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Thermal energy storage systems use thermal energy to elevate the temperature of a storage substance, enabling the release of energy during a discharge cycle. The storage or retrieval of energy occurs through the heating or cooling of either a liquid or a solid, without undergoing a phase change, within a sensible heat storage system. In a sensible packed bed thermal energy storage system, the structure comprises porous media that form the packed solid material, while fluid occupies the voids. Thus, a cavity, partially filled with a fluid layer and partially with a saturated porous layer, has become important in the investigation of natural convection heat transfer, carrying significant relevance within thermal energy storage systems. Motivated by these insights, the current investigation delves into the convection heat transfer driven by buoyancy and entropy generation within a partially porous cavity that is differentially heated, vertically layered and filled with a hybrid nanofluid.

The investigation encompasses two distinct scenarios. In the first instance, the porous layer is positioned next to the heated wall, while the opposite region consists of a fluid layer. In the second case, the layers switch places, with the fluid layer adjacent to the heated wall. The system of equations for fluid and porous media, along with appropriate initial and boundary conditions, is addressed using the finite difference method. The Tiwari–Das model is used in this investigation, and the viscosity and thermal conductivity are determined using correlations specific to spherical nanoparticles.

Comprehensive numerical simulations have been performed, considering controlling factors such as the Darcy number, nanoparticle volume fraction, Rayleigh number, bottom slit position and Hartmann number. The visual representation of the numerical findings includes streamlines, isotherms and entropy lines, as well as plots illustrating average entropy generation and the average Nusselt number. These representations aim to provide insight into the influence of these parameters across a spectrum of scenarios.

The computational outcomes indicate that with an increase in the Darcy number, the addition of 2.5% magnetite nanoparticles to the GO nanofluid results in an enhanced heat transfer rate, showing increases of 0.567% in Case 1 and 3.894% in Case 2. Compared with Case 2, Case 1 exhibits a 59.90% enhancement in heat transfer within the enclosure. Positioning the porous layer next to the partially cooled wall significantly boosts the average total entropy production, showing a substantial increase of 11.36% at an elevated Rayleigh number value. Positioning the hot slit near the bottom wall leads to a reduction in total entropy generation by 33.20% compared to its placement at the center and by 33.32% in comparison to its proximity to the top wall.

Scientists have been conducting trials to find ways to reduce fuel consumption and enhance heat transfer rates to make heating systems more efficient and cheaper. Adding solid nanoparticles to conventional liquids may greatly improve their thermal conductivity, according to the available evidence. This study aims to examine the influence of external magnetic flux on the flow of a mixed convective Maxwell hybrid non-Newtonian nanofluid over a linearly extending porous flat plate. The investigation considers the effects of thermal radiation, Dufour and Soret.

The mathematical model is formulated based on the fundamental assumptions of mass, energy and momentum conservation. The implicit models are epitomized by a set of interconnected nonlinear partial differential equations, which include a suitable and comparable adjustment. The numerical solution to these equations is assessed for approximate convergence by the Runge−Kutta−Fehlberg method based on the shooting technique embedded with the MATLAB software.

The findings are presented through graphical representations, offering a visual exploration of the effects of various dynamic parameters on the flow field. These parameters encompass a wide range of factors, including radiation, thermal and Brownian diffusion parameters, Eckert, Lewis and Soret numbers, magnetic parameters, Maxwell fluid parameters, Darcy numbers, thermal and solutal buoyancy factors, Dufour and Prandtl numbers. Notably, the authors observed that nanoparticles with a spherical shape exerted a significant influence on the stream function, highlighting the importance of nanoparticle geometry in fluid dynamics. Furthermore, the analysis revealed that temperature profiles of nanomaterials were notably affected by their shape factor, while concentration profiles exhibited an opposite trend, providing valuable insights into the behavior of nanofluids in porous media.

A distinctive aspect of the research lies in its novel exploration of the impact of external magnetic flux on the flow of a mixed convective Maxwell hybrid non-Newtonian nanofluid over a linearly extending porous flat plate. By considering variables such as solar radiation, external magnetic flux, thermal and Brownian diffusion parameters and nanoparticle shape factor, the authors ventured into uncharted territory within the realm of fluid dynamics. These variables, despite their significant relevance, have not been extensively studied in previous research, thus underscoring the originality and value of the authors’ contribution to the field.

Thermophoresis deposition of particles is a crucial stage in the spread of microparticles over temperature gradients and is significant for aerosol and electrical technologies. To track changes in mass deposition, the effect of particle thermophoresis is therefore seen in a mixed convective flow of Williamson hybrid nanofluids upon a stretching/shrinking sheet.

The PDEs are transformed into ordinary differential equations (ODEs) using the similarity technique and then the bvp4c solver is employed for the altered transformed equations. The main factors influencing the heat, mass and flow profiles are displayed graphically.

The findings imply that the larger effects of the thermophoretic parameter cause the mass transfer rate to drop for both solutions. In addition, the suggested hybrid nanoparticles significantly increase the heat transfer rate in both outcomes. Hybrid nanoparticles work well for producing the most energy possible. They are essential in causing the flow to accelerate at a high pace.

The consistent results of this analysis have the potential to boost the competence of thermal energy systems.

It has not yet been attempted to incorporate hybrid nanofluids and thermophoretic particle deposition impact across a vertical stretching/shrinking sheet subject to double-diffusive mixed convection flow in a Williamson model. The numerical method has been validated by comparing the generated numerical results with the published work.

This study aims to introduce a novel machine learning feature vector (MLFV) method to bring machine learning to overcome the time-consuming computational fluid dynamics (CFD) simulations for rapidly predicting turbulent flow characteristics with acceptable accuracy.

In this method, CFD snapshots are encoded in a tensor as the input training data. Then, the MLFV learns the relationship between data with a rod filter, which is named feature vector, to learn features by defining functions on it. To demonstrate the accuracy of the MLFV, this method is used to predict the velocity, temperature and turbulent kinetic energy fields of turbulent flow passing over an innovative nature-inspired Dolphin turbulator based on only ten CFD data.

The results indicate that MLFV and CFD contours alongside scatter plots have a good agreement between predicted and solved data with R² ≃ 1. Also, the error percentage contours and histograms reveal the high precisions of predictions with MAPE = 7.90E-02, 1.45E-02, 7.32E-02 and NRMSE = 1.30E-04, 1.61E-03, 4.54E-05 for prediction velocity, temperature, turbulent kinetic energy fields at Re = 20,000, respectively.

The method can have state-of-the-art applications in a wide range of CFD simulations with the ability to train based on small data, which is practical and logical regarding the number of required tests.

The paper introduces a novel, innovative and super-fast method named MLFV to address the time-consuming challenges associated with the traditional CFD approach to predict the physics of turbulent heat and fluid flow in real time with the superiority of training based on small data with acceptable accuracy.