Machine learning approach of Casson hybrid nanofluid flow over a heated stretching surface

Nomenclature

1.   Introduction

Artificial neural networks (ANN) have played a significant role in the development of cooling equipment in the past few years. Artificial connections are frequently implemented in the heating industry for the purpose of analyzing and boiling, estimating the effectiveness of radiator mechanisms, or controlling the behaviour of heating equipment. Consequently, the investigators used the artificial neural network (ANN) in order to do an analysis of the temperature transfer information, determine the coefficient of warmth transfer, and manage the heating element. The movement of heat from one location to another is a critical step in the process of either central heating or central cooling an item. The considerable heat that is generated must be reduced to a reasonable level or dissipated in order for a device or system to function at its highest possible standard. In order to minimize heat on equipment, such as processors, and in a number of sectors, such as automotive and technology, fluid refrigerants have historically been employed ^[1]. However, traditional liquid refrigerants have a weak heat transfer ^[2,3,4]. Because of this, there is a greater potential for creativity and technological improvement in the field of cooling. The use of nanofluid, which is a fluid that includes nanoparticles, is one of the methods that have been established to improve the efficiency of heat transfer in liquid refrigeration systems ^{[3,4,5,6,7,8]}. The issue with fluid containing dispersed nanometre particles is coagulation, deposition, and flow line blockage. The aforementioned issues might be partly addressed by employing nanomaterials embedded as micro particles. Nanofluids are extensively employed in semiconductors, automotive, power, and universal healthcare industries, among others, owing to their appealing high-temperature, visual, and electromagnetic characteristics ^[9,10,11].

An innovative approach of mixing two or three kinds of metallic nanoparticles within the base fluid to boost thermal performance has been developed. Hybrid nanofluid mixes metallic nanoparticles in base fluid. Using hybrid nanofluids to chill or warm machines enhances heat transmission ^{[12,13,14,15,16,17,18]}. Many researchers have studied nanoparticles combinations in base fluids to create a refrigerant with superior thermal conductivity. Temperature distribution and heat capacity of hybrid nanofluid are influenced by particle diameter, volume fraction, thermal efficiency, and viscosity ^[19,20]. Nanofluid improves material impact energy and is an excellent refrigerant. Among the several metals that are used in the study of nanofluids, alumina, due to its chemical stability and increased mechanical power, is often employed in the investigation of hybrid nanofluids. Other types of cobalt alloys are put to use in devices like jet turbines, which need high hardness. Electroplating is sometimes performed using cobalt metal because of its excellent standards, high level of hardness, and excellent corrosion resistance.

The investigation of electromagnetic flows in liquids subjected to a magnetic field is known as magnetohydrodynamics, or simply MHD. The scientific name for this phenomenon is magneto-fluid dynamics. Examples of magneto-fluids include plasmas, ions, seawater, and liquid crystals. Hanne's Alfvin was given the Nobel Prize in Physics in 1970 for his ground-breaking work in the field of MHD. The magneto-nanofluid combines the characteristics of magnets and liquids into a single substance. The magnetic impacts not only radicalize the particles in the liquid system but also create a modification in heat transfer. Additionally, the magnetic effects cause the absorption to be reconstructed. Because they contain more viscous body cells than non-malignant nanoparticles, and because the cells improve blood circulation, magnetic nanoparticles are used in the biomedical industry ^[21,22,23]. MHD has received a lot of attention in astronomy, healthcare, optical transplants, metal processing, geophysical disciplines, and petrochemical engineering ^{[24,25,26,27]}.

According to the current literature, no research has been done to analyse the concept of magnetohydrodynamics (MHD) over a stretched surface with the presence of a porous medium and linear thermal radiation. The equations that are created are in the form of PDEs. To convert the PDE into ODEs, a self-similarity transformation is used. After applying transformations, for the graphical purpose, we have used the MATLAB solver in the Fehlberg scheme. In this study, we employ blood as a base fluid and Gold-Fe₃O₄ as a hybrid nanofluid. This model plays an essential role in aerospace machinery, nuclear reactor cooling and vehicles. In the results and discussion section, graphs for different physical significance are given. As a result, the researchers are confident that the new study is unique, will have a considerable influence in the domains of engineering and mathematics, and has the opportunity to encourage new investigators.

2.   Mathematical model

We consider an incompressible Casson type viscous nanofluid flow over a heated stretching surface. In addition to a stretched surface, a steady two-dimensional flow is examined in a hybrid nanofluid. The velocity components u and v are measured along the x-axis and the y-axis, respectively; U = axis the stretching velocity. Furthermore, the temperature of the sheet as well as the temperature of the free stream is represented by the symbols T_f and T_∞, respectively, which are presented in Figure 1. Within the framework of the velocity equation and temperature equation, Casson fluid, porous medium, MHD, heat generation and thermal radiation are taken into account. Two distinct kinds of nanoparticles, namely Gold-Fe₃O₄ are suspended in the base fluid blood, which is demonstrated in Table 1.

Figure 1.  Flow geometry of the problem.

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Table 1.  Blood/Gold-Fe₃O₄ nanoparticles of thermophysical properties ^[34,35,36].

Property	Blood	Gold	Fe3O4
Density $\rho \; (kg{m^{ - 3}})$	1050	19,300	5200
Specific heat ${C_p}\; (Jk{g^{ - 1}}{K^{ - 1}})$	3617	129	670
Heat conductivity ${k_f}\; (W{m^{ - 1}}{K^{ - 1}})$	0.52	318	6
Electrical conductivity $\sigma \; {(\Omega m)^{ - 1}}$	0.8	$4.1 \times {10^6}$	25,000
Prandtl number (Pr)	21

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The governing flow equations are constructed as ^[28,29,30]:

By the Rosseland approach, we have

By applying the Taylor's series expansion of ${T^4}$ about ${T_\infty }$ and neglecting terms having higher order, we obtain

Putting Eq (5) into Eq (3), we get

The boundary conditions are:

The self-similarity variables are defined as:

Thermophysical properties of hnf are

The modified dimensionless ODEs, Eqs (2) and (6) are converted by using Eq (8).

The boundaries of the change are described as:

Note that $M = \frac{{{\sigma _f}{B^2}}}{{{\rho _f}a}}$ is the magnetic field parameter, $Pr = \frac{{{\mu _f}{{\left({{c_p}} \right)}_f}}}{{{k_f}}}$ is the Prandtl number, $Rd = \frac{{4{\sigma ^*}T_\infty ^3}}{{{k^*}{k_f}}}$ is the radiation parameter, $Ec = \frac{{{U^2}}}{{{c_p}\left({{T_f} - {T_\infty }} \right)}}$ is the Eckert number, $K = \frac{{{\upsilon _f}}}{{a{K^ * }}}$ is the porosity parameter, and $Q = \frac{{{Q_0}}}{{a{{\left({\rho {C_p}} \right)}_f}}}$ is the heat absorption/generation.

The dimensional form of ${C_f}$, and Nu are defined as

where shear stress ${\tau _w}$ is

where heat flux ${q_w}$ is

The non-dimensional form of Eqs (13) and (14) are:

Where $R{e_r}$ is the local Reynolds number.

3.   Numerical scheme

In order to solve the nonlinear ODE system consisting of Eqs (10) and (11) and the boundary conditions, Eqs (12) all that is required is to enter data into the Fehlberg method function that is included inside MATLAB. At the beginning of the procedure, Eqs (10)–(12) are transformed into a collection of related first-order equations. This is done in order to make it possible for the Fehlberg technique to be used in the MATLAB (R2020b) solver. The numerical findings are shown for a variety of values of the factors that have been taken into consideration in this particular research endeavor. In the present model, we used the step size $\varDelta \eta = \mathit{0.01}$ and which is the standard used for convergence. When performing the inner iteration, criteria for convergence are consistently employed. In the present investigation, a relative tolerance of error ${10^{ - 8}}$ is taken into account.

4.   Design of artificial neural network modelling

An artificial neural network (ANN) is a type of concurrently processed design made up of various levels of relatively straightforward producers named neurons that are all highly linked to each other. An ANN is a computational framework and high-tech computer instrument which employs neuro-computing for handling data. The ANN is capable of both matching patterns and neural network learning. The human nervous system served as the inspiration for the development of the aforementioned model, which has computational features that are identical to those of the actual physical cell. Our brain has the capacity to perform enormous computations and makes use of the trillions of sensory neurons that are present in the skull. In contrast with neurons in real life, artificial neural networks retain input as weighted interconnections, rather than storing it in storage. When employing this method, the establishment of structural characteristics may be accomplished more quickly and with a reduced number of calculations. Both input and output levels, in addition to a secret layer that comprises components that turn information into output so that the layer that generates results may utilize the outcome to make decisions, constitute the components that make up artificial neural networks (ANNs). Artificial brains have the ability to extract significant information from erroneous data as they are used to identify patterns and generate characteristics which are difficult for both computers and people to characterize. The design of the ANN is presented in Figure 2(a). The ANN algorithm was tested, proven, and trained using different sets of data. Seventy percent of the information provided was employed for training, fifteen percent for validation, and fifteen percent for testing the findings of the present model. Figure 2(b) demonstrates a back propagation of the neural network. Figure 3 illustrates the outcomes of the ${C_f}Re_r^{{\raise0.7ex\hbox{$\mathit{1}$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$\mathit{2}$}}}$ measurements that were performed on the ANN model in the ANN performance, initial training and Regression lines for skin friction. Figure 4 shows that the outcomes of the $N{u_r}Re_r^{ - {\raise0.7ex\hbox{$\mathit{1}$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$\mathit{2}$}}}$ measurements that were performed on the ANN model in the ANN performance, initial training and Regression lines for Nusselt number. All of the necessary parameters are provided to the machine learning models so that they may represent the intricate relationship among the two sets of data. The ANN algorithm's outputs are quite consistent with the computed numbers ^[31,32,33].

Figure 2.  (a) The design of the ANN; (b) Back propagation of the neural network.

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Figure 3.  (a) ANN performance; (b) ANN training; (c) Regression lines for ${C_f}Re_r^{{\raise0.7ex\hbox{$\mathit{1}$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$\mathit{2}$}}}$.

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Figure 4.  (a) ANN performance; (b) ANN training; (c) Regression lines for $N{u_r}Re_r^{ - {\raise0.7ex\hbox{$\mathit{1}$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$\mathit{2}$}}}$.

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At the moment, the ANN model MLP that is both the most popular and exciting is the feed-forward neural network (FFNN). This model is based on a perception architecture that has numerous layers. During the process of determining the error in the network's output, the method known as back propagation may be used to rearrange the relative weights of each of the cells. Back propagation delivers the greatest outcomes when it comes to training FFNNs. This is the case by a considerable margin and by a significant amount of success. In general, it may assist in reducing the amount of mistakes that occur.

The j^th hidden neuron receives the following as its net input:

In this notation, x_i represents the input layer of the i^th node, a_j represents the hidden layer of the j^th node, and W1_ji stands for the linking weight between x_i and a_j.

Here is a representation of the j^th hidden node in the output:

The next equation pertains to the layer's k^th output node:

If node k in the output layer is connected to node j in the hidden layer, then the weight W2_kj indicates the strength of that relationship. In addition, the word "bias" is denoted by the variable "b_k" and is located at the k^th node of the output layer.

5.   Results and discussion

The non-dimensional Eqs (10) and (11), along with the boundary Eq (12), might be answered numerically using the help provided by the MATLAB solver according to the Fehlberg method. The numerical values of the non-dimensional variables were taken and used for evaluation. The precise results are shown in Tables 2–4, which demonstrate the variations in the skin friction coefficient along with ANN outputs. The findings of both investigations were determined to be fairly accurate.

Table 2.  Comparison table for various Pr of the current study.

Pr	Ali et al. [28]	Present results
0.7	0.4560	0.455322
2.0	0.9113	0.910337

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Table 3.  Values of the ${C_f}Re_r^{{\raise0.7ex\hbox{$\mathit{1}$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$\mathit{2}$}}}$${\mathrm{C}}_{\mathrm{f}}\mathrm{R}{\mathrm{e}}_{\mathrm{x}}^{1/2}$ for various parameters.

M	K	Ec	Numerical model ${C_f}Re_r^{{\raise0.7ex\hbox{$\mathit{1}$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$\mathit{2}$}}}$	ANN model
0.1	0.3	0.5	1.528030	1.52803002
0.1	0.3	0.5	1.259748	1.25974804
0.1	0.3	0.5	1.791082	1.79108292
0.1	0.3	0.5	1.125601	1.12560183
0.1	0.3	0.5	1.033058	1.03305894
1.0	1.0	1.0	1.041283	1.04128301
0.1	0.3	0.5	1.053002	1.05300224
0.1	0.3	0.5	1.010293	1.01029305
0.1	0.3	0.5	1.088258	1.08825804
0.1	0.3	0.5	1.564001	1.56400162
0.1	0.3	0.5	1.033502	1.03350294
0.1	0.3	0.5	1.932038	1.93203882

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Table 4.  Values of $N{u_r}Re_r^{ - {\raise0.7ex\hbox{$\mathit{1}$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$\mathit{2}$}}}$ for various parameters.

Pr	M	Rd	Q	Numerical model $N{u_r}Re_r^{ - {\raise0.7ex\hbox{$\mathit{1}$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$\mathit{2}$}}}$	ANN model
0.62	0.5	0.7	0.9	0.0436	0.043602
0.62	0.5	0.7	0.9	0.0772	0.077273
0.62	0.5	0.7	0.9	0.3800	0.380065
0.62	0.5	0.7	0.9	0.1024	0.102449
0.62	0.5	0.7	0.9	0.8025	0.802538
1.0	1.0	1.0	1.0	0.3900	0.390002
0.62	0.5	0.7	0.9	0.5993	0.599330
0.62	0.5	0.7	0.9	0.1811	0.181125
0.62	0.5	0.7	0.9	0.4326	0.432602
0.62	0.5	0.7	0.9	0.7381	0.738116
0.62	0.5	0.7	0.9	0.8324	0.832436
0.62	0.5	0.7	0.9	0.9380	0.938016

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The influence of M on the velocity sketch is seen in Figure 5(a). It can be seen that the velocity profile has a decreasing trend for higher M values, which indicates that the value of M is increasing. Physically speaking, an increase in M causes a Lorentz force to be produced, which retards the movement of the liquid. As a consequence of the fact that the Lorentz force is antagonistic to the motion of fluids, the flow velocity decreases as a consequence of the increased resistance, which in turn leads to a declined in the velocity profile. When increases in the K values improves the velocity outline, which is demonstrated in Figure 5(b), physically by raising the K values, a porous substance becomes more difficult to flow across, retarding the flow of fluids. As a direct and immediate consequence of this, the dimension of the outermost boundary layer is decreased.

Figure 5.  (a) Pictogram of $M$ on $f'\left(\eta \right)$; (b) Pictogram of on $K$$f'\left(\eta \right)$.

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The effect of the magnetic field on the temperature profile is seen in Figure 6(a). It states that growing the M parameter increased the energy outline. This enhancement is due to the fact that the addition of M into an electrically conducting material produces a resistive Lorentz force. This kind of force has the ability to raise the fluid's temperature. So, this is the reason that the temperature profile enhances. Figure 6(b) shows the radiation parameter temperature profile features. Physically, thermal radiation is the radiant energy released by all generators above zero. Electromagnetic radiation from heated surfaces rotates atoms in matter, generating kinetic energy. The graph demonstrates how increasing radiation enhances heat transmission. When Rd is raised, more heat is transmitted into the liquid, strengthening the thermal barrier. Since the mean absorption coefficient drops as radiation growths, temperature also increases. The effect of the Eckert number on the energy profile is seen in Figure 6(c). It states that growing the Ec values enhances the energy outline. Physically, Ec generates the frictional force that causes the fluid temperature increased. The influence of Pr on the energy profile is seen in Figure 6(d). It states that increasing the Pr values enhanced the temperature profile. The thermal dissipation of the liquid essentially reduces, resulting in a declined in the energy profile.

Figure 6.  (a) Pictogram of $M$ on $\theta \left(\eta \right)$; (b) Pictogram of $Rd$ on $\theta \left(\eta \right)$; (c) Pictogram of $Ec$ on $\theta \left(\eta \right)$; (d) Pictogram of $Pr$ on $\theta \left(\eta \right)$.

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Figures 7(a) and (b) display the influence of $\Gamma$, K, and M on the ${C_f}Re_r^{{\raise0.7ex\hbox{$\mathit{1}$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$\mathit{2}$}}}$. It shows that the skin friction factor is decreasing for the greater values of K, while we observed the reverse nature in Figure 7(b). Figures 7(c) and (d) demonstrate the inspiration of the Ec, Rd and magnetic field on the $N{u_r}Re_r^{ - {\raise0.7ex\hbox{$\mathit{1}$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$\mathit{2}$}}}$ profile. It shows that the Nu enhanced in all cases with greater values of Rd and the magnetic field. Streamlines, when applied to the study of fluid dynamics and the representation of flow, have a number of advantages that, considered together, allow them to be efficient tools for comprehending and studying fluid movement. Figures 8(a) and (b) exhibit magnetic parameters for various values of M = 0.2, 0.4 influences on streamline plots. Magnetic parameter strength draws electrical conductivity molecules more toward the main stream.

Figure 7.  (a) Sway of $M\; and\; \Gamma$ on ${C_f}Re_r^{{\raise0.7ex\hbox{$\mathit{1}$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$\mathit{2}$}}}$; (b) Sway of $M\; and\; K$ on ${C_f}Re_r^{{\raise0.7ex\hbox{$\mathit{1}$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$\mathit{2}$}}}$; (c) Sway of $M\; and\; Rd$ on $N{u_r}Re_r^{ - {\raise0.7ex\hbox{$\mathit{1}$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$\mathit{2}$}}}$; (d) Sway of $M\; and\; Ec$ on $N{u_r}Re_r^{ - {\raise0.7ex\hbox{$\mathit{1}$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$\mathit{2}$}}}$.

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Figure 8.  (a) Stream lines for $M = 0.2$; (b) Stream lines for $M = 0.4$.

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6.   Conclusions

An investigation of a numerical model used in the MHD of a hybrid nanofluid composed of Gold-Fe₃O₄ and blood was carried out in the present investigation. The mathematical system's PDEs are converted into matching ODEs that show the issue's dynamics and the results are numerically solved by using the Fehlberg technique. The findings were displayed in a various visual style, such as streamlines, surface plots in three dimensions and 2-dimensional figures. The research produced a number of interesting findings, which are listed below:

● Obtaining accurate forecasts for reality via the integration of neural networks and artificial intelligence.

● Velocity profile decreases for the greater values of the magnetic field and porosity parameter values.

● The temperature profile improved with an improvement in the radiation parameter.

● The skin friction factor slowly increased for the larger values of the porous parameter.

● The Nu profile enhanced, for the increasing values of M and Rd.

● Streamlines are oscillating character, which is necessary for magnifying the magnetic field parameter.

Author contributions

Conceptualization, GR and NAS; Data curation, GR and SA, Formal Analysis, SA; Investigation, NAS; Methodology, SA and NAS; Software, SA; Validation, NAS; Writing–original draft, GR; Writing–review & editing, SA and NAS.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This project was supported by Researchers Supporting Project number (RSPD2024R909), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

All authors declare no conflicts of interest in this paper.