Artificial intelligence driven heuristics approach to analyze entropy optimized MHD flow of non-linear radiative hybrid nanofluids considering vertical thin needle

Abstract

The current study aims to investigate entropy generation in a two-dimensional magnetic Williamson hybrid nanofluid flow that contains titanium oxide and cobalt ferrite nanoparticles and is subjected to surface-catalyzed reactions via a thin vertical needle by using Levenberg-Marquardt backpropagated neural networks. The properties of heat transport are elaborated by considering the effects of viscous dissipation and joule heating. Additionally, the effects of homogeneous-heterogeneous response, thermal radiation, and thermal stratification are considered. The system of coupled ordinary differential equations is dimensionless by the use of suitable similarity variables. By using “ND-solve” method in Mathematica software the graphical results with matrix data set is generated for $f^{\prime }\left(\eta \right)$ , $\theta \left(\eta \right)$, $g_1\left(\eta \right)$ and $N_G\left(\eta \right)$. Further, the obtained matrix data set from Mathematica software is used in MATLAB software to achieve the required graphical for $f^{\prime }\left(\eta \right)$ , $\theta \left(\eta \right)$, $g_1\left(\eta \right)$ and $N_G\left(\eta \right)$. The 86 samples are obtained by using artificial intelligence neural networks on Williamson hybrid nanofluid. The total 86 samples are divided into three types of data with 60 samples are used for training, 13 samples for testing and 13 samples for validation. The increase in the $f^{\prime }\left(\eta \right)$ profile with rising values of $\lambda$ is attributed to enhanced stretching or surface tension effects, which increase the momentum gradient near the boundary, and the moderate absolute error values reflect the artificial intelligence neural networks’ ability to handle such sharp gradients. The observed decrease in $f^{\prime }\left(\eta \right)$ with increasing $P_m$ is due to the influence of magnetic fields, which introduce Lorentz forces that resist fluid motion, and the consistently low absolute error shows that the model accurately captures this Magnetohydrodynamics behavior. The decline in $\theta \left(\eta \right)$ as $\mathrm{Pr}$ increases is explained by reduced thermal diffusivity at higher Prandtl numbers, leading to thinner thermal boundary layers, and the slightly higher absolute error reflects the stronger nonlinearity in thermal conduction. Conversely, the increase in $\theta \left(\eta \right)$ with greater $R_d$ values indicates enhanced internal heat generation or radiative effects, which elevate the temperature field; the wider absolute error range in this case results from the compound effects of heat generation and diffusion. The decrease in the concentration profile $g_1\left(\eta \right)$ with increasing $S_c$ is consistent with reduced mass diffusivity, leading to sharper concentration gradients, and the small absolute error confirms the model’s effectiveness in resolving mass transport dynamics. Similarly, the decreasing trend of $g_1\left(\eta \right)$ with higher $K_c$ arises from intensified chemical reactions that consume species and lower concentration levels, and the very low absolute error illustrates the artificial intelligence neural networks’ ability to model chemically reactive flows. The increase in entropy generation $N_G\left(\eta \right)$ with growing $B_r$ is due to viscous dissipation effects that contribute additional irreversibility to the system, and the relatively larger absolute error reflects the complexity in modeling entropy dynamics. Lastly, the rise in $N_G\left(\eta \right)$ with increasing $P_m$ is a consequence of stronger magnetic-induced Joule heating, and the absolute error remains within a tight bound, verifying the artificial intelligence neural networks’ capability to handle thermodynamic influences from electromagnetic effects. Overall, the absolute error values across scenarios indicate robust artificial intelligence neural networks generalization and precise modeling of highly nonlinear coupled physical phenomena.