Integration of artificial intelligence into computational fluid dynamics has significantly enhanced the simulation of complex transport phenomena. This review presents a detailed analysis of artificial neural network (ANN) techniques, namely Levenberg Marquardt (LM), Bayesian Regularization (BR), and Scaled Conjugate Gradient (SCG), in the simulation of magnetohydrodynamic (MHD) hybrid nanofluid flows with bio-convection and internal heat generation. Such flows are characterized by strong nonlinearities arising from the synergistic interaction of magnetic forces, nanoparticle-particle interactions, convective effects caused by microorganisms, and heat gradients. Traditional numerical methods, though well-refined, typically face convergence issues, computational costs, and adaptability to highly nonlinear cases. ANN-based models, on the other hand, show remarkable characteristics toward the approximation of intricate physical correlations with enhanced convergence and generalization. This review discusses, in a detailed manner, the theoretical basis, training behavior, prediction accuracy, and computational efficiency of LM, BR, and SCG algorithms in maintaining critical fluid properties like velocity, temperature, and concentration. Special focus is given to the role played by bio-convection in enhancing transport properties and to how ANN techniques effectively model these dynamics with minimal residual error and computational intensity. A comparative study highlights the applicability and uniformity of such neural network algorithms in advanced heat and mass transfer operations with MHD hybrid nanofluids.

This article explores the Ricci tensor of slant submanifolds within locally metallic product space forms equipped with a semi-symmetric metric connection (SSMC). Our investigation includes the derivation of the Chen–Ricci inequality and an in-depth analysis of its equality case. More precisely, if the mean curvature vector at a point vanishes, then the equality case of this inequality is achieved by a unit tangent vector at the point if and only if the vector belongs to the normal space. Finally, we have shown that when a point is a totally geodesic point or is totally umbilical with $n=2$, the equality case of this inequality holds true for all unit tangent vectors at the point, and conversely.

We explore the structure and dynamics of compact stellar structures by assuming anisotropic fluid solutions within the framework of general relativity. Assuming that the interior geometry of compact stellar structures follows the Durgapal–Fuloria spacetime, we obtain exact analytic solutions of Einstein’s field equations under physically consistent boundary conditions. The resulting models satisfy all essential physical requirements, including regularity at the center, positive and decreasing pressure and density, energy and causality conditions, and dynamical stability criterion. To test the viability of our model, we examine ten well-known compact stellar structure candidates-LMC X-4, SMC X-1, Cen X-3, Vela X-1, 4U 1608-52, 4U 1820-30, SAX J1808.4-3658, Her X-1, PSR J1614-2230, and PSR J1903+327-assuming their interiors are described by the Durgapal-Fuloria geometry. Furthermore, we extending the analysis to the slow rotation regime, we incorporate first-order rotational effects to compute the moment of inertia and investigate its dependence on stellar mass and radii. The results confirm that the Durgapal-Fuloria spacetime provides physically acceptable and stable descriptions for realistic anisotropic compact stellar structures, consistent with observed systems.

The present study investigates the generalized dispersion of a solute in an incompressible MHD flow via a rectangular channel with injectable or suctioned walls. The mathematical model of dispersion suggests a distinct type of mass flux expressed as a fractional partial differential equation based on the time-fractional Caputo derivative. The mass flow in the model under investigation is determined by both the concentration gradient and its historical evolution. A constant external magnetic field is provided transverse to the flow direction. The analysis and discussion of the analytical solution for the advection velocity are performed in relation to the Hartmann number and the suction/injection Reynolds number. To determine the solute concentration in space and time, the unstable fractional convection–diffusion equation is analytically solved. The polynomial in the geographic variable y that has coefficients that depend on the spatial variable x and the time t is the analytical solution of the concentration. The effects of the fractional order of the Caputo derivative, Reynolds number, Hartmann number, and Peclet number on the advection–diffusion process are examined using numerical simulations of the analytical solution of the solute concentration.

This study conducts a comparative analysis of nano, hybrid, and ternary nanofluids in an Oldroyd-B fluid model within a Darcy–Forchheimer porous medium, focusing on heat transfer efficiency. It examines the effects of thermal radiation, heat source/sink, viscous dissipation, relaxation, and retardation time under different thermophysical conditions. Nanofluids are widely used in medicine, electronic cooling, heat exchangers, and renewable energy systems. However, the comparative performance of nano, hybrid, and ternary nanofluids in an Oldroyd-B framework remains unexplored. This study fills this gap by evaluating TiO ₂ , Fe ₃ O ₄ , and CoFe ₃ O ₄ nanoparticles and optimizing their thermal efficiency using numerical and statistical modeling. The governing nonlinear partial differential equations are transformed into ordinary differential equations using similarity variables and solved with the BVP4C method. Response surface methodology (RSM) is applied for parameter optimization, while multilinear regression analysis provides further insights. Graphical and tabulated results show that nano nanofluids achieve the highest heat transfer, followed by hybrid, while ternary nanofluids perform the lowest due to increased resistance. Thermal radiation improves heat transfer, whereas the Forchheimer number reduces it. The obtained findings can guide the selection of nanofluids for high-performance heat exchangers, magnetohydrodynamic (MHD) cooling systems, solar-thermal collectors, nuclear and industrial cooling, and biomedical thermal regulation technologies. The integration of RSM further provides a practical optimization framework for engineering design and performance improvement in energy-efficient systems. These findings offer valuable insights into optimizing nanofluid-based thermal systems, aiding advancements in heat exchangers, MHD cooling, and energy-efficient technologies.