H	$-(1+\frac{4}{3}\mathit{R}){\mathit{\theta }}^{\prime }(0)$
	R = 0	R = 0, ANN Values	R = 0.2	R = 0.2, ANN Values
0.2	6.4551	6.350771	8.1557	8.143513
0.3	6.4617	6.448727	8.1642	8.304773
0.4	6.4684	6.537417	8.1727	8.306825
0.5	6.4750	6.534251	8.1811	8.173728
0.6	6.4817	6.493596	8.1895	8.010682
0.7	6.4883	6.458979	8.1981	8.020188
0.8	6.4950	6.452199	8.2064	8.113884
0.9	6.5016	6.473475	8.2149	8.224964
1.0	6.5082	6.509865	8.2233	8.322432
2.0	6.5741	6.526886	8.3072	8.290738

,

Table 11. Impact of curvature parameter on Nusselt number for non-radiative and radiative flow fields.

$\mathit{\gamma }$	$-(1+\frac{4}{3}\mathit{R}){\mathit{\theta }}^{\prime }(0)$
	R = 0	R = 0, ANN Values	R = 0.2	R = 0.2, ANN Values
0.2	1.0819	1.094894	1.2766	1.31689
0.3	1.3301	1.308242	1.5966	1.549922
0.4	1.5890	1.606951	1.9305	1.875747
0.5	1.8563	1.885001	2.2747	2.356478
0.6	2.1179	2.149441	2.6266	2.726628
0.7	2.3970	2.436858	2.9842	3.0233
0.8	2.6798	2.652202	3.3463	3.329046
0.9	2.9657	2.915744	3.7118	3.680747
1.0	3.2541	3.197967	4.0801	4.090512
2.0	6.2061	6.133482	7.8373	7.837882

and

Table 12. Impact of Casson fluid parameter on Nusselt number for radiative and non-radiative flow fields.

$\mathit{\beta }$	$-(1+\frac{4}{3}\mathit{R}){\mathit{\theta }}^{\prime }(0)$
	R = 0	R = 0, ANN Values	R = 0.2	R = 0.2, ANN Values
0.2	3.4247	3.490627	4.1927	4.248408
0.3	3.3868	3.328621	4.1547	4.058267
0.4	3.3672	3.306248	4.1336	3.976712
0.5	3.3551	3.321303	4.1212	4.025897
0.6	3.3467	3.288144	4.1125	4.085468
0.7	3.3406	3.372717	4.1064	4.130031
0.8	3.3359	3.383158	4.1017	4.159121
0.9	3.2683	3.33232	4.0979	4.176746
1.0	3.2658	3.242111	4.0948	4.18649
2.0	3.2541	3.267779	4.0802	4.031526

are recorded in this regard. For the case of calculations in Table 1, Table 2, Table 3, Table 5 and Table 6, we fixed the radiation parameter R = 0.2 while, for the case of calculations in Table 7, Table 8, Table 9, Table 11 and Table 12, we fixed the heat generation parameter H = 0.2. In detail, for two different values of H, namely H = 0 and H = 0.2, towards a positive fluctuation in the Casson fluid parameter, Nusselt number evaluation at the cylindrical surface is observed. Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6 are offered in this manner. The impact of varied thermal conductivity on the Nusselt number is seen for both thermal fields, namely thermal flow fields with and without the effect of heat generation. In this direction, Table 1 is given. The Nusselt number decreases with highly variable thermal conductivity, as can be seen. The effect of the Eckert number on Nusselt is investigated for two unique scenarios, a thermal flow field with heat generation and a thermal flow field without heat generation; see Table 2. Our observations show that the Nusselt number grows as the Eckert number increases in value. Table 3 provides the observation of the Nusselt number toward a positive variation in the Prandtl number. The Nusselt number rises both in the presence and absence of the heat generating effect. Table 4 displays the observation on the Nusselt number in favor of a rise in the thermal radiation parameter. With increased heat radiation parameter, the Nusselt number increases. Both the presence and absence of the heat-generating effect result in values being observed. Furthermore, it is demonstrated that the magnitude of the Nusselt numbers increases slightly in the presence of the heat-generating effect. Table 5 displays the effect of the curvature fluid parameter on the Nusselt number for two various values, H = 0 and H = 0.2. We have demonstrated that the Nusselt number grows as the curvature parameters rise. Additionally, the heat-generating action has a slightly stronger Nusselt number. Table 6 illustrates the impact of the Casson fluid parameter on the Nusselt number in more detail. In this case, the presence of the heat generation effect is shown by the value of H = 0.2, while its absence is indicated by H = 0, respectively. It can be demonstrated that, in both cases, the Nusselt number and the Casson fluid parameter display an inverse relationship.

Table 7, Table 8, Table 9, Table 10, Table 11 and Table 12 show the fluctuation in the Nusselt number for two different values of the thermal radiation parameter, R = 0 and R = 0.2, with respect to various flow factors. Table 7 displays the temperature-dependent variable thermal conductivity’s favorable variation influence on the Nusselt number. Both when heat radiation is present and when it is not, these observations are made. It has been demonstrated that the Nusselt number falls off at an increasing thermal conductivity parameter. When the thermal radiation parameter is zero, the magnitude of the Nusselt number is higher. The impact of the Eckert number on the Nusselt number is evident for both the existence and nonexistence situations of thermal radiations; see Table 8. According to our observations, the Nusselt number increases as the Eckert number increases. The effect of the Prandtl number on the Nusselt number is shown for both the presence and absence of thermal radiations; see Table 9. For higher Pr, we have seen an increase in the Nusselt number. We have also observed that, for thermal radiations, the Nusselt number is greater in magnitude. When considering both non-radiative and radiative fluid flow, the Nusselt number is seen to trend toward a higher heat generation parameter. Table 10 provides this case. As can be seen, for a higher heat generation parameter, the Nusselt number rises substantially. A higher Nusselt number is noticed for thermal radiations cases. The Nusselt number significantly increases for positive modifications in the curvature parameter, as seen in Table 11. The magnitude of the Nusselt number is larger when a thermal radiation parameter is present. Table 12 displays how the Casson fluid parameter affects the Nusselt number. In this case, the presence of the heat generation effect is shown by the value of H = 0.2, while its absence is indicated by H = 0, respectively. It can be demonstrated that, in both cases, the Nusselt number and the Casson fluid parameter display an inverse relationship.

5. Artificial Neural Networking Model

In order to estimate the values of the Nusselt Number (NN) at cylindrical surface manifested with Casson fluid flow towards various flow parameters, we have constructed four different Artificial Neural Networking (ANN) models against thermal flow regimes without and with thermal radiations, with heat generation effect, and without heat generation effect. Mathematically, with thermal radiations the flow regime is evaluated by selecting R = 0.2, and without thermal radiations the flow regime is obtained by selecting R = 0. The flow regime with a heat generation effect is evaluated by selecting H = 0.2, and the thermal flow regime without heat generation is marked as H = 0. In the developed ANN models, the multilayer perceptron (MLP) structure, which is widely used due to its strong structural features, is preferred [22]. MLP networks are structurally composed of layers, and all of the layers are interconnected. Since the input factors affecting each estimated NN value are different, four different ANN models were developed and different input and output values were obtained for each. The MLP architecture showing the layered structure of the developed ANN models is shown in Figure 1.

The input and output parameters for each developed model are given in

Table 13. The input and output parameters for each developed model.

Model	Inputs						Output
Model 1 (R = 0)	β	γ	H	Pr	E	ε	NN
Model 2 (R = 0.2)	β	γ	H	Pr	E	ε	NN
Model 3 (H = 0)	β	γ	R	Pr	E	ε	NN
Model 4 (H = 0.2)	β	γ	R	Pr	E	ε	NN

. In the construction of ANN models, it is important to ideally optimize the data set being used [23]. A total of 70% of the data set used for the four different models developed is reserved for training the model, and 15% for validation and testing each. The lack of a rule for calculating the number of neurons in the hidden layer is one of the challenges in designing ANN models [24]. Because of this, the performance of ANN models with various numbers of hidden layer neurons has been examined, and the number of neurons in the hidden layer has been optimally tuned. The structural topologies of four different ANN models developed are shown in Figure 2a–d. Particularly, the structural topology of ANN model without heat generation effect is shown in Figure 2a while the structural topology of the ANN model with the heat generation effect is given in Figure 2b. Further, Figure 2c offers the structural topology of the ANN model without the radiation effect and Figure 2d offers the structural topology of the ANN model with the radiation effect. Information about the data set used in each ANN model and the number of neurons in the hidden layer of each model is given in

Table 14. Information about the data set used in each ANN model and the number of neurons.

Model	Number of Neuron	Training	Validation	Test	Total
Model 1 (R = 0)	16	42	9	9	60
Model 2 (R = 0.2)	15	42	9	9	60
Model 3 (H = 0)	10	42	9	9	60
Model 4 (H = 0.2)	9	42	9	9	60

.

MLP network models use the Levenberg–Marquardt training method, one of the most used training algorithms because of its high learning performance [25]. The hidden and output layers both contain Tan-Sig and Purelin transfer functions as transfer functions. The transfer functions’ mathematical expressions are provided below [26]:

$$f(x)=\frac{1}{1+\exp (-x)},$$

(31)

$$\mathrm{purelin}(x)=x.$$

(32)

In order to evaluate the performance of four different ANN models, mean squared error (MSE), coefficient of determination (R_d), and margin of deviation (MoD) parameters, which are widely used in the literature, were selected [27]. The mathematical expressions used in the calculation are given below [28,29,30]:

$$\mathrm{MSE}=\frac{1}{N}{\displaystyle \sum _{i=1}^N(X_{\mathrm{targ}\left(\mathrm{i}\right)}}-X_{\mathrm{pred}\left(\mathrm{i}\right)}{)}^2,$$

(33)

$${\mathrm{R}}_{\mathrm{d}}=\sqrt{1-\frac{{\displaystyle \sum _{i=1}^N(X_{\mathrm{targ}\left(\mathrm{i}\right)}}-X_{\mathrm{pred}\left(\mathrm{i}\right)}{)}^2}{{\displaystyle \sum _{i=1}^N{(X_{\mathrm{targ}\left(\mathrm{i}\right)})}^2},}},$$

(34)

$$\mathrm{MoD}(\%)=\left[\frac{X_{\mathrm{targ}}-X_{\mathrm{pred}}}{X_{\mathrm{targ}}}\right]\times 100.$$

(35)

The training accuracy of the ANN models developed to predict the NN values in four different situations has been extensively investigated. In Figure 3a–d, training performance graphs for each ANN model are presented. In detail, Figure 3a shows the training performance graph of the ANN model without the heat generation effect, and Figure 3b shows the training performance graph of the ANN model with the heat generation effect. The training performance graph of the ANN model without the radiation effect is given in Figure 3c,d, which offer the training performance graphs of the ANN model with the radiation effect. The graphs show the training cycle (epoch) that takes place in an MLP network.

We have seen that the MSE values, which were high at the beginning of the training phase of each ANN model, decreased with the advancing stages. It is seen that the training phases of the ANN model came to an end with the MSE values recorded for each data set meeting with the most ideal point. Figure 4a–d show the error histograms for the each ANN model. To be more specific, Figure 4a offers an error histogram of the ANN model without heat generation and Figure 4b offers an error histogram of the ANN model with heat generation. Figure 4c shows an error histogram of the ANN model without a radiation effect, and Figure 4d depicts the error histogram of the ANN model with a radiation effect. It should be noted that in error histograms, the calculated error rates for data sets are located very close to the zero error line. It is also seen that the error values in the error histograms are very low. The findings obtained from the performance and error histograms show that the training stages of the ANN models developed for estimating NN values are ideally completed. Actual values of NN parameters under different conditions and values obtained from ANN models are shown in Figure 5a–d. Figure 5a offers the predicted and target values according to the data number without the heat generation effect, while Figure 5b offers the predicted and target values according to the data number with the heat generation effect. Figure 5c shows the predicted and target values according to data number without radiation effect, while Figure 5d gives outcomes for the case of the presence of radiation effect. When the results for each data point are evaluated, it is clearly seen from the graphs that the results obtained from the ANN models are in very good agreement with the target values. This ideal fit of the target and ANN outputs shows that the developed ANN models can predict NN values with high accuracy. The deviation ratio between the NN values obtained from four different ANN models with the target data was calculated and shown in Figure 6a–d. Figure 6a offers the estimated MoD values for each data point without the heat generation effect.

Figure 6b offers the calculated MoD values for each data point with the heat generation effect. The MoD values without radiation effect are offered in Figure 6c,d, which also offer the estimated MoD values for each data point with radiation effect. From MoD values, we have seen that the data are generally concentrated around the zero-error line. The nearness of the points expressing the MoD data to the zero-error line shows that the deviation rates of the ANN outputs are low. In order to make the error analysis of the ANN models in more detail, the differences between the outputs obtained from the ANN model and the target data were calculated for each data point and are shown in Figure 7a–d. In detail, the differences between ANN outputs and target values without the heat generation effect are offered in Figure 7a, and 7b depicts the differences between ANN outputs and target values with the heat generation effect. Figure 7c,d gives the differences between target values and ANN outputs for both the absence and presence of thermal radiations. When the different values given for four different ANN models are taken into consideration, the difference values are very low. The results obtained from the analysis of MoD and difference values show that all four ANN models can predict NN values with very low error values. In Figure 8a–d, target and prediction values are provided for four different cases, namely without heat generation effect, with heat generation effect, without thermal radiations, and with thermal radiations effect. The target values are on the x-axis and the ANN outputs are on the y-axis. When the data given for four different ANN models are examined, it is seen that the data points are located very close to the zero error line. It is also seen from the figures that the data points are in the +10% error band range. The performance parameters calculated for four different ANN models developed to estimate the NN values under different conditions are given in

Table 15. The performance parameters calculated for four different ANN models developed.

Model	MSE	Rd	MoDmin (%)	MoDmax (%)
Model 1 (R = 0)	3.23 × 10−2	0.99722	−0.03	−1.96
Model 2 (R = 0.2)	1.41 × 10−1	0.98935	−0.007	3.92
Model 3 (H = 0)	7.54 × 10−2	0.99147	0.06	3.25
Model 4 (H = 0.2)	1.58 × 10−2	0.99941	−0.03	3.52

. We have seen that the MoD values calculated for the ANN models are quite low.

The low MoD values indicate that the deviation rates of the outputs obtained from the ANN model are very low. The closeness of the R_d values to 1 and the low MSE values confirm that each ANN model can forecast with high accuracy. The obtained results show that each ANN model can calculate the output parameters NN with high accuracy.

In the absence of concentration, heat generation/absorption, and externally applied magnetic field our problem reduced to Hayat et al. [31]. Two different surface quantities, namely Nusselt number and skin friction coefficient, are considered for comparison.

Table 16. Comparison of skin friction coefficient with Ref. [31].

$\mathit{\beta }$	$\mathit{\gamma }$	Ref. [31]	Present Values
1.0	0.1	1.2347	1.2135
1.5	0.1	1.1082	1.1030
2.1	0.1	1.0310	1.0150
2.0	0.0	0.9966	0.9643
2.0	0.1	1.0409	1.0214
2.0	0.2	1.0850	1.0413
2.0	0.1	1.2165	1.2032
2.0	0.1	1.0976	1.0743
2.0	0.1	0.9311	0.9101

and

Table 17. Comparison of Nusselt number with Ref. [31].

$\mathit{\epsilon }$	$\mathit{\beta }$	$\mathit{\gamma }$	Ref. [31]	Present Values
0.0	1.0	0.2	0.5276	0.5054
0.0	1.4	0.2	0.5316	0.5203
0.0	1.8	0.2	0.5336	0.5124
0.0	2.0	0.0	0.5442	0.5220
0.0	2.0	0.12	0.5336	0.5213
0.0	2.0	0.19	0.5279	0.5016
0.0	2.0	0.19	0.5739	0.5216
0.2	2.0	0.19	0.5308	0.5124
0.3	2.0	0.19	0.5123	0.5061

are evident in this direction. One can see that we have an excellent match with the existing literature, which yields the surety of the present work.

6. Conclusions

The thermally magnetized stagnation point Casson [32,33] fluid flow is considered towards a cylindrical inclined surface. The energy equation is formulated in the presence of mixed convection, heat generation, viscous dissipation, thermal radiations, variable thermal conductivity, and first-order chemical reaction. The flow equations are solved by using the shooting method [34,35,36]. The Nusselt number is evaluated at the surface for four different cases and, for better estimation, the corresponding artificial neural networking models are constructed. The key outcomes are as follows:

• The estimation of the coefficient of determination values Mean squared error values guarantees that each ANN model predicts the accuracy in the estimation of the Nusselt number.
• For each artificial neural networking model of the Nusselt number, as an error, the data points lie within band rage.
• The Casson fluid parameter and variable conductivity are found to have a diminishing relationship with the Nusselt number, but curvature, Eckert number, Prandtl number, and the thermal radiation parameter have an opposite relationship.
• When the heat generation impact is taken into consideration, the Nusselt number is slightly larger in relation to the Casson fluid parameter, Eckert number, Prandtl number, curvature, and thermal conductivity parameter.
• For the presence of thermal radiations, the Nusselt number strength is higher towards variations in Casson fluid parameter, Prandtl number, heat generation, curvature, and Eckert number.