Artificial neural network paradigm of magneto-thermal behavior in tangent hyperbolic hybrid-nanofluid flow

Abstract

The study considers the flow behavior of a Magnetohydrodynamic (MHD) tangent-hyperbolic hybrid-nanofluid as it flows on an exponentially stretched surface. Boundary slippage, Joule heating, changes in thermal radiation and convective effects have impacts on thermal exchange rate. The fluid used for these experiments is ethylene glycol (EG) and copper with alumina nanoparticles to enhance heat transfer. The dictating partial differential equations (PDEs) transformed into ordinary differential equations (ODEs) employing suitable similarity techniques. Artificial Intelligence (AI) based Machine Learning (ML) Levenberg Marquardt Algorithm (LMA) is used to determine the impact of parameters involved in MHD tangent hyperbolic nanofluid flow. The influence of flow rate and thermal heat transfer are studied through graphical analyses. A rise in magnetic parameter and ratio of elastic to viscous drags causes decline in flow. While the Lorentz parameter and ambient temperature difference boosts the temperature profile on rising. The ANN-LMA had mean-squared-error in the range of 7.62 × 10^–11 to 4.59 × 10^–10 on the six cases and also converged within 79–1000 epochs based on the experiment and good regression fits were displayed. This research is useful in paper production, cooling metal sheets, and crystal growth. The novelty lies in applying LMA-based ANN to approximate the model which has not been reported previously.

Introduction

Recent studies have looked at various non-Newtonian fluid (NNF) models to find ways to enhance flow in engineering and science areas. What is important about the tangent hyperbolic fluid model is that it portray pseudoplastic and non-Newtonian behavior. Being a member of the class of rate-type fluids, this model supports equations under high and moderate tensile stresses. The tangent hyperbolic fluid model has extensive applications in industrial and laboratory use, ranging from food items like sauces, melted cheese, whipped cream, ketchup, to biomedical materials like blood.

Hybrid nanofluids (dispersions of a base (one or more) fluid with two or more nanoparticle species) can have enhanced thermal conductivity and controllable viscosity than single-particle nanofluids, making them useful in heat transfer engineering applications. Ethylene glycol (EG) mixed with high conductivity (metallic) particles like copper (Cu) with aluminum oxide provide a realistic working fluid in cooling applications; experimental evidence has demonstrated that with hybrids, thermal transport increases with loading levels compared to single-component nanofluids with the same loading levels. The reason we select as a prototype hybrid nanofluid to explore the impacts of MHD and radiant heat. Several applications of nanofluids are mentioned in Fig. 1.

Fig. 1.

Applications of HNF.

Significant works in this field involve studies by researcher^1–4 who computed the flow of tangent hyperbolic fluid along a vertically stretched barrel. Scholars studied the impact of the Dufour and Soret effects on the second-grade nanofluid into a turbulent MHD system^5,6.

Conductive fluid creates an electric current when exposed to a magnetic force⁷, which can be utilized to control fluid flow. MHD finds many applications ranging from the generation of power to medical procedures such as cancer therapy and treatment of asthma⁸. Rashidi and Reddy et al.^9,10 studied the convective heat transfer of MHD in a squeezed sheet and while other evaluated thermal exchange impact on MHD flow caused by an inclined plate. El-Aziz et al.¹¹ discussed thermal diffusion and heat flow with radiation while Ashornejad et al.¹² analyzed nanofluid over stretching cylinder. Raptis et al.¹³ worked on the hydro-flow between porous plates.

Momentum and energy transfer in peristaltic flow over straighten surfaces is of vital importance in industries involving polymer, metallurgy, and glass production. Pop et al.^14,15 investigated thermal radiation in peristaltic flow, and studied electromagnetic radiation effects on MHD flow over moving plates. Classic fluids like ethylene glycol, water, and oil are poor thermal conductors, which hinders their effectiveness. To counter this, nanofluids containing particles that have dimensions in nanometers have been engineered to promote heat transfer, as discussed in^16,17.

Nanofluids provide enhanced thermal performance and are applied in microelectronics, fuel cells, and in biomedical uses. Hybrid nanofluids, which are a combination of two or more types of nanoparticles, have been investigated in¹⁸ to further enhance heat transfer characteristics. Researches in^19–23 have used similar AI-based computational methodology for evaluation of complex nonlinear models. Dynamic systems tend to have uncertainties caused by measurement errors, environmental factors, and limited data.

Bartwal et al. extensively explored tangent fluids for various computational and physical frameworks. Their studies includes modified Fourier’s law in fuzzy²⁵, magnetic effect and anisotropic slip over rotating disk²⁶, MLP-based analysis of thermo-material exchange²⁷ and MHD stagnation flow in fuzzy domains²⁸. Shamshuddin et al. explored Casson ternary nanofluid²⁹, HNF in rocket nozzles using adomians³⁰, Darcy-Forchheimer radiated flow³¹, chemical reaction on couple-stress flow³² and flow in microchannels³³. Bafe et al. studied MHD Carreau nanofluid in³⁴, Cattaneo-Christove thermal flux in Maxwell in³⁶ and non-Forier-Fick bioconvective nanofluid in³⁷. Raja et al.³⁵ utilized back-propagated neural network for tangent hyperbolic fluids. Ali et al.³⁸ implemented Levenberg–Marquardt for MHD tangent hyperbolic flow.

Although tangent hyperbolic and hybrid-nanofluid models have already been worked out, there are still relatively few studies on turbulent tangent hyperbolic hybrid nanofluid flow over an exponentially stretched surface, taking into consideration at the same time MHD and Joule heating, slip, and thermal radiation. This combination has a direct application to real-world processes that require the use of electromagnetic control and an increase in thermal transport, including metal-sheet cooling and crystal growth. This paper thus (i) finds similarity ODEs of an hybrid nanofluid in MHD and thermal radiation; (ii) uses an ANN trained by Levenberg–Marquardt algorithm to estimate the resultant velocity and temperature fields; and (iii) measures performance our trained ANN is within the range of to and with a healthy convergence diagnostic plot (see Table 3, and Figs. 5, 6, 7, 8, 9 and 10). These findings are a computationally efficient surrogate in parametric studies, and they also point to the physical trends (magnetic, elastic and radiation effects) that are of interest in engineering applications.

Table 3.

Convergence parameters.

Scenario	M.S.E. data			Performance Grids	Gradient Grids	Mu Grids	Closing Epoch	T/Sec
	Training	Validation	Testing
1	1.9E-10	2.23E-10	3.149E-10	1.9E-10	9.97E-08	1.00E-09	1000	01
2	4.59E-10	4.86E-10	7.59E-10	4.59E-10	9.79E-08	1.00E-09	80	01
3	7.62E-11	2.81E-10	2.61E-10	7.62E-11	9.83E-08	1.00E-09	79	01
4	3.45E-10	9.12E-10	4.91E-10	3.45E-10	9.73E-08	1.00E-09	90	01
5	2.06E-10	7.51E-10	3.37E-10	2.06E-10	9.74E-08	1.00E-09	92	01
6	1.98E-10	5.63E-10	9.59E-09	1.98E-10	9.94E-08	1.00E-09	85	01

Fig. 5.

Training, Validation, and Testing curves for HNF flow.

Fig. 6.

Training states of LMA for HNF flow.

Fig. 7.

Error histograms of LMA for HNF flow.

Fig. 8.

Regression analysis of LMA for HNF flow over stretched porous sheet.

Fig. 9.

Comparison curves for fitness of LMA for HNF flow.

Fig. 10.

AI-generated Solutions and difference curves of HNF.

The principal novelties and objectives of this study are: (1) to come up with an unsteady MHD tangent-hyperbolic hybrid-nanofluid flow over exponentially spreading sheet involving slip, Joule heating, and radiations; (2) to generate an ANN-LMA surrogate that can perfectly replicate velocity and temperature fields in the parameter space; and (3) to examine and interpret the physical effects of magnetic, elastic (Weissenberg), and radiations parameter on flow and heat transfer and give it advice in terms of the engineering application.

Mathematical modeling

The study looks at two-dimensional (2D) fluid flow when a nanofluid moves through the equilibrium zone of a tangent hyperbolic along a sheet. The x-axis runs parallel to the stretch on the surface, and the y-axis runs perpendicularly to it. Flow of viscous fluid is observed within the zone . The magnetic field, given by , is used in opposite to the flow, as shown in Fig. 2. Because of the low Reynolds assumption, the resulting magnetic field due to induction is assumed negligible, and there is no electric field. Nevertheless, the Joule heating effects are included in the analysis. With these assumptions, the dictating equations for heat and boundary constraints flow are expressed as²⁴.

1

2

3

Fig. 2.

Geometry of the problem.

The Rosseland approximation is used for radiative heat flux :

Limitations are given accordingly,

4

The starting expression in Eqs. (1)–(3) are transient (unsteady) and convectional transport, the second are viscous diffusion of momentum by the viscosity of the nanofluid, and the third are the non-Newtonian non-hyperbolic constitutive terms that represent the dependence of the stresses on the shear. M in the term containing M (Lorentz force) is that causes resistance to the flow in MHD and Joule heating appears in the energy equation as internal heat generation proportional to the electrical conductivity, the square of magnetic field. The Rosseland assumption is added to include the radiation term to take into consideration the effects of radiation on the temperature. The interpretations of similar terms are applied in the earlier research on MHD and tangent-hyperbolic fluids^1,3,7,24.

The velocity components along the x and y-axis are u and v. Here, represents the electrical conductivity of hybrid nanofluid, σ ∗ is the Stefan–Boltzmann parameter, and is ambient thermal reading. The thermal conductivity is given by , while T denotes the fluid temperature. The constant refers to a material property, is the density, and is the kinematic viscosity. denotes the reference velocity, and denotes the specific heat capacity of hybrid nanofluid. is the convective fluid temperature under the sheet, and k* is the mean ratio of the absorption coefficients. The convective heat transfer coefficient is denoted as .

For transformation of the governing equations (Eqs. (1)–(3)) and their respective boundary conditions into non-dimensional ODEs, the following similarity transformations are employed:

The initial stretching rate symbolized by c and similarity conditions are given²⁴,

5

The transformation is chosen to collapse PDEs into system of ODEs^7,24.

Definitions and non-dimensionalization. In order to prevent confusion, we clearly define dimensional variables and the similarity scales employed: . The non-uniform magnetic strength is in the form of the Lorentz body force in the momentum equation. and (z) at least in the assumptions of one-dimensional transverse field leads to . We establish the non-uniform magnetic parameter which is a non-uniform part of the magnet which is dimensionless , and non-dimensionalization defines the magnetic interaction parameter (which is often a number of the type of a Hartmann number) H as .

The boundary conditions are no-slip velocity along with partial slip (e) along the sheet, a prescribed surface temperature ( Bi at the wall) and ambient conditions as . These are physically accounted by finite slip as a consequence of surface coating, applied thermal conditions, and treatment to free-stream velocity and ambient temperature well out of the sheet.

By introducing appropriate dimensionless parameters that represent the volume fractions of , the mathematical model is reformulated and is symbolized by , respectively. Below are the converted reformations.

6

7

where the conditions are:

8

The condition is that of a partially-slipped stretching sheet (under the influence of surface coats or lubrication layers) on a wall; a specified surface temperature (or thin-film convection characterized by Biot number Bi) at the wall; approached by the ambient velocity/temperature as . Such conditions are comparable to experimental arrangements of polymer sheet cooling and thin-film crystal growth in the limit where finite slip and convective heat transfer is taking place at the interface.

The variables are as,

9

The parameters used are defined as: Power law index, unsteady parameter, slip parameter,

Wessinberg number, Biot characteristic, Magneticconstant, Eckert ratio,

Radiation parameter, and temperature difference. We also define the.

were,

The values are taken from²⁴ with reported uncertainties of ± 3% in thermal conductivity and ± 1% in density. Numerical values of thermophysical properties are detailed in Table 1.

Table 1.

Copper thermo-physical properties with Aluminum Oxide and Ethylene Glycol²⁴.

Physical properties
Ethylene Glycol (EG)	1114	2415	0.252
	8933	385	401
	3970	765	40

The dimensionless form will be:

where the local Reynolds number is represented by . Further, the values are.

Solution methodology

The research depends on a stochastic computation process that enhances performance. LMA is relied upon for analyzing Artificial Intelligence. By putting in similarity parameters, the model of P.D.E.s transforms into a nonlinear system of ODEs. This is then coded into Mathematica. A set of artificial data was made. By varying velocity and temperature parameters, the study gets four resultant velocity and temperature profiles for each influencer. A total of six scenarios are tested, and each consists of 3 for every combination of flow rate portfolio and temperature record. In order to use MATLAB for artificial neural networks, the data is separated into three groups named test, validation and training sets. 15% of each separate set is set aside for testing and validation and the rest 70% goes toward training the neural network. There are 10 input neurons and 16 output layers, input neurons corresponds the dimensionality of the input data and these were selected to achieve optimal performance. The output depicts the velocity and temperature profiles. The design of the neural scheme used in A.I. evaluation is shown in Fig. 3.

Fig. 3.

LMA Neural Network.

Table 2 describes the different scenarios considered in this article along with their variations with respect to the velocity and temperature. In total six scenarios are considered with four cases in each scenario. The table reflects the variations for magnetic parameter M, Weissenberg number We and non-uniform magnetic field B for the velocity while the variations in Temperature T, B and We are studied against the temperature difference .

Table 2.

Numerical scenarios with cases.

Scenarios	Cases	Parameters
		M	We	B
S-1 Variation of for profile	1	0.5	0.7	0.1	0.9
	2	1.0	0.7	0.1	0.9
	3	1.5	0.7	0.1	0.9
	4	2.0	0.7	0.1	0.9
S-2 Variation of for profile	1	0.4	0.4	0.1	0.9
	2	0.4	0.6	0.1	0.9
	3	0.4	0.8	0.1	0.9
	4	0.4	1.0	0.1	0.9
S-3 Variation of for profile	1	0.4	0.1	0.1	1.0
	2	0.4	0.3	0.1	1.0
	3	0.4	0.6	0.1	1.0
	4	0.4	0.9	0.1	1.0
S-4 Variation of B for profile	1	0.4	0.4	0.1	0.9
	2	0.4	0.4	0.6	0.9
	3	0.4	0.4	1.1	0.9
	4	0.4	0.4	1.6	0.9
S-5 Variation of for profile	1	0.4	0.4	0.1	0.9
	2	0.4	0.4	0.6	0.9
	3	0.4	0.4	1.1	0.9
	4	0.4	0.4	1.6	0.9
S-6 Variation of for profile	1	0.5	0.4	0.1	0.3
	2	0.5	0.4	0.1	0.5
	3	0.5	0.4	0.1	0.7
	4	0.5	0.4	0.1	0.9

The table shows convergence parameters and mean square errors that LMA generates. Errors from order 10^–10 to 10^–13, the number of epochs and the Mu and gradient grids in order 10^–08 can be used to judge how close the procedure is to the right outcome. The schematic flowchart is demonstrated in Fig. 4.

Fig. 4.

Flow chart of problem evaluation.

Details of implementation A Levenberg–Marquardt algorithm (LMA) trained feed-forward neural network was used to estimate the mapping of the solution between the nondimensional parameters and the velocity and temperature profiles. The data set was created by the transfiguration of the ODE solutions found in MATHEMATICA (the cases presented at Table 2) and was then fed into the ANN toolbox of MATLAB where they were trained. The data were distributed into training (70%), validation (15%) and testing (15%) sets. The loss function was the mean squared error (MSE) and the activation of the hidden layer(s) was the sigmoid, and the early stopping was defined as the failure of validation as indicated in the training-state plots (Fig. 6). During training, regularization and LMA damping parameter (mu) were observed (see Table 3). This is the procedure that enables the ANN-LMA to learn the nonlinear mapping effectively without necessarily having domain discretization, and diagnostics (error histograms, regression plots) that indicate the discrepancy of the model.

Strengths of ANN-LMA: ANN-LMA method has a number of practical benefits.

• (i)The ANN-LMA method directly approximates the parametric mapping of control parameters to profiles without the need of mesh generation or repeated solution of ODEs per parameter set.
• (ii)The ANN-LMA is sensitive to strong nonlinearities and noisy data.
• (iii)The parametric study can be rapidly evaluated by the ANN-LMA model itself (no re-integration is necessary)

The parametric model can be assessed using diagnostics Such benefits have been observed in recent ANN-LMA and ML-based heat-transfer papers.

Results and discussion

This study seeks to answer: How magnetic and elastic (Weissenberg) effects interact in an exponentially stretched sheet tangent-hyperbolic hybrid nanofluid? Does ANN-LMA surrogate predictable to solve ODEs over large parameter regimes and give useful parametric sensitivity in cost effective computation? How does this impact on thermal management processes (e.g. crystal growth, thin-film cooling)?

Figs. 5, 6, 7, 8, 9 and 10 introduces this article by showing hyperparametric plots and the performance of AI computation. All the neural network training plots represent significant aspects of model training. Each scenario is depicted by an individual training curve or bars.

A comparison of training testing and validation curves is presented in Fig. 5 (a-f). Training, testing, and validation are represented by blue, green and red curves, respectively. All the curves follow the same pattern in each scenario and overlap in most cases which is the representation of best training of the neural network (Fig. 6).

Figure 7 (a-f) represents the error histograms of 20 different bins. Each bin shows the data distribution for training, validation and testing with blue, green, and red portions. Minimum or zero error is represented by a yellow line. In all the scenarios, the whole data is symmetric about the zero error which is the best optimization scenario of the neural network evaluation process.

Regression analysis for each case is presented in Fig. 8 (a-f). All the scenarios have data along unit gradient which is an ideal training algorithm. Each case has a different plot for training, validation and testing in blue, green, and red gradients. The fourth slope in this graph is the overall average training gradient.

Figure 9 (a-f) indicates that the target and output curves are close enough, which shows that the ANN recreates the global trends and local variations of the reference solutions, which facilitates faster sweeps of parametric (advantage: fast surrogates to design optimization). On the physical level, low velocity with increasing M (and We) means thicker thermal boundary layers and increased viscous/elastic dissipation; hence, in the field where it is needed to control the rate of flow by using field of magnetism, such findings measure the compromise between flow-stopping and heat generation. These parametric trends are not an artefact of the surrogate but are associated with the governing physics of the PDE as they are determined by a small error of the ANN (see Fig. 10 error panels and Table 3).

Figure 10 (a-l) are graphical results of different scenarios, showcasing AI generated solutions and error estimations. In total six scenarios are illustrated covering four different cases. Each scenario is examined against either the velocity or temperature and their respective error graphs are also presented Table 4.

Table 4.

Validation of Results.

β	Siddique et al.5	Nadeem et al.24	Present
0	1.281809	1.281809	1.281808
0.1	1.253580	1.253580	1.253578
0.2	1.195120	1.195118	1.195117
0.5	0.879835	0.879833	0.879831
0.8	0.397771	0.397767	0.397764
1.2	0.451571	0.451568	0.451565

Physical interpretation: The reason why velocity decreases with strengthening magnetic coefficient M is because the Lorentz force is a resistive body force (against motion) that transforms kinetic energy into thermal energy through Joule heating and not only slows the flow but also increases local temperatures. Improving the Weissenberg number (We) increases the elastic stress levels in the tangent hyperbolic fluid that increases the resistance to flow and thus decreases the thickness of the velocity boundary-layer, at the same time the elastic dissipation can also alter the temperature fields. The non-uniform parameter of the magnetic field B causes spatially varying Lorentz forcing; this can either heating (or localized Joule heating) of the temperature even though it decreases the velocity locally. Such mechanistic explanations are in agreement with the existing literature on MHD and hybrid-nanofluids^7,24 and can be numerically described by our ANN-LMA model.

Conclusion

HNF flow over an exponentially spreading surface is discussed here. The numerous applications of HNF in biomedical and engineering sciences are well renowned. The main observations point out that upward variations in the magnetic parameter M, declines the fluid velocity. The same pattern is depicted with the higher variations of the Wisenberg number We, in both the cases of the fluid velocity and temperature. However, non-uniform magnetic field B presents an opposite behavior for fluid temperature. The decreasing trend of B for fluid temperature with higher values but aligns with fluid velocity at higher rates. Lastly, as the temperature rises, the fluid temperature difference also enhances. Quantitatively, overall values of MSE under the ANN-LMA approach ranged between 7.62 × 10–11 and 4.59 × 10–10 (Table 3) with consistent convergence over 79–1000 epochs across scenarios and thus, the ANN-LMA approach provides very accurate approximations of velocity and temperature profiles of the tangent hyperbolic hybrid-nanofluid model.

Future researches might take the following directions:

• Magnetically-driven (MHD) heat-transfer heat transfer enhancement of EG-Cu-Al2O3 hybrid nanofluids.

• Comparison with computational cost and accuracy of other data-driven solvers (e.g., deep neural networks, Gaussian processes) and classical numerical solvers (bvp4c, spectral methods).

• Generalization to three-dimensional and unsteady geometries, such as porous media, and non-uniform wall temperatures.

• Entropy-generation and exergy analysis to measure thermodynamic irreversibilities of the hybrid-nanofluid systems. Such instructions will enhance practical applicability as well as widen the modelling framework.

This research has been restricted to two-dimensional similarity solutions, and it does not address nanoparticle agglomeration, slip induced turbulence and experimental validation. Future studies are to be done on (i) three-dimensional and transient hybrid-nanofluid flows, (ii) entropy generation and exergy, (iii) ANN architecture optimization against other machine-learning methods, and (iv) laboratory validation of thermal properties and flow predictions.

Abbreviations

u,v

Flow rate constituents (m/s)

ρ

Density (kg/m³)

T

Thermal reading (K)

Cp

Specific thermal capacity (J/kg.K)

σ

Electrical conductivity (S/m)

k_f

Heat conductivity of engine oil (W/m.K)

T_∞

Ambient Temperature (K)

Dimensionless Parameters

ϕ₁,ϕ₂

Volume ratios

B

Non-uniform magnetic field

Slip parameter

M

Magnetic parameter

α

Power law index

B_i

Biot number

σ^∗

Stefan Boltzmann constant

We

Weissenberg number

β

Unsteady parameter

θ_w

Temperature difference

Pr

Prandtl Number

Ec

Eckert Number

N_r

Radiation parameter

f(h)

Position function

HNF

Hybrid Nano Fluids

M.S.E.

Mean Square Error

E.H.

Error Histogram

NNFF

Neural Network Feed Forward

T.V.T.

Training, Validation, and Testing

L.M.A.

Levenberg–Marquardt Algorithm

Author contributions

Tazeen Athar: Conceptualization, Investigation, Visualization Hamid Qureshi: Methodology, Writing draft, Validation Taseer Muhammad: Revision and editing.

Funding

There are no funding details at present.

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Consent to Publish

The authors confirm that this manuscript has not been published elsewhere and is not under consideration by any other journal.