Thermal Enhancement in the Ternary Hybrid Nanofluid (SiO₂+Cu+MoS₂/H₂O) Symmetric Flow Past a Nonlinear Stretching Surface: A Hybrid Cuckoo Search-Based Artificial Neural Network Approach

Abstract

In this article, we considered a 3D symmetric flow of a ternary hybrid nanofluid flow (THNF) past a nonlinear stretching surface. The effect of the thermal radiation is considered. The THNF nanofluid SiO${}_2$+Cu+MoS${}_2$/H${}_2$O is considered in this work, where the shapes of the particles are assumed as blade, flatlet, and cylindrical. The problem is formulated into a mathematical model. The modeled equations are then reduced into a simpler form with the help of suitable transformations. The modeled problem is then tackled with a new machine learning approach known as a hybrid cuckoo search-based artificial neural network (HCS-ANN). The results are presented in the form of figures and tables for various parameters. The impact of the volume fraction coefficients ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_3$, and the radiation parameter is displayed through graphs and tables. The higher numbers of the radiation parameter $\left(Rd\right)$ and the cylinder-shaped nanoparticles, ${\varphi }_3$, enhance the thermal profile. In each case, the residual error, error histogram, and fitness function for the optimization problem are presented. The results of the HCS-ANN are validated through mean square error and statistical graphs in the last section, where the accuracy of our implemented technique is proved.

1. Introduction

Fluid flows can be symmetrical under specific conditions. However, if such circumstances are modified, such a symmetric flow is not always attained. The parabolic symmetry retains in the flat Poiseuille flow for smaller values of the Reynolds number. This flow describe the exact symmetry in the solution of the Navier–Stokes equations. But, when the Reynolds number increases the symmetry can not be retained anymore. This type of flow transformation from a constant symmetric condition to a more complicated state is found in natural fluid flow phenomena as well as in fluid flow experiments or analysis. The breakdown of the symmetry in the flow field has great importance in heat transfer analysis. It has been studied from both theoretical and experimental point of view in the field of fluid mechanics [1]. Fluids which contain two different types of nanoparticles are called hybrid nanofluids (HNFs) [2]. On a stretching sheet, such kind of fluids have higher thermal conductivity as compared to nanofluids. A THNF is the combination of three distinct nanoparticles, having a wide range of chemical and physical intersections, all existing within a base liquid [3]. This article examines the inclusion of three different nanomaterials, namely, molybdenum disulfide (MoS${}_2$), copper (Cu), and silicon dioxide (SiO${}_2$), within water. As a consequence, the MoS${}_2$-SiO${}_2$-Cu-water combination formed. The MoS${}_2$-SiO${}_2$-Cu-water amalgamation offers a smart solution for addressing the degradation of toxic materials, environmental purification, and efficient cooling in diverse equipment setups. The main function of automobile coolants, including vehicle coolants, is to prevent engines from experiencing overheating. MoS${}_2$ is widely used for lubrication purposes, copper (Cu) has the ability of its high thermal conductivity, while SiO${}_2$ has a stable molecular structure. Additionally, SiO${}_2$ exhibits a high dielectric strength, which makes it suitable for use in insulators and semiconductor materials. When compared with the base fluid, HNF has its own physical and dynamical properties that have been explained by many authors with suitable models [4,5,6]. The current and prospective applications of NFs were explored by Wong and De Leon [7]. The utilization, limitations, and challenges associated with HNFs in the field of solar energy were investigated by Shah and Ali [8]. Sidik et al. [9] provides a detailed analysis of the preparation methods, stability, and applications of HNFs. Moldoveanu et al. [10] used experimental investigation on the thermal conductivity of stable Al${}_2$O${}_3$ and SiO${}_2$ NFs and their hybrids. Babar and Ali [11] conducted a study on the production, thermophysical characteristics, applications, and challenges of hybrid nanofluids (HNFs). Sahoo [12] experimentally studied the THNFs viscosity and developed a novel functional. The effect of the volume fraction for various nanoparticles of the THNF flow was studied by Sahoo [13]. The impact of these nanoparticles on the temperature profile by taking advantage of the second law was presented in detail. The sensitivity of the particles to the variation in the state variable is further briefly described by Xuan et al. [14]. In this study, the impact of the thermal profile with variable volume fraction was described. The rheological relation for THNFs (CuO/MgO/TiO${}_2$/water) was studied by Mousavi et al. [15]. Furthermore, in this work, the volume fraction and concentration of these particles were derived for the thermal enhancements. A comparative analysis of NFs, HNFs, and THNFs was presented by Adun et al. [16]. This work explained the synthesis of THNFs and their widely used application in industry. Kashyap et al. [17] studied the cooling effects of THNFs by taking suitable investigative approaches. Manjunatha et al. [18] explained the 2D THNF flow past a linearly stretching sheet. This work theoretically investigated the transfer of heat by convection. Nazir et al. [19] further used the ionization impact for a 3D THNF flow past a stretching surface. In this work, they studied the Brownian motion and explained its impact on the thermal profile. Ahmed et al. [20] explained experimentally the heat transfer enhancement of the THNF ZnO${}_4$+Al${}_2$O${}_3$+TiO${}_2$/distilled water in a square flow. Sahoo [21] explained various nanomaterials’ shape effects for the thermohydraulic characteristics of a radiator by utilizing the THNF. Khan et al. [22] presented a mathematical model for three-dimensional flow and heat transfer on a linear stretching surface. More literature can be found in the references [23,24,25]. Shoaib et al. [26] studied the MHD Casson nanofluid flow by using a supervised neural network approach. First, they used a numerical method to solve the given problem and find the data set, and then implemented the supervised neural network. However, the supervised learning always needed a solution, which is always not available with us. For the solution, one has to perform a complex calculation, and in many cases in, highly nonlinear problems, this makes the problem too difficult to solve. For this purpose we need to implement a suitable approach that is easy to implement and requires less computational work.

The literature discussed above encompasses both experimental and theoretical investigations on nanofluids (NFs), hybrid nanofluids (HNFs), and ternary hybrid nanofluids (THNFs) across various applications. Literature is available on linearly stretching sheets with THNFs; the primary aims of this work is to explore the theoretical aspects of heat and mass transfer in three-dimensional THNFs (MoS${}_2$-SiO${}_2$-Cu-water) on a nonlinear stretching sheet. In this work, a new HCS-ANN technique is implemented to solve the proposed problem. This technique uses a basic machine learning approach, which is an emerging area for engineering and technology these days.

In this article, the formulation is given in Section 2 that explains the geometrical description of the problem. A general scheme of the nonlinear differential equations for approximating the solution by using an ANN is presented in Section 3. The $L_2$ norms for minimizing the objective functions are presented in Section 4. The main algorithm that will be combined with the ANN is presented in Section 5. Section 7 shows the results obtained and discusses them in detail. The obtained results with an error analysis are numerically tabulated and discussed here. The obtained results are validated in this particular section. The conclusions and future work are discussed in Section 8.

2. Problem Formulation

Assume a three-dimensional steady hybrid nanofluid (SiO${}_2$+Cu+MoS${}_2$/H${}_2$O) flow on non-linear stretched sheet. The impact of the generalized thermal radiation and viscous dissipation are considered in the flow field. The wall temperature is considered as $T_w$, while $T_{\infty }$ is the reference temperature, as presented in Figure 1. The main equations for this analysis are considered by Hasnain and Abid [27] as shown below:

$$-\left(\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\right)=\frac{\partial u}{\partial x},$$

(1)

$$\begin{array}{c}{\rho }_{\mathit{THNF}}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)={\mu }_{\mathit{THNF}}\left(\frac{{\partial }^{2}u}{\partial z^2}\right),\end{array}$$

(2)

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

(3)

$$\begin{array}{c}{\left(\rho C_p\right)}_{THNF}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}\right)=k_{\mathit{THNF}}\frac{{\partial }^{2}T}{\partial z^2}+{\mu }_{THNF}\left({\left(\frac{\partial u}{\partial z}\right)}^2+{\left(\frac{\partial v}{\partial z}\right)}^2\right)+\frac{16{\sigma }^{*}}{3k^{*}}\frac{\partial }{\partial z}\left(T^3\frac{\partial T}{\partial z}\right),\end{array}$$

(4)

The B.C.’s, as per the problem formulation, take the following form.

$$\begin{array}{c}v=v_w=b(x+y{)}^n,u=u_w=a(x+y{)}^n,w=0,T=T_w\mathrm{at}z=0,\\ u\to 0,v\to 0,w=0,T\to T_{\infty }\mathrm{as}z\to \infty .\end{array}$$

(5)

Introducing the following transformations to reduce the complexity of the system of Equations (1)–(5), we have [27]:

$$\begin{array}{cc}\hfill & \eta ={\left(\frac{b}{{\nu }_f}\right)}^{\frac{1}{2}}{(x+y)}^{\frac{n-1}{2}}z,v=a(x+y{)}^n{g}^{{}^{\prime }}\left(\eta \right),u=a(x+y{)}^n{f}^{{}^{\prime }}\left(\eta \right),\hfill \\ \hfill & w=-{\left(a{\nu }_f\right)}^{\frac{1}{2}}{(x+y)}^{\frac{n-1}{2}}\left[\frac{n+1}{2}\{g\left(\eta \right)+f\left(\eta \right)\}+\frac{n-1}{2}\eta \left\{g^{{}^{\prime }}\left(\eta \right)+f^{{}^{\prime }}\left(\eta \right)\right\}\right],\hfill \\ \hfill & T=\left(T_w-T_{\infty }\right)\theta \left(\eta \right)+T_{\infty }.\hfill \end{array}$$

(6)

Using the above system (6) in Equations (1)–(5), we obtain

$$\left(\frac{{\mu }_{THNF}}{{\mu }_F}\right)\left(\frac{{\rho }_F}{{\rho }_{THNF}}\right)f^{{}^{‴}}\left(\eta \right)+\frac{n+1}{2}{f}^{{}^{″}}\left(\eta \right)[g\left(\eta \right)+f\left(\eta \right)]-n{f}^{{}^{\prime }}\left(\eta \right)\left[g^{{}^{\prime }}\left(\eta \right)+f^{{}^{\prime }}\left(\eta \right)\right]=0,$$

(7)

$$\left(\frac{{\mu }_{THNF}}{{\mu }_F}\right)\left(\frac{{\rho }_F}{{\rho }_{THNF}}\right)g^{{}^{‴}}\left(\eta \right)+\frac{n+1}{2}{g}^{{}^{″}}\left(\eta \right)[g\left(\eta \right)+f\left(\eta \right)]-n{g}^{{}^{\prime }}\left(\eta \right)\left[g^{{}^{\prime }}\left(\eta \right)+f^{{}^{\prime }}\left(\eta \right)\right]=0,$$

(8)

$$\begin{array}{c}\left(\frac{k_{THNF}}{k_F}+R_d\right){\theta }^{{}^{″}}\left(\eta \right)+\left\{\begin{array}{c}{\left({\theta }_w-1\right)}^3\left(3{\theta }^{{}^{\prime }2}\left(\eta \right){\theta }^2\left(\eta \right)+{\theta }^{{}^{″}}\left(\eta \right){\theta }^3\left(\eta \right)\right)\hfill \\ +3\left({\theta }_w-1\right)\left({\theta }^{{}^{\prime }2}\left(\eta \right)+\theta \left(\eta \right){\theta }^{{}^{″}}\left(\eta \right)\right)\hfill \\ +3{\left({\theta }_w-1\right)}^2\left({\theta }^2\left(\eta \right){\theta }^{{}^{″}}\left(\eta \right)+2\theta \left(\eta \right){\theta }^{{}^{\prime }2}\left(\eta \right)\right)\hfill \end{array}\right\}{R}_d\\ +\mathrm{Pr}Ec\frac{{\mu }_{THNF}}{{\mu }_F}\left\{f^{{}^{″}2}\left(\eta \right)+g^{{}^{″}2}\left(\eta \right)\right\}+\mathrm{Pr}\frac{{\left(\rho C_p\right)}_{THNF}}{{\left(\rho C_p\right)}_F}\frac{n+1}{2}\{g\left(\eta \right)+f\left(\eta \right)\}{\theta }^{{}^{\prime }}\left(\eta \right),\end{array}$$

(9)

$$\begin{array}{ccc}\hfill f\left(0\right)=0,f^{{}^{\prime }}\left(0\right)=1,g^{{}^{\prime }}\left(0\right)=S,g\left(0\right)=0,\theta \left(0\right)=1& \mathrm{when}\hfill & \eta =0,\hfill \\ \hfill f^{{}^{\prime }}(\infty )\to 0,g^{{}^{\prime }}(\infty )\to 0,\theta (\infty )\to 0& \mathrm{when}\hfill & \eta \to \infty .\hfill \end{array}$$

(10)

Now, the system of equations given in (7)–(10) are dimensionless ODEs. Here, $S=\frac{b}{a}$ is the stretching parameter, $\mathrm{Pr}=\frac{{\mu }_F{\left(C_p\right)}_F}{k_F}$ and $Ec=\frac{u_w^2}{\left(T_w-T_{\infty }\right)}$ are the Prandtl and the Eckert numbers respectively, while $R_d=\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}{k}_F}$ denotes the radiation parameter.

The physical and dynamic properties together with the nanoparticles and the base fluid used in this work are defined in the following tables. In

Table 1. The THNF models and their physical properties [13].

Property	Ternary Hybrid Nanofluid
Density	${\rho }_{THNF}={\varphi }_1{\rho }_1+{\varphi }_2{\rho }_2+{\varphi }_3{\rho }_3+\left(1-{\varphi }_1-{\varphi }_2-{\varphi }_3\right){\rho }_F$,
Heat capacity	${\left(\rho C_p\right)}_{THNF}=$
	${\varphi }_1{\left(\rho C_p\right)}_1+{\varphi }_2{\left(\rho C_p\right)}_2+{\varphi }_3{\left(\rho C_p\right)}_3+\left(1-{\varphi }_1-{\varphi }_2-{\varphi }_3\right){\left(\rho C_p\right)}_F$,
Dynamic viscosity	${\mu }_{THNF}=\left({\mu }_{N{F}_1}{\varphi }_1+{\mu }_{N{F}_2}{\varphi }_2+{\mu }_{N{F}_3}{\varphi }_3\right)/\left({\varphi }_1+{\varphi }_2+{\varphi }_3\right)$,
	${\mu }_{N{F}_1}={\mu }_F\left(1+14.5{\varphi }_1+123.3{\varphi }_1^2\right),{\mu }_{N{F}_2}={\mu }_F\left(1+13.5{\varphi }_2+904.4{\varphi }_2^2\right)$,
	${\mu }_{N{F}_3}={\mu }_F\left(1+37.1{\varphi }_3+612.6{\varphi }_3^2\right)$,
	$k_{THNF}=\left(k_{N{F}_1}{\varphi }_1+k_{N{F}_2}{\varphi }_2+k_{N{F}_3}{\varphi }_3\right)/\left({\varphi }_1+{\varphi }_2+{\varphi }_3\right)$,
	$\frac{K_{NFi}}{K_F}=\frac{k_i+\left(N^{*}-1\right)k_F-\left(N^{*}-1\right)\left(k_F-k_i\right){\varphi }_i}{k_i+\left(N^{*}-1\right)k_F-\left(k_i-k_F\right){\varphi }_i},N^{*}=\frac{3}{\psi }$
	Blade-shaped nanoparticles $\psi =0.36$,
	Cylindrical nanoparticles $\psi =0.61$,
	Platelet-shaped nanoparticles $\psi =0.52$

, the volume fraction of the first, second, and third nanoparticles are denoted by ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_3$, respectively. The density, dynamic viscosity, kinematic viscosity, and specific heat are also presented here. Furthermore, the numerical values of these physical and dynamical properties of the nanoparticles and base fluid are presented in

Table 2. Thermophysical properties of base fluid and nanoparticles [28].

Base Fluid/Nanoparticles	$\mathit{\rho }\left(\mathbf{kg}{\mathbf{m}}^{-3}\right)$	${\mathit{C}}_{\mathit{p}}\left({\mathbf{J}}^{-1}{\mathbf{K}}^{-1}\right)$	$\mathbf{k}\left({\mathbf{Wm}}^{-1}{\mathbf{K}}^{-1}\right)$	Shape
Water	997.1	4179	0.613	-
${\mathrm{SiO}}_2$	2270	730	1.4013	Blade
$\mathrm{Cu}$	8933	385	401	Platelet
${\mathrm{MoS}}_2$	5060	397.746	34.5	Cylinder

.

3. The Approximate Solution and the Neural Network Modeling

Machine learning approaches toward daily life problems are a need of the day. For this purpose, a variety of approaches have been adopted in the literature [29,30]. ANNs have been utilized to solve a variety of mathematical programming problems due to their good approximation qualities [31]. Even with massive data sets, ANN models are capable of approximating the non-linear functions. The network is made up of three layers, input, output and hidden layers. The neural network receives X as input and gives h as output. The NN models seek to make a relation between the input and output by choosing the best weights and biases of the network nodes [32]. Before characterizing a physical system, it must first be expressed mathematically. For a description of certain phenomena, a mathematical model always paly a key role. It has a wide range of practical applications. One of the primary goals of the resultant mathematical model is to select the best unknown parameters to be included in a model solution to best describe the system. Optimization methods are essential for this aim.

In this study, we use the intelligence of the artificial neural network (ANN) approach to simulate the solutions for a THNF flow past a nonlinear stretching surface. The neural network series solution together with its derivatives for Equations (7)–(9) can be represented as:

$$\widehat{f}\left(\eta \right)=\sum _{i=1}^m{\xi }_i\left[f\left({\gamma }_i\eta +{\beta }_i\right)\right]$$

(11)

$$\frac{d\widehat{f}\left(\eta \right)}{d\eta }=\sum _{i=1}^m{\xi }_i\frac{d}{d\eta }\left[f\left({\gamma }_i\eta +{\beta }_i\right)\right],$$

(12)

$$\frac{d^2\widehat{f}\left(\eta \right)}{d{\eta }^2}=\sum _{i=1}^m{\xi }_i\frac{d^2}{d{\eta }^2}\left[f\left({\gamma }_i\eta +{\beta }_i\right)\right],$$

(13)

$$\frac{d^3\widehat{f}\left(\eta \right)}{d{\eta }^3}=\sum _{i=1}^m{\xi }_i\frac{d^3}{d{\eta }^3}\left[f\left({\gamma }_i\eta +{\beta }_i\right)\right],$$

(14)

$$\widehat{g}\left(\eta \right)=\sum _{i=1}^m{\xi }_i\left[g\left({\gamma }_i\eta +{\beta }_i\right)\right]$$

(15)

$$\frac{d\widehat{g}\left(\eta \right)}{d\eta }=\sum _{i=1}^m{\xi }_i\frac{d}{d\eta }\left[g\left({\gamma }_i\eta +{\beta }_i\right)\right],$$

(16)

$$\frac{d^2\widehat{g}\left(\eta \right)}{d{\eta }^2}=\sum _{i=1}^m{\xi }_i\frac{d^2}{d{\eta }^2}\left[g\left({\gamma }_i\eta +{\beta }_i\right)\right],$$

(17)

$$\widehat{\theta }\left(\eta \right)=\sum _{i=1}^m{\xi }_i\left[\theta \left({\gamma }_i\eta +{\beta }_i\right)\right]$$

(18)

$$\frac{d\widehat{\theta }\left(\eta \right)}{d\eta }=\sum _{i=1}^m{\xi }_i\frac{d}{d\eta }\left[\theta \left({\gamma }_i\eta +{\beta }_i\right)\right],$$

(19)

$$\frac{d^2\widehat{\theta }\left(\eta \right)}{d{\eta }^2}=\sum _{i=1}^m{\xi }_i\frac{d^2}{d{\eta }^2}\left[\theta \left({\gamma }_i\eta +{\beta }_i\right)\right],$$

(20)

where $W=(\xi ,\gamma ,\beta )=\left({\xi }_1,{\xi }_2,\cdots ,{\xi }_m,{\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_m,{\beta }_1,{\beta }_2,\cdots ,{\beta }_m\right)$ are the weights to be chosen by the HCS with m neurons.

In this mapping, $\widehat{f}$, $\widehat{g}$, and $\widehat{\theta }$ represent solutions to the proposed problem with continuous input. Introducing the activation function $f_{rb}\left(t\right)=\frac{1}{1+exp(-\eta )}$ into Equations (11)–(20), we find improved approximate solutions to the problem by employing altered combinations of ANNs and choosing appropriate weights with the help of HCS. The geometrical description of the ANN is explained in Figure 2.

4. The Optimization Problem

In this, we will construct a functional for Equations (7)–(10) that will reduce the mean square error ($L_2$ norm) when approximating the solution $f\left(\eta \right)$, $g\left(\eta \right)$, and $\theta \left(\eta \right)$. Therefore, we introduce four new objective functions for Equations (7)–(10). We have

$$Minimize\mathcal{E}={\mathcal{E}}_1+{\mathcal{E}}_2+{\mathcal{E}}_3+{\mathcal{E}}_4,$$

(21)

where

$${\mathcal{E}}_1=\frac{1}{N}\sum _{i=1}^N{\left[\left(\frac{{\mu }_{THNF}}{{\mu }_F}\right)\left(\frac{{\rho }_F}{{\rho }_{THNF}}\right)f^{{}^{‴}}\left(\eta \right)+\frac{n+1}{2}{f}^{{}^{″}}\left(\eta \right)[f\left(\eta \right)+g\left(\eta \right)]-n{f}^{{}^{\prime }}\left(\eta \right)[f^{{}^{\prime }}\left(\eta \right)+g^{{}^{\prime }}\left(\eta \right)]\right]}^2$$

(22)

$${\mathcal{E}}_2=\frac{1}{N}\sum _{i=1}^N{\left[\left(\frac{{\mu }_{THNF}}{{\mu }_F}\right)\left(\frac{{\rho }_F}{{\rho }_{THNF}}\right)g^{{}^{‴}}\left(\eta \right)+\frac{n+1}{2}{g}^{{}^{″}}\left(\eta \right)[f\left(\eta \right)+g\left(\eta \right)]-n{g}^{{}^{\prime }}\left(\eta \right)[f^{{}^{\prime }}\left(\eta \right)+g^{{}^{\prime }}\left(\eta \right)]\right]}^2$$

(23)

$$\begin{array}{c}\hfill {\mathcal{E}}_3=\frac{1}{N}\sum _{i=1}^N\begin{array}{c}[\left(\frac{k_{THNF}}{k_F}+R_d\right){\theta }^{{}^{″}}\left(\eta \right)+R_d\left\{\begin{array}{c}{\left({\theta }_w-1\right)}^3\left(3{\theta }^{{}^{\prime }2}\left(\eta \right){\theta }^2\left(\eta \right)+{\theta }^{{}^{″}}\left(\eta \right){\theta }^3\left(\eta \right)\right)\hfill \\ +3\left({\theta }_w-1\right)\left({\theta }^{{}^{\prime }2}\left(\eta \right)+\theta \left(\eta \right){\theta }^{{}^{″}}\left(\eta \right)\right)\hfill \\ +3{\left({\theta }_w-1\right)}^2\left({\theta }^2\left(\eta \right){\theta }^{{}^{″}}\left(\eta \right)+2\theta \left(\eta \right){\theta }^{{}^{\prime }2}\left(\eta \right)\right)\hfill \end{array}\right\}\\ +\mathrm{Pr}Ec\frac{{\mu }_{THNF}}{{\mu }_F}\left\{f^{{}^{″}2}\left(\eta \right)+g^{{}^{″}2}\left(\eta \right)\right\}+\mathrm{Pr}\frac{{\left(\rho C_p\right)}_{THNF}}{{\left(\rho C_p\right)}_F}\frac{n+1}{2}\{f\left(\eta \right)+g\left(\eta \right)\}{\theta }^{{}^{\prime }}\left(\eta \right){]}^2\end{array}\end{array}$$

(24)

$${\mathcal{E}}_4=min\left[{(f_0^{\prime }-1)}^2+{\left(f_0\right)}^2+{(g_0^{\prime }-S)}^2+{\left(f_0\right)}^2+{({\theta }_0-1)}^2+{\left(f_{\infty }^{\prime }\right)}^2+{\left(g_{\infty }^{\prime }\right)}^2+{\left({\theta }_{\infty }\right)}^2\right].$$

(25)

In the above equations, ${\xi }_j=jh$, $j=1,2,\cdots ,N$, $N=\frac{1}{h}$, and $\xi \ge 0$. Here, h is the size of the step, and $N\in (0,N)$, where N denotes the sub-intervals $\xi \in ({\xi }_0=0,{\xi }_1,{\xi }_2,\cdots ,{\xi }_j=N)$.

5. The Cuckoo Search Algorithm

Cuckoo search is based on the population strategy for simulating the brood, a parasitic relationship found in some cuckoo species [33]. This technique includes Lévy flights and random search equations. It has been tested in studies with modern algorithms, such as PSO and GA, for its unique approach towards the optima. In terms of difficult optimization problem solutions, CS has been found to outperform its counterpart algorithms [33,34].

For the local and global search, we introduce the following functional.

$$x_i^{t+1}=x_i^t+\alpha s\otimes H\left(p_a-ϵ\right)\otimes \left(x_j^t-x_k^t\right).$$

(26)

Now, introducing to the Lévy walk functional the enhancement of the global search, we have

$$x_i^{t+1}=x_i^t+\alpha \mathcal{L}\left(s\lambda \right),$$

(27)

where $\mathcal{L}\left(s\lambda \right)={\displaystyle \frac{\lambda \Gamma \left(\lambda \right)sin\left(\frac{\pi \lambda }{2}\right)}{\pi s^{1+\lambda }}}$, in which $s>0$ is the step size, and $p_{\alpha }>0$, $x_i^t$, $x_j^t$, $ϵ$, $H\left(u\right)$, ⊗, $\alpha$, and $\mathcal{L}\left(s\lambda \right)$ denote the witching parameter, two distinct solutions, a random variable, the Heaviside function, element-wise product, scaling factor, and the random steps of the Lévy distribution, respectively. The diagramatic representation of this work is presented in the form of the flow chart in Figure 3.

Hybrid Cuckoo Search

The biogeography-based revelation operator: The impact of modifying search operators is employed in the second phase of the HCS to build new solutions. A biogeography-based migration operator allows the host bird to identify foreign eggs with high accuracy, depart old nests, and build new nests. The population is then evaluated from best to worst, and emigration rates, $\mu$, are assigned to each response. The emigration rates are defined as follows:

$${\mu }_i=\frac{E{S}_i}{MP}$$

(28)

The maximum emigration rate in the preceding equation is $E=1$, and the number of species in the solution is $S_i$ = $MPI$. The first algorithm defines the biogeography-based operator used to find the $ith$ answer. Higher fitness solutions may share more features with other solutions in the biogeography-based discovery operator, which is advantageous for exploitation improvement. Figure 3 describes the whole procedure applied in this work to handle the THNF problem and obtain more accurate profiles for the state variables.

6. Procedure

Our proposed methodology can be implemented for the given THNF model as follows:

• First, we define a neural network series solution which approximates the given THNF model.
• In the series solution we have random weights to be determined.
• To obtain a better approximate solution, we need to find the best set of weights.
• We define a fitness function as the mean squared error to convert the given system with the boundary conditions to an optimization problem, so that we obtain the best set of weights.
• To tackle the fitness function, we apply the HCS-ANN.

7. Results and Discussion

The results obtained by solving the system of Equations (22)–(25) are presented in the form of Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 and

Table 3. Variation in the state variables with the blade-shaped nanoparticles (${\varphi }_1=0.01$).

$\mathit{\eta }$	${\mathit{f}}_{\mathit{Ann}}^{\prime }$	${\mathit{g}}_{\mathit{Ann}}$	${\mathit{\theta }}_{\mathit{Ann}}$	Error	Error	Error
0	0.999997	1.000194	0.999962	3.52 × 10${}^{-8}$	2.49 × 10${}^{-8}$	1.37 × 10${}^{-8}$
0.5	0.64857	0.641643	0.639839	4.28 × 10${}^{-8}$	8.64 × 10${}^{-10}$	1.45 × 10${}^{-8}$
1	0.420625	0.411592	0.409277	2.59 × 10${}^{-9}$	3.49 × 10${}^{-9}$	3.24 × 10${}^{-9}$
1.5	0.272804	0.264031	2.62 × 10${}^{-1}$	1.15 × 10${}^{-8}$	2.52 × 10${}^{-12}$	1.41 × 10${}^{-9}$
2	0.176854	0.169352	0.167258	1.69 × 10${}^{-8}$	2.37 × 10${}^{-9}$	8.01 × 10${}^{-9}$
2.5	0.114714	0.108549	0.106688	6.15 × 10${}^{-11}$	4.05 × 10${}^{-10}$	3.73 × 10${}^{-11}$
3	0.074399	0.069509	0.067887	1.85 × 10${}^{-8}$	5.32 × 10${}^{-10}$	8.44 × 10${}^{-9}$
3.5	0.048098	0.04448	0.043128	8.22 × 10${}^{-9}$	1.42 × 10${}^{-9}$	5.69 × 10${}^{-9}$
4	0.030924	0.028459	0.027344	1.98 × 10${}^{-11}$	5.93 × 10${}^{-10}$	1.08 × 10${}^{-10}$
4.5	0.019761	0.018205	0.017213	1.05 × 10${}^{-9}$	3.85 × 10${}^{-13}$	8.36 × 10${}^{-9}$
5	0.012541	0.011623	0.010611	1.04 × 10${}^{-9}$	5.77 × 10${}^{-10}$	1.82 × 10${}^{-8}$
5.5	0.007884	0.007369	0.006217	5.75 × 10${}^{-10}$	1.59 × 10${}^{-9}$	2.03 × 10${}^{-8}$
6	0.004884	0.004592	0.003229	3.82 × 10${}^{-10}$	2.26 × 10${}^{-9}$	1.64 × 10${}^{-8}$
6.5	0.002955	0.002753	0.001159	3.63 × 10${}^{-10}$	2.37 × 10${}^{-9}$	1.09 × 10${}^{-8}$
7	0.001722	0.001514	−0.00029	3.98 × 10${}^{-10}$	2.06 × 10${}^{-9}$	6.23 × 10${}^{-9}$
7.5	0.000943	0.000665	−0.00132	4.18 × 10${}^{-10}$	1.56 × 10${}^{-9}$	3.15 × 10${}^{-9}$
8	0.000459	7.33 × 10${}^{-5}$	−0.00205	3.99 × 10${}^{-10}$	1.06 × 10${}^{-9}$	1.39 × 10${}^{-9}$

,

Table 4. Variation in the state variables with the platelet-shaped nanoparticles (${\varphi }_2=0.01$).

$\mathit{\eta }$	${\mathit{f}}_{\mathit{Ann}}^{\prime }$	${\mathit{g}}_{\mathit{Ann}}$	${\mathit{\theta }}_{\mathit{Ann}}$	Error	Error	Error
0	0.999869	0.999942126	1.000046539	8.48 × 10${}^{-9}$	2.49 × 10${}^{-8}$	1.35 × 10${}^{-7}$
0.5	0.657548	0.648361105	0.652267958	1.13 × 10${}^{-10}$	8.64 × 10${}^{-10}$	1.48 × 10${}^{-9}$
1	0.432323	0.420373492	0.425345515	2.49 × 10${}^{-11}$	3.49 × 10${}^{-9}$	2.13 × 10${}^{-9}$
1.5	0.284134	0.272460669	0.277227263	2.75 × 10${}^{-9}$	2.52 × 10${}^{-12}$	1.77 × 10${}^{-8}$
2	0.186657	0.176493933	0.180648726	3.35 × 10${}^{-9}$	2.37 × 10${}^{-9}$	1.26 × 10${}^{-8}$
2.5	0.122446	0.114370487	0.117500726	3.41 × 10${}^{-9}$	4.05 × 10${}^{-10}$	1.19 × 10${}^{-8}$
3	0.080115	0.074177006	0.076153604	3.80 × 10${}^{-11}$	5.32 × 10${}^{-10}$	7.03 × 10${}^{-10}$
3.5	0.052232	0.048075896	0.049165752	1.07 × 10${}^{-9}$	1.42 × 10${}^{-9}$	9.59 × 10${}^{-9}$
4	0.033899	0.031018745	0.031632304	1.91 × 10${}^{-9}$	5.93 × 10${}^{-10}$	1.02 × 10${}^{-8}$
4.5	0.021866	0.019809821	0.020281361	1.24 × 10${}^{-9}$	3.85 × 10${}^{-13}$	5.37 × 10${}^{-9}$
5	0.013974	0.0124322	0.012941831	3.30 × 10${}^{-10}$	5.77 × 10${}^{-10}$	1.54 × 10${}^{-9}$
5.5	0.00879	0.007598363	0.008187146	3.09 × 10${}^{-14}$	1.59 × 10${}^{-9}$	7.49 × 10${}^{-11}$
6	0.005373	0.004468614	0.005088242	2.66 × 10${}^{-10}$	2.26 × 10${}^{-9}$	2.54 × 10${}^{-10}$
6.5	0.003101	0.002481237	0.00304435	8.46 × 10${}^{-10}$	2.37 × 10${}^{-9}$	1.20 × 10${}^{-9}$
7	0.001572	0.001251771	0.001669306	1.49 × 10${}^{-9}$	2.06 × 10${}^{-9}$	2.33 × 10${}^{-9}$
7.5	0.000522	0.000513104	0.000716345	2.04 × 10${}^{-9}$	1.56 × 10${}^{-9}$	3.32 × 10${}^{-9}$
8	−0.00022	0.0000790103	0.0000288704	2.45 × 10${}^{-9}$	1.06 × 10${}^{-9}$	4.05 × 10${}^{-9}$

and

Table 5. Variation in the state variables with the cylinder-shaped nanoparticles (${\varphi }_3=0.01$).

$\mathit{\eta }$	${\mathit{f}}_{\mathit{Ann}}^{\prime }$	${\mathit{g}}_{\mathit{Ann}}$	${\mathit{\theta }}_{\mathit{Ann}}$	Error	Error	Error
0	1.000207	0.999984463	1.000160487	1.29 × 10${}^{-8}$	2.64 × 10${}^{-11}$	4.29 × 10${}^{-8}$
0.5	0.607373	0.622932087	0.64851126	3.48 × 10${}^{-8}$	2.10 × 10${}^{-10}$	4.20 × 10${}^{-9}$
1	0.36874	0.387983682	0.420428542	4.99 × 10${}^{-9}$	1.52 × 10${}^{-11}$	1.14 × 10${}^{-9}$
1.5	0.223817	0.241550879	0.272454235	1.84 × 10${}^{-9}$	1.71 × 10${}^{-9}$	2.35 × 10${}^{-9}$
2	0.135761	0.15030047	0.176513107	3.23 × 10${}^{-10}$	3.85 × 10${}^{-12}$	1.19 × 10${}^{-8}$
2.5	0.082288	0.093391042	0.114374775	1.90 × 10${}^{-9}$	1.01 × 10${}^{-9}$	2.76 × 10${}^{-10}$
3	0.04976	0.057843178	0.074069561	1.80 × 10${}^{-9}$	7.90 × 10${}^{-10}$	1.09 × 10${}^{-8}$
3.5	0.029948	0.035616546	0.047797612	3.69 × 10${}^{-11}$	9.41 × 10${}^{-12}$	2.48 × 10${}^{-8}$
4	0.017906	0.021730061	0.030568072	2.21 × 10${}^{-10}$	5.25 × 10${}^{-10}$	2.01 × 10${}^{-8}$
4.5	0.010611	0.013081778	0.019220147	3.26 × 10${}^{-10}$	1.44 × 10${}^{-9}$	8.06 × 10${}^{-9}$
5	0.006204	0.007723781	0.01174661	1.89 × 10${}^{-10}$	1.47 × 10${}^{-9}$	8.03 × 10${}^{-10}$
5.5	0.003546	0.004422034	0.006857646	7.20 × 10${}^{-11}$	7.26 × 10${}^{-10}$	7.79 × 10${}^{-10}$
6	0.001943	0.002390821	0.003711368	2.03 × 10${}^{-11}$	9.74 × 10${}^{-11}$	5.39 × 10${}^{-9}$
6.5	0.000975	0.001132019	0.001749424	4.13 × 10${}^{-12}$	3.89 × 10${}^{-11}$	1.15 × 10${}^{-8}$
7	0.000389	0.000335618	0.000596071	4.60 × 10${}^{-13}$	2.76 × 10${}^{-10}$	1.71 × 10${}^{-8}$
7.5	0.0000345	−0.00018474	−0.0000047669	5.75 × 10${}^{-16}$	3.78 × 10${}^{-10}$	2.12 × 10${}^{-8}$
8	−0.00018	−0.00053577	−0.0002295	5.78 × 10${}^{-14}$	2.41 × 10${}^{-10}$	2.35 × 10${}^{-8}$

. The results are validated in Figure 9, Figure 10, Figure 11 and Figure 12. The validation section provides a more comprehensive analysis of the present results, where the statistical approach is adopted. The impact of various parameters (nanofluid volume fraction and radiation parameter) on the state variables together with their residuals, fitness functions, and error analysis is presented through graphs.

The impact of the varying volume fraction of the first nanoparticle over the velocity gradient $\left(f^{\prime }\right)$, $y-$direction velocity g, and the thermal profile $\theta$ is analyzed in Figure 4. It is clear from Figure 4a that the increasing values of ${\varphi }_1$ enhance the velocity gradient. These nanoparticles describe SiO${}_2$. When these are added, they start colliding with each other and a haphazard motion takes place, which further causes the increase in the velocity profile. One important point is the shape of the particle, which is a blade in this case. A similar trend for the velocity profile on the y-axis can be seen in Figure 4b. The increasing values of ${\varphi }_1$ cause a decline in the thermal profile, as depicted in Figure 4c. The blade-shaped nanoparticles have $N^{*}=8.333$ when substituted into the nanofluid thermal conductivity model for various increasing values of ${\varphi }_1$, which causes a decline in the hybrid nanofluid’s thermal conductivity. Physically, the larger shape of the nanoparticles does not result in Brownian motion as compared to the smaller ones. The smaller the interaction, the less the transfer of heat. Thus, for larger values of the volume fraction, the thermal profile falls down. These variations in performance are presented in Figure 4d,e. In Figure 4d, the residual error for various states of the volume fraction is presented. An impressive result for ${\varphi }_1=0.3$ is shown that nearly approaches ${10}^{-14}$. On the other hand, the fitness function is presented in Figure 4d, where the minimum fitness for the optimization function approaches ${10}^{-12}$. This proves that our technique searches for the best in the minimum of all the available weights.

The effect of the second nanoparticle’s (Cu) volume fraction is studied in Figure 5a–e. The velocity gradient for the increasing values of the fraction is displayed in Figure 5a. A very small decrease can be seen in the velocity gradient for larger values of ${\varphi }_2$. These nanoparticles are platelets in shape and have the same effect on the y-axis velocity too, as depicted in Figure 5b. The temperature profile also shows a decrease in the thermal profile as ${\varphi }_2$ increases. When the platelet-shaped nanoparticles size increases, it causes the flow and interaction of the particles to slow down. As a result, the transfer of heat from one particle to another slows down, which further causes a decline in the thermal profile. A quite impressive result for the intermediate and initial values of ${\varphi }_2$ can be seen in Figure 5d,e. For ${\varphi }_2=0.02$, the fitness function is at its minimal value and approaches ${10}^{-12}$. On the other hand, the residual has the minimum values at the initial value ${\varphi }_2=0.01$, that varies up to ${10}^{-14}$.

The impact of the cylinder-shaped nanoparticles ${\varphi }_3$ is depicted in Figure 6a–e. The high thermal conductivity of MoS${}_2$ and the shape factor causes an increase in the thermal conductivity of the THNF. As ${\varphi }_2$ increases, the velocity gradient increases. This thermal increase causes the fluid to flow more rapidly due to the faster interaction of the nanoparticles. The increase in the velocity gradient and the motion of the THNF can be seen in Figure 6a,b, respectively. In Figure 6c, the thermal profile shows an increase due to the larger values of ${\varphi }_3$. In this case, the nanofluid becomes less dense due to the shape factor and the Brownian motion takes place in a more rapid way that causes the transfer of heat from one particle to another, which as a result improves the thermal profile. The residual error and the fitness function approach ${10}^{-16}$ and ${10}^{-6}$, respectively, for the boundary points of ${\varphi }_2$. This shows the accuracy of our technique and the minimal values in each iteration that the optimization function achieves.

The impact of the radiation parameter $Rd$ is described in Figure 7a. As the radiation parameter increases, the thermal profile jumps up. The larger the thermal radiation, the greater the thermal profile. The larger values of $Rd$ make the fluid more electrically conductive and cause a decrease in the fluid’s thermal conductivity. The residual errors and fitness functions are also computed in Figure 7b,c. For 30 independent runs, the fitness function achieves its minimal value, up to ${10}^{-6}$, for the intermediate value of $Rd$. The residual error behaves the same for various values but achieves minimal values up to ${10}^{-13}$ for $Rd=0.5$ and 1.

The optimization technique, and specifically the neural network approach, utilizes data at specified nodes. Three plots each for $f^{\prime }$, g, and $\theta$ are presented in Figure 8, labeled with (a), (b), and (c), respectively. These are actually the bar graphs in which the blue color shows the trained data of the neural network, the green shows the validated data, the red color shows the tested data, and the line drawn depicts the zero error. The data trained by the neural network has been validated at various points in the mainstream and the results are presented in a green color. Furthermore, some tests are performed for these analyses that are presented on the bar graph in a red color. In each case, the data are equally dispersed about the mean position. Each plot shows 20 bins, while the error for each case has been plotted on the x-axis. The minimal error is found for the third and first cases, and approaches ${10}^{-7}$.

Here, the impact of the changing volume fractions ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_3$ over the state variables have been described numerically in the form of tables. Table 3 shows the impact of ${\varphi }_1=0.01$ at various step size. As the step size increases, the state variables approach 0 from 1. The corresponding errors for each state variable have been computed. It is clear that the error in each case tends to $E-11$, $E-13$, and $E-11$ for $f^{\prime }$, g, and $\theta$, respectively. These values prove the accuracy of our implemented technique over other techniques. In Table 4, the impact of Cu nanoparticles ${\varphi }_2$ has been presented. The state variables start with the initial step size from 1 and move rapidly toward zero as $\eta \to 8$. This decrease is very rapid as compared to Table 3. The error in this case is very small and is predicted at $E-14$ for some intermediate values. These numerical results show the efficiency of our implemented technique. A quite similar trend for the velocity profile can be seen in Table 5 for the cylinder-shaped nanoparticles. These nanoparticles show that the larger values of $\eta$ cause the state variables to decrease from 1 to zero, with an error of $E-14$ in some cases. This smaller amount of error shows the stability of the method in each case.

Validation of Results

This section is specifically constructed for the validation of the obtained results for the above problems. A more statistical approach is adopted in this section. The mean square error and the gradient impact for 562 epochs are described in Figure 9a,b. In Figure 9a, the validation performance is presented against the tested and trained data. The validation performance is ${10}^{-9}$ for the selected epochs. The gradient, $\mu$, and wall fail for these epochs is presented in Figure 9b. The values are predicted at ${10}^{-8}$ and ${10}^{-9}$. All these steps are further elaborated with the well-known regression model. Each step of training, validation, and testing is presented in Figure 9c. All three cases as a sum are carried out in the last figure of the regression model. In all three cases, the $100\%$ data lies on the regression line. This is the validation of the tested data using our implemented technique.

The mean square error in Figure 10a shows quite impressive results that are achieved at eight epochs. The training, testing, and validation have a value equal to 0.00014938 at 2 for each epoch. The results vary by up to $0.00033075$, ${10}^{-8}$, and the validation checks equal 8, as presented in Figure 10b. The results are validated with a regression model up to $99.99\%$. The combined effect of training, validation, and testing are presented in Figure 10c. A quite similar trend is observed in Figure 11 for all three cases. This impact is for the cylinder-shaped nanoparticles ${\varphi }_3$. A more impressive result for the radiation parameter $Rd$ is presented in Figure 12. The obtained results are displayed for the temperature profile, where the results are obtained at eight epochs. The target results are achieved at two epochs and are equal to $0.00014938$. This result is achieved at a ${10}^{-4}$ mean square error. The regression analysis and the gradient and mu, etc., are plotted here. In the last subfigure, the validation, testing, and training have a total of $99.99\%$, which is quite impressive. The regression analysis shows that our total data fits on the linear regression line, which proves the performance of the implemented technique.

8. Conclusions

We studied the THNF problem through the HCS-ANN in this paper.The obtained results are displayed with tables and graphs. Based on our analysis, we conclude the following:

• The impact of the cylinder-shaped nanoparticles play a key role in the heat transfer enhancements.
• The blade-shaped nanoparticles are opposing the heat transfer, while the gradient of the velocity (f) and the fluid motion (g) in the boundary layer increases with increasing values.
• The radiation parameter enhances the thermal profile with increasing values. A similar trend may be observed for the cylinder-shaped nanoparticles ${\varphi }_3$.
• The radiation parameter enhances the thermal profile.
• The implemented technique’s (HCS-ANN) efficiency has been proved through graphs and tables.
• The results of the proposed problem are validated through statistical graphs, such as regression analysis.
• The validation of the results show that HCS-ANN is the best for the solution of nonlinear problems.

Based on the above analysis, we recommend HCS-ANN for the solution of nonlinear problems.