Heat and Mass Transfer Analysis for Unsteady Three-Dimensional Flow of Hybrid Nanofluid Over a Stretching Surface Using Supervised Neural Networks

The application of hybrid nanomaterials for the improvement of thermal efficiency of base fluid has increasingly gained attention during the past few decades. The basic purpose of this study is to investigate the flow characteristics along with heat transfer in an unsteady three-dimensional flow of hybrid nanofluid over a stretchable and rotatory sheet (3D-UHSRS). The flow model in the form of PDEs was reduced to the set of ordinary differential equations utilizing the appropriate transformations of similarity. The influence of the rotation parameter, unsteadiness parameter, stretching parameter, radiation parameter, and Prandtl number on velocities and thermal profile was graphically examined. A reference solution in the form of dataset points for the 3D-UHSRS model are computed with the help of renowned Lobatto IIIA solver, and this solution is exported to MATLAB for the proper implementation of proposed solution methodology based on the Levenberg–Marquardt supervised neural networks. Graphical and numerical results based on the mean square error (MSEs), time series response, error distribution plots, and regression plots endorses the precision, validity, and consistency of the proposed solution methodology. The MSE up to the level of 10–12 confirms the accuracy of the achieved results.

Introduction

In the recent past, improvement in heat transfer for various engineering and industrial applications remains the main topic of research for scientists. Improvement in the heat transfer ability can be done through different methods such as use of microchannel, extended surface and through vibrating the surface. Thermal conductivity plays the most important role among all other characteristics of fluid for judging its heat transfer ability. Thermal conductivity of any nanofluid depends on the type, as well as the shape of nanoparticles. There will be less energy loss with the improved thermal efficiency of the fluid, which can further reduce the cost and increase the production in industrial applications. Various conventional fluids, such as water, kerosene oil, and ethylene glycol, are commonly used in many industrial and engineering applications (e.g., chemical manufacturing, water distillation, HVAC systems, medicine, and power-generation systems), but due to inadequate thermal transfer ability, these ordinary fluids lack good efficiency in systems having large thermal transfer requirements. As it is well known, the ability to conduct heat for metals is hundred times greater than that in liquids, which is why it is appropriate to use them in thermal systems [1–5]. Thus, the use of nanoparticles <100 nm not only improved the thermal capabilities of the fluid, but also positively improved the other rheological properties. Initially [6] was the first work to discuss the nature and characteristics of nanofluids. [7] compared the performance of heat transfer between the nanofluid comprised of a single wall and multiwall carbon nanotubes in a magnetohydrodynamic (MHD) flow under the influence of radiative flux.

A new type of nanofluids known as “hybrid nanofluids” is made by dispersing two different types of nanoparticles into base fluid to obtain various synergetic effect of both types of nanomaterial. The development of hybrid nanofluidic system is currently undergoing and under the evaluation to enhance the strength of thermal effects for different flow dynamics. Performance in terms of thermal energy utilization is being tested and assessed by various researchers. Due to its marvelous heat transfer abilities, this new type of nanofluid attracted the attention of researchers and scientists to investigate it in many of the industrial and engineering problems. Hybrid nanofluid have extensive range of application in scientific, industrial, engineering, and medical fields like medicine manufacturing, heating systems, transfer cooling, electronic chips, and solar panels [8]. Of late, several studies have been conducted to analyze the improvement of heat transfer capabilities for hybrid nanofluids as compared to conventional single-particle nanofluids [9–13]. The flow over rotating surfaces has many practical applications such as electrical appliances, cutting discs, data-storing devices, and heavy machinery parts. Song et al. [14] implemented a numerical shooting methodology to analyze the effects of various physical parameters on the velocity and thermal fields in the stagnation point flow of a hybrid nanofluid over a rotatory disc under radiative and activation energy effects. [15] numerically evaluated the three-dimensional MHD flow of hybrid nanofluid over a rotating and stretching sheet. In addition, the behavior of velocity and temperature profile depending on different physical factors has been highlighted through graphs and numerical results.

Any flow in which all of its parameters are independent of time is known as steady flow, whereas the time-dependent flow is known as unsteady flow. Due to excessive applicability in engineering and industrial applications, steady flow has gained a lot of concentration of researchers and scholars. However, recently many studies have been conducted on the behavior and characteristics of unsteady fluid flow systems, automobiles, power-generation systems, and aviation industry [16], explored the influence of variable thermal conductivity, viscosity, and Joule heating on the velocity and thermal field in the unsteady flow of 3D Maxwell nanofluid over a stretching surface. An enhancement in the velocity of the fluid has been observed with increased viscosity. [17] numerically investigated the biconvection flow of cross nanofluid and studied the effects of thermal radiation along with melting phenomenon over a cylinder. [18] studied the considerable effect of external magnetic field acting at an inclined angle on Williamson’s nanofluid over a rotating stretchable surface. [19] conducted a comparative analysis of heat transfer over a porous stretching surface between nanofluid containing nanoparticles of Graphene oxide and the combination of Ag–graphene oxide in kerosene oil as base fluid.

[20] described experimental research of various factors responsible for achieving better thermophysical properties and more stable results in terms of heat transfer in nanofluids and ionanofluids. There are a variety of nanoparticles available for manufacturing of nanofluids and hybrid nanofluid. Each type of material has its own benefits and limitations depending on the material’s characteristics. [21] presented the importance and significance of Aluminum nanoparticles in the industrial and engineering fields and their various advantages over other renowned nanomaterials. [22] explicated an analysis to show the physical and chemical stability and thermophysical properties based on thermal conductivity for the nanofluid comprising TiO₂ nanoparticles. Furthermore, different experimental results showing the improvement in thermal properties have also been tabulated in the research. [23] presented a numerical investigation to examine the flow behavior, as well as heat transmission capabilities of a hybrid fluid over a gyrating surface under the influence of a uniform external magnetic field. [24] numerically explained the behavior of mass and heat transfer for the MHD flow of a nanofluid over stretched surface under the heating effects through the radiation phenomenon. [25] used state-of-the-art supervised neural network to solve the hybrid nanofluid flow model with Joule heating and MHD effects. A comparison with an already available solution has also been made to show the accuracy and performance of solution methodology. Among several other key factors, the concentration of nanoparticles also plays an important part in deciding the thermal capability of nanofluids. [26] presented experimental research showing a rapid increase in thermal efficiency of an oil-based nanofluid. In a similar manner, [27] explained the fact that for a nanofluid with TiO₂ nanoparticles in water, the negative effect of using a lower concentration of nanoparticles can be overcome by the other important properties such as critical heat flux. [28] utilized the Laplace transformation method to solve the flow model of an incompressible non-Newtonian hybrid nanofluid over permeable surface revolving with uniform acceleration with velocity and thermal slip effects. [29] conducted a numerical analysis of activation energy for the two-dimensional flow of a hybrid nanofluid under the effect of buoyancy force and thermal radiation. Furthermore, the influence of key parameters, such as the Nusselt number and Sherwood number, on the heat and velocity profiles has also been investigated.

Most of the researcher used conventional solution methodologies to explain various fluid models describing the effects of entropy generation, rotating flow problems, and Joule heating [30–36]. The application of modern solution methodologies based on artificial neural networks to solve such problems is an inventive work. Of late, researcher applied these modern solutions depending on artificial intelligence to resolve the problem related to various field such as financial trading [37], rainfall prediction models [38], bioinformatics [39, 40], fluid dynamics [41, 42], energy, HIV virus spread models [40, 43, 44], and coronavirus perdition model [45] (also see [46, 47, 49]). In this study, the authors intended to solve the 3D-UHSRS fluid problem for the first time, to the best of our knowledge, by utilizing the Levenberg–Marquardt supervised neural networks (LM-SNNs) bases solution technique. Theses modern solution approaches were based on state-of-the-art computational algorithms that can easily tackle the nonlinear behavior of flow model’s equations.

The basic purpose of this research work is to investigate and explain the flow feature along with the heat and mass transfer in 3D-UHSRS. The basic feature of purposed solution methodology for 3D-UHSRS are as follows:

• A state-of-the-art mathematical model for the 3D-UHSRS problem has been developed and expressed in terms of PDEs, which are further converted into a set of nonlinear ordinary differential equations (ODEs) using dimensionless similarity variables.

• A numerical solution of the 3D-UHSRS problem was obtained by implementing the Lobatto IIIA solution methodology based on the bvp4c solver in MATLAB. The solution datasets of the 3D-UHSRS problem is then subjected for LM back propagation to carry out the training, validation, and testing of the data set points.

• Different plots describing the effect of physical constants on velocity as well as thermal profile have been presented. The performance, convergence, and accuracy of the solution approach were validated thorough residual error, the number of grid points, ODEs, and BCs evaluations.

• Numerical and graphical results in the form of the MSE curve, error histograms, regression and error plots were authenticated from the performance, precision, and convergence of the proposed LM-SNNs methodology.

Problem Formulation

Assume a 3D unsteady flow of a hybrid nanofluid over a stretchable surface, as shown as Figures 1and 2. The surface is stretched in the $xy$-plane of the Cartesian coordinate system. The surface is rotating with a uniform velocity “$w$” about an axis for which $z=0$. Here, $\left(u,v,w\right)$ are the components of velocity in the direction along $\left(x,y,z\right).$ Stretching velocities of the surface and in $x$ and $y$ direction are represented by $u$ and $v$. Temperature of fluid at stretching surface is $T_w$, whereas the temperature of the ambient fluid is $T_{\infty }$.

FIGURE 1

FIGURE 1. Geometrical interpretation for magnetohydrodynamic-HNRD.

FIGURE 2

FIGURE 2. Working flow chart.

According to abovementioned assumptions, described flow model can be expressed in term of following set of mathematical equations (50) and (51):

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0,(1)$

${\rho }_{hnf}\left(\frac{\partial u}{\partial t}-2\omega v+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)={\mu }_{hnf}\frac{{\partial }^{2}u}{\partial z^2},(2)$

${\rho }_{hnf}\left(\frac{\partial v}{\partial t}+2\omega u+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right)={\mu }_{hnf}\frac{{\partial }^{2}v}{\partial z^2},(3)$

${\rho }_{hnf}\left(\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\right)={\mu }_{hnf}\frac{{\partial }^{2}w}{\partial z^2},(4)$

$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}=\frac{k_{hnf}}{{\left(\rho C_p\right)}_{hnf}}\left[\frac{{\partial }^{2}T}{\partial z^2}\right]+\frac{1}{{\left(\rho C_p\right)}_{hnf}}\left[\frac{\partial q_r}{\partial z}\right].(5)$

In the above system of equations, $q_r$ exhibits radiative heat flux known as Rosseland approximation, which can be mathematically expressed as

$q_r=-\frac{4{\sigma }^{\ast }}{3k^{\ast }}\frac{\partial T^4}{\partial y}$

Here $k^{\ast }$ is the Stephen–Boltzmann constant, and ${\sigma }^{\ast }$ represents the mean absorption coefficient. Expending $T^4$ and ignoring the higher order terms, we get $T^4=4T{T}_{\infty }^3-3T_{\infty }^4$.

Using the value of $T^4$ in Eq. (5), we obtain

$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}=\frac{1}{{\left(\rho C_p\right)}_{hnf}}\left(k_{hnf}+\frac{16{\sigma }^{\ast }{T}_{\infty }^3}{3k^{\ast }}\right)\frac{{\partial }^{2}T}{\partial z^2}(6)$

Along with the following boundary conditions of the system:

$\begin{array}{l}\mathrm{When} z=0\mathrm{Then}u=u_w=\frac{cx}{1-\alpha t} , v=0,w=0,T=T_w\\ \mathrm{When} z\to \infty \mathrm{Then}u\to 0,v\to 0,T\to T_{\infty }\end{array}(7)$

Here, “c” is the stretching coefficient of the surface, $q_r$ is the amount of radiative flux, $\omega$ is angular velocity, and T_∞ is temperature of the ambient fluid. Undermentioned linear transformation is designed to transmute the modeling ODEs (1–5) into a nonlinear equivalent system of PDEs.

$\begin{array}{l}u=\frac{ax}{1-\alpha t} f\prime \left(\eta \right), v=\frac{ax}{1-\alpha t} g\left(\eta \right), w=-\sqrt{\frac{a{\nu }_f}{1-\alpha t} }f\left(\eta \right),\\ \eta =\sqrt{\frac{\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{${\nu }_f$}\right.}{1-\alpha t} }z, \theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }}.\end{array}\}(8)$

In the above equations, ${\rho }_{hnf }$ is density, ${\mu }_{hnf}$ is dynamic viscosity, and ${\left(\rho c_p\right)}_{hnf}$ is the heat capacity of the hybrid nanofluid. Whereas ${\varphi }_1$ represents the concentration of Al nanoparticles, and ${\varphi }_2$ represents the concentration of nanoparticles of TiO₂. As “hybrid nanofluid” is a combination of two different types of nanoparticles, ${\varphi }_{hnf}$ is the total concentration of nanoparticle in the base fluid that can be calculated as the sum of concentrations of both types of nanoparticles, i.e., ${\varphi }_1+{\varphi }_2$. All mathematical relationships expressing thermophysical properties are expressed in Table 1 [51]:

TABLE 1

TABLE 1. Mathematical expression of various thermophysical quantities.

Values regarding density, electrical, and thermal conductivity and specific heat against used nanoparticles and base fluid has been mentioned in Table 2.

TABLE 2

TABLE 2. Numerical values for various physical and chemical properties [48, 49].

Following important physical terminologies representing the skin friction and Nusselt number can be written as:

$C_{fx}=\frac{{\mu }_{hnf}}{{\rho }_f{u}_w^2}{\left(\frac{\partial u}{\partial z}\right)}_{z=0},C_{fy}=\frac{{\mu }_{hnf}}{{\rho }_f{u}_w^2}{\left(\frac{\partial v}{\partial z}\right)}_{z=0},N{u}_x=-\frac{r{k}_{hnf}}{k_f\left(T_f-T_{\infty }\right)}{\left(\frac{\partial T}{\partial z}\right)}_{z=0}+x{\left(q_r\right)}_z,C_{fx}(9)$

The mathematical evaluation of Eqs. 1 and 2)–5 gives

$\frac{\raisebox{1ex}{${\mu }_{hnf}$}\!\left/ \!\raisebox{-1ex}{${\mu }_f$}\right.}{\raisebox{1ex}{${\rho }_{hnf}$}\!\left/ \!\raisebox{-1ex}{${\rho }_f$}\right.}f\prime \prime \prime -f^{\prime 2}+f{f}^{″}+2\text{Ω}g-\beta \left(f\prime +\frac{\eta }{2}{f}^{″}\right)=0,(10)$

$\frac{\raisebox{1ex}{${\mu }_{hnf}$}\!\left/ \!\raisebox{-1ex}{${\mu }_f$}\right.}{\raisebox{1ex}{${\rho }_{hnf}$}\!\left/ \!\raisebox{-1ex}{${\rho }_f$}\right.}g″-f\prime g+fg\prime -2\text{Ω}f\prime -\beta \left(g+\frac{\eta }{2}g\prime \right)=0,(11)$

$\frac{1}{Pr}\frac{{\left(\rho C_p\right)}_f}{{\left(\rho C_p\right)}_{hnf}}\left(\frac{k_{hnf}}{k_f}+\frac{4}{3}Rd\right)\theta ″-\beta \frac{\eta }{2}\theta \prime +f\theta \prime =0,(12)$

Here, corresponding BCs are

$\begin{array}{l}f\left(\eta \right)=0, f\prime \left(\eta \right)=\lambda , g\left(\eta \right)=0, \theta \left(\eta \right) =1 \mathrm{at} \eta =0,\\ f\prime \left(\eta \right)\to 0,g\left(\eta \right)\to 0 ,\theta \left(\eta \right)\to 0 \mathrm{when} \eta \to \infty .\end{array}\}(13)$

The physical parameters of 3D-UHSRS flow problem involved in Eqs. 8–12 are

$\lambda =\frac{c}{a},\mathrm{\Omega }=\frac{{\omega }^{\ast }}{c}, Rd=\frac{4{\sigma }^{\ast }{T}_{\infty }^2}{k^{\ast }{k}_f}, \mathrm{\beta }=\frac{\alpha }{a},Pr=\frac{{\upsilon }_f{\left(\rho c_p\right)}_f}{k_f}=\frac{{\upsilon }_f}{{\alpha }_f} (14)$

Here, $Rd$ represents the radiation parameter, $Pr$ is the Prandtl number, $\beta$ is the unsteadiness parameter, $\text{Ω}$ is the rotation parameter, $\lambda$ decides the stretching or shrinking ability of the sheet, and $\lambda =0$ corresponds to the static nature of the revolving surface. The nondimensionalized form of Skin friction coefficient and Nusselt number are as follows:

$C_{fx}R{e}_x^{1/2}=\left[\frac{{\mu }_{hnf}}{{\mu }_f}\right]f^{″\left(0\right)}, C_{fy}R{e}_x^{\frac{1}{2}}=\left[\frac{{\mu }_{hnf}}{{\mu }_f}\right]g^{\prime \left(0\right)}N{u}_x{\mathrm{Re}}_x^{-\frac{1}{2}}=-\left(\frac{k_{hnf}}{k_f}+\frac{4}{3}Rd\right){\theta }^{\prime \left(0\right)},C_{fx}R{e}_x^{1/2}(15)$

in which $R{e}_r=\frac{ux}{{\upsilon }_f}$ represent the local rotational Reynold number.

Results and Discussion

Solution of the system of ODE resenting the flow model has been accomplished in two major phases, in the first phase, set of nonlinear ODEs along with their relevant boundary conditions (Eqs. (1)–(7) are transformed into the first-order ODEs for the solution by using the renowned solution technique “Lobatto IIIA” with the use of MATLAB software. Graphical variation of velocity and temperature profile against various important physical parameters are shown in Figures 3 and 4. Various scenarios and cases are generated based on the variation of involved physical parameters, as described in Table 3.

FIGURE 3

FIGURE 3. Variation of velocity and temperature profile against $\mathit{\beta }$, $\mathit{\omega }$, and, $\mathit{Rd}$.

FIGURE 4

FIGURE 4. Variation of velocity and temperature profile against $\mathit{\lambda }$ and$\mathit{Pr}$.

TABLE 3

TABLE 3. Variation of parameters for the unsteady three-dimensional flow of hybrid nanofluid over a stretchable and rotatory sheet problem.

Figure 3A ∼ c) presented the impact of $\beta$ (unsteadiness parameter) on the velocities in x and y directions, as well as on the temperature of the fluid. It was observed that, initially, the velocity of the fluid showed a decreasing trend with the increase in the values of $\beta$; after that, a further increase in $\beta$ will produce a direct change in the velocity of the fluid because an increase in $\beta$ will result in thickening of the velocity boundary layer. In addition, higher values of $\beta$ will also results in the fluid temperature increase. Figure 3D ∼ e) revealed the rising trend of velocities against the increasing values of $\omega$ (the rotation parameter). In reality, centrifugal forces increase with the increase in $\omega$, which tends the particles of fluid to move into the y direction. Figure 3F exposed the influence of $\omega$ (the rotation parameter) on the temperature profile of the fluid. As with the increase in the rotation parameter, the radial velocity and hence the kinematic energy of the fluid rises, which produces more heat; therefore, the overall temperature of the fluid rises.

Figures 4A,B presents the influence of $\lambda$ (the stretching parameter) on the velocity and thermal profile of the fluid. It was observed that initially velocity of the fluid increases with the increase in values of stretching parameter and the temperature of the fluid shows a slight decline against the rising values of stretching parameter. Figure 4C portrayed the variation in temperature profile of the fluid against the variable values of Rd (the radiation parameter). It is understood that radiation parameter expresses the amount of heat contributed to fluid via thermal radiation. Due to this fact, increasing the radiation parameter will generate more heat in the fluid. Figure 4D was drawn to study the effect of the Prandtl number (Pr) on the temperature profile of the fluid, and it has been observed that with the rise in the Prandtl number, the overall temperature of the fluid shows a declining behavior. As the Prandtl number varies inversely to the thermal diffusivity of the fluid, therefore, with the rise in the Prandtl number, the viscosity of the fluid also rises, causing the temperature of the fluid to drop.

In the second phase, numerical results of the solution containing datasets for each involved variable against all scenarios and cases are generated between a fixed domain, and these dataset points are further subjected to the proper execution of innovative LM-SNNs-based solution. During the operation of proposed methodology out of total points, 80% points were subjected under the network training, whereas the leftover 10% dataset points were used for the operation of validation and testing processes. The number of neuron and delay steps in the calculations were adjusted as per complexity of the problem and required level of accuracy. A two-layer internal structure of the LM-SNNs is presented in Figure 5. Table 4–8 presents the numerical performance indicators comprising of MSE, gradient, Mu, and the number of epochs for scenario (I–V), respectively.

FIGURE 5

FIGURE 5. Internal structure of NAR.

TABLE 4

TABLE 4. Numerical performance indicators for scenario I.

TABLE 5

TABLE 5. Numerical performance indicators for scenario II.

TABLE 6

TABLE 6. Numerical performance indicators for scenario III.

TABLE 7

TABLE 7. Numerical performance indicators for scenario IV.

TABLE 8

TABLE 8. Numerical performance indicators for scenario V.

Figure 6A ∼ d) presents the graphical details of the time series response, error distribution plot, MSE, and gradient plots against the second case of the first scenario. Furthermore, MSE for any computational methodology represents the mean of square of differences between the actual value and estimated value. Accuracy and stability of any method can be judged through the MSE value encountered during the computational process. Smaller MSE values correspond to a better solution and an accurate and reliable solution technique. Figures 7A–D were designed to illustrate the MSE plots of the first case of each scenario for training, validation, and testing to compare performances based on MSE for each case.

FIGURE 6

FIGURE 6. Performance and accuracy plots of C-2 S-1.

FIGURE 7

FIGURE 7. Mean square error bases performance of second cases for all scenarios.

The reference solution of the problem is available in the form of dataset points that are further categorized into training, validation, and testing processes at a specific ratio. In time series response, a close relationship between target and output values of training, validation, and testing depict the accuracy and precision of the solution methodology. Time series response of any variable is the statistical measure of any characteristics with respect to time. In addition, time series response help the researchers to estimate and understand the performance of a variable through any calculated data. Through these fitness plots one can easily observe the separate accuracy of all training, validation, and testing points as compared to available graphical solution. Figures 8A–D illustrate the fitness plots for first cases of all scenarios.

FIGURE 8

FIGURE 8. Fitness performance plots of second cases for all scenarios.

Error histograms are another way to measure the closeness of predicted values with the reference values. These histograms are actually the distribution of errors of all these computed values from a zero error point. Errors in term of all the achieved values are classified into 20 portions, which are aligned across a line representing the zero error line. More values that lie close to the zero error line indicate more accuracy and precision of the solution methodology. Figures 9A–D were designed to illustrate a comparison of error histograms for first cases of all scenarios.

FIGURE 9

FIGURE 9. Error distribution plots of second cases of all scenarios.

Regression is a graphical way to present the precision of the predicted values to the reference values separately for training, validation and testing points. In these plots, an available reference solution has been shown by a straight line, whereas the predicted values are shown by dots or small circles. Accuracy and precision of the computation can also be judged through the numerical value of regression. R = 1 means that the predicted values are very close to reference values, and R = 0 means there is a very poor relationship between reference and predicted values. Figures 10A–D exhibit a comparison of regression plots between first cases of all scenarios.

FIGURE 10

FIGURE 10. Regression plots of second cases of all scenarios.

Gradient is actually a vector responsible for guiding the network in the right direction with an accurate magnitude to reach the required solution as early as possible, whereas mu is a factor that controls a certain algorithm. The value of mu directly portraits the convergence of the solution. Figures 11A–D reflect a comparison between the gradient and mu plots of first cases of all scenarios.

FIGURE 11

FIGURE 11. Plot for gradient and mu of second cases of all scenarios.

Figure 12 portrays the graphic comparison of proposed LM-SNNs base solutions with the already available solutions for all scenarios, and additional comparison of errors for all cases of each scenario are also placed opposite showing the accuracy and precision of the proposed solution methodology.

FIGURE 12

FIGURE 12. Comparison of the Levenberg–Marquardt supervised neural network solution with reference solution along with relative errors for various scenarios. (a), (b).

Conclusion

Here, we employed a numerical study using a novel LM-SNNs–based methodology to investigate the 3D-UHSRS by modeling it in terms of PDEs, which were further reduced to ODEs. A well-renowned solution technique “Lobatto IIIA” was implemented to solve these sets of equation. The influence of various involved physical parameters on velocity and thermal performance are visualized and studied. The solution in respect of each variable in the form of dataset points are acquired and placed in MATLAB for proper operation of the LM-SNNs solution by training, validation, and testing of these dataset points at 80%, 10%, and 10%, respectively. Accuracy, precision, and cogency of the solution were validated through various graphical results consisting of time series, MSE, error distribution, and regression plots.

Followings are few important research outcomes:

• Higher values of $\omega$ (rotational velocity) will make the fluid velocity to decrease, whereas temperature of the fluid shows an increasing trend for a similar change.

• An increase in velocity and decline in the temperature field were observed with the increasing values of $\mathit{\lambda }$ (the stretching parameter).

• A boost in the fluid temperature was observed with increasing values of $Rd$ (the radiation parameter) whereas reverse behavior has been noted for the $Pr$.

Solution methodologies working on the principle of artificial intelligence and machine learning can be more beneficial and valuable in solving the problems related to nano [52–54] and micro fluids [55–61].

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material. Further inquiries can be directed to the corresponding authors.

Author Contributions

MS, MR, and MK contributed to conception and design of the study. SS organized the database. II performed the statistical analysis. MS and II wrote the first draft of the manuscript. KN, MR, and SS wrote sections of the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

The work in this study was supported, in part, by the Open Access Program from the American University of Sharjah”. This study represents the opinions of the author(s) and does not mean to represent the position or opinions of the American University of Sharjah.

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Nomenclature

Symbols

$\text{Ω}$ (rad s⁻¹) Angular velocity

${\text{ρ}}_{}$ (kgm⁻³) Density

u, v, w (ms⁻¹) Velocity components

C_P (m²s⁻²K⁻¹) Specific heat

λ (m⁻¹) Stretching coefficient

T (K) Temperature

K (mkgs⁻³K⁻¹) Thermal conductivity

$\text{μ }$ (kgm⁻¹s⁻¹) Viscosity

Abbreviations

LM Levenberg–Marquardt

SNN Supervised neural networks

C Case

S Scenario

nf Nanofluid

hnf Hybrid nanofluid

f, g Dimensionless velocity components

$\text{θ}$ Dimensionless temperature

Nu Nusselt number

$\varphi$ Nanoparticle concentration

Re Reynolds number

$\text{η}$ Transformed coordinate

$\text{Ω}$ (rad s⁻¹) Angular velocity Transformed angular velocity

SWCNTs Single wall CNTs

MWCNTs Multiwall CNTs

MSE Mean square error

MHD Magnetohydrodynamics

ODEs Ordinary differential equations

PDEs Partial differential equations