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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.

Fluid flows through rotatory disks are encountered in industrial and practical engineering processes, such as computer storage devices, gas turbine rotators, rotating machinery, air cleaning machines, etc. The primary purpose of this research is to examine the combined aspects of variable electrical conductivity, thermal radiation, Soret and Dufour effects on a magnetohydrodynamic Maxwell single-walled carbon nanotubes–graphene oxide–multi-walled carbon nanotubes–copper (SWCNT–GO–MWCNT–Cu)/sodium alginate tetra-hybrid nanofluid flow through a stretchable rotatory disk.

The modeled administrative equations of the present flow problem are converted to a non-dimensional system of ordinary differential equations by applying suitable similarity conversion and then solved numerically by implementing the bvp4c method. The impressions of noteworthy dimensionless parameters on velocity, temperature, concentration distributions, Nusselt number, skin friction and Sherwood number are reported via graphs and tables.

The authors figured out that the developed values of the rotation parameter diminish the temperature but enhance both the radial and angular velocities. Further, the mass and heat transmission rates are better for tetra-hybrid nanofluids than for ternary and hybrid nanofluids.

The present study emphasizes a special type of fluid called the tetra-hybrid nanofluid. The existing literature has not discussed the Maxwell tetra hybrid nanofluid flow through a stretchable rotatory disk with variable electrical conductivity. Besides, the novel aspects of magnetohydrodynamics, thermal radiation, Soret and Dufour effects are also incorporated into the present flow problem.

To improve the thermal performance of base fluid, nanoparticles of three types are dispersed in the base fluid. A novel theory of non-Fourier heat transfer is used for design and development of models. The thermal performance of sample fluids is compared to determine which types of combination of nanoparticles are the best for an optimized enhancement in thermal performance of fluids. This article aims to: (i) investigate the impact of nanoparticles on thermal performance; and (ii) implement the Galerkin finite element method (GFEM) to thermal problems.

The mathematical models are developed using novel non-Fourier heat flux theory, conservation laws of computational fluid dynamics (CFD) and no-slip thermal boundary conditions. The models are approximated using thermal boundary layer approximations, and transformed models are solved numerically using GFEM. A grid-sensitivity test is performed. The accuracy, correction and stability of solutions is ensured. The numerical method adopted for the calculations is validated with published data. Quantities of engineering interest, i.e. wall shear stress, wall mass flow rate and wall heat flux, are calculated and examined versus emerging rheological parameters and thermal relaxation time.

The thermal relaxation time measures the ability of a fluid to restore its original thermal state, called thermal equilibrium and therefore, simulations have shown that the thermal relaxation time associated with a mono nanofluid has the most substantial effect on the temperature of fluid, whereas a ternary nanofluid has the smallest thermal relaxation time. A ternary nanofluid has a wider thermal boundary thickness in comparison with base and di- and mono nanofluids. The wall heat flux (in the case of the ternary nanofluids) has the most significant value compared with the wall shear stresses for the mono and hybrid nanofluids. The wall heat and mass fluxes have the highest values for the case of non-Fourier heat and mass diffusion compared to the case of Fourier heat and mass transfer.

An extensive literature review reveals that no study has considered thermal and concentration memory effects on transport mechanisms in fluids of cross-rheological liquid using novel theory of heat and mass [presented by Cattaneo (Cattaneo, 1958) and Christov (Christov, 2009)] so far. Moreover, the finite element method for coupled and nonlinear CFD problems has not been implemented so far. To the best of the authors’ knowledge for the first time, the dynamics of wall heat flow rate and mass flow rate under simultaneous effects of thermal and solute relaxation times, Ohmic dissipation and first-order chemical reactions are studied.

This investigation delves into the rationale behind the preferential applicability of the non-Newtonian nanofluid model over alternative frameworks, particularly those incorporating porous medium considerations. The study focuses on analyzing the mass and heat transfer characteristics inherent in the Williamson nanofluid’s non-Newtonian flow over a stretched sheet, accounting for influences such as chemical reactions, viscous dissipation, magnetic field and slip velocity. Emphasis is placed on scenarios where the properties of the Williamson nanofluid, including thermal conductivity and viscosity, exhibit temperature-dependent variations.

Following the use of the OHAM approach, an analytical resolution to the proposed issue is provided. The findings are elucidated through the construction of graphical representations, illustrating the impact of diverse physical parameters on temperature, velocity and concentration profiles.

Remarkably, it is discerned that the magnetic field, viscous dissipation phenomena and slip velocity assumption significantly influence the heat and mass transmission processes. Numerical and theoretical outcomes exhibit a noteworthy level of qualitative concurrence, underscoring the robustness and reliability of the non-Newtonian nanofluid model in capturing the intricacies of the studied phenomena.

Available studies show that no work on the Williamson model is conducted by considering viscous dissipation and the MHD effect past over an exponentially stretched porous sheet. This contribution fills this gap.

A novel type of heat transfer fluid known as hybrid nanofluids is used to improve the efficiency of heat exchangers. It is observed from literature evidence that hybrid nanofluids outperform single nanofluids in terms of thermal performance. This study aims to address the stagnation point flow induced by Williamson hybrid nanofluids across a vertical plate. This fluid is drenched under the influence of mixed convection in a Darcy–Forchheimer porous medium with heat source/sink and entropy generation.

By applying the proper similarity transformation, the partial differential equations that represent the leading model of the flow problem are reduced to ordinary differential equations. For the boundary value problem of the fourth-order code (bvp4c), a built-in MATLAB finite difference code is used to tackle the flow problem and carry out the dual numerical solutions.

The shear stress decreases, but the rate of heat transfer increases because of their greater influence on the permeability parameter and Weissenberg number for both solutions. The ability of hybrid nanofluids to strengthen heat transfer with the incorporation of a porous medium is demonstrated in this study.

The findings may be highly beneficial in raising the energy efficiency of thermal systems.

The originality of the research lies in the investigation of the Darcy–Forchheimer stagnation point flow of a Williamson hybrid nanofluid across a vertical plate, considering buoyancy forces, which introduces another layer of complexity to the flow problem. This aspect has not been extensively studied before. The results are verified and offer a very favorable balance with the acknowledged papers.

The purpose of this paper is to showcase the utilization of the magnetohydrodynamics-microrotating Casson’s nanofluid flow model (MHD-MRCNFM) in examining the impact of an inclined magnetic field within a porous medium on a nonlinear stretching plate. This investigation is conducted by using neural networking techniques, specifically using neural networks-backpropagated with the Levenberg–Marquardt scheme (NN-BLMS).

The initial nonlinear coupled PDEs system that represented the MRCNFM is transformed into an analogous nonlinear ODEs system by the adoption of similarity variables. The reference data set is created by varying important MHD-MRCNFM parameters using the renowned Lobatto IIIA solver. The numerical reference data are used in validation, testing and training sets to locate and analyze the estimated outcome of the created NN-LMA and its comparison with the corresponding reference solution. With mean squared error curves, error histogram analysis and a regression index, better performance is consistently demonstrated. Mu is a controller that controls the complete training process, and the NN-BLMS mainly concentrates on the higher precision of nonlinear systems.

The peculiar behavior of the appropriate physical parameters on nondimensional shapes is demonstrated and explored via sketches and tables. For escalating amounts of inclination angle and Brinkman number, a viable entropy profile is accomplished. The angular velocity curve grows as the rotation viscosity and surface condition factors rise. The dominance of friction-induced irreversibility is observed in the vicinity of the sheet, whereas in the farthest region, the situation is reversed with heat transfer playing a more significant role in causing irreversibilities.

To improve the efficiency of any thermodynamic system, it is essential to identify and track the sources of irreversible heat losses. Therefore, the authors analyze both flow phenomena and heat transport, with a particular focus on evaluating the generation of entropy within the system.

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 work aims to concentrate on the mixed convection of the stagnation point flow of ternary hybrid nanofluids towards vertical Riga plate. Aluminum trioxide (Al₂O₃), silicon dioxide (SiO₂) and titanium dioxide (TiO₂) are regarded as nanoparticles, with water serving as the base fluid. The mathematical model incorporates momentum boundary layer and energy equations. The Grinberg term for the viscous dissipation and the wall parallel Lorentz force coming from the Riga plate are taken into consideration in the context of the energy equation.

Through the use of appropriate nonsimilar transformations, the governing system is transformed into nonlinear nondimensional partial differential equations (PDEs). The numerical method bvp4c (built-in package for MATLAB) is used in this study to simulate governing equations using the local non-similarity (LNS) approach up to the second truncation level.

Numerous graphs and numerical tables expound on the physical properties of the nanofluid temperature and velocity profiles. The local Nusselt correlations and the drag coefficient for pertinent parameters have been computed in tabular form. Additionally, the temperature profile drops while the velocity profile increases when the mixed convection parameter is included to oppose the flow.

The fundamental goal of this work is to comprehend how ternary nanofluids move towards a vertical Riga plate in a mixed convective domain with stagnation point flow.

The steady laminar wall jet flow over a stretching/shrinking surface in the presence of lateral suction or injection with a convective boundary condition is considered.

The partial differential equations for mass, momentum and energy conservation are changed to the system of ordinary differential equations through similarity solution transformations. Solutions, both numerical and asymptotic, to these similarity equations are found in some new ranges of parameters in the governing equations.

The equations are solved both asymptotically and numerically for a range of the transpiration parameter S and the flow parameter λ given in Mahros et al. (2023), thus greatly extending the range of these previous solutions. Asymptotic solutions for both large and small values of the Prandtl number σ are derived, showing good agreement with additional numerical integrations. It should be noted that in Mahros et al. (2023), only the case when σ=1 was treated. A solution for large λ when S=1 is obtained, showing a different asymptotic form to the case when S>0 in Mahros et al. (2023). Multiple solutions were seen by them for S<0 and the nature of the lower solution branch as S→0 from below is discussed. The question as to whether the lower branch solutions join as λ>0 when S<0 is resolved through obtaining an asymptotic solution λ small.

The accuracy of the solutions has been checked through a detailed comparison between the solutions obtained numerically and analytically, where excellent agreement has been found. This study is important for scientists working in the area of jet flows to become familiar with the flow properties and behaviour of jets.

The purpose of the paper is to conduct a numerical and experimental investigation into the properties of nanofluids containing spherical nanoparticles of random sizes flowing through a porous medium. The study aims to understand how the thermophysical properties of the nanofluid are affected by factors such as nanoparticle volume fraction, permeability of the porous medium, and pore size. The paper provides insights into the behavior of nanofluids in complex environments and explores the impact of varying conditions on key properties such as thermal conductivity, density, viscosity, and specific heat. Ultimately, the research contributes to the broader understanding of nanofluid dynamics and has potential implications for engineering and industrial applications in porous media.

This paper investigates nanofluids with spherical nanoparticles in a porous medium, exploring thermal conductivity, density, specific heat, and dynamic viscosity. Studying three compositions, the analysis employs the classical Maxwell model and Koo and Kleinstreuer’s approach for thermal conductivity, considering particle shape and temperature effects. Density and specific heat are defined based on mass and volume ratios. Dynamic viscosity models, including Brinkman’s and Gherasim et al.'s, are discussed. Numerical simulations, implemented in Python using the Langevin model, yield results processed in Origin Pro. This research enhances understanding of nanofluid behavior, contributing valuable insights to porous media applications.

This study involves a numerical examination of nanofluid properties, featuring spherical nanoparticles of varying sizes suspended in a base fluid with known density, flowing through a porous medium. Experimental findings reveal a notable increase in thermal conductivity, density, and viscosity as the volume fraction of particles rises. Conversely, specific heat experiences a decrease with higher particle volume concentration.xD; xA; The influence of permeability and pore size on particle volume fraction variation is a key focus. Interestingly, while the permeability of the medium has a significant effect, it is observed that it increases with permeability. This underscores the role of the medium’s nature in altering the thermophysical properties of nanofluids.

This paper presents a novel numerical study on nanofluids with randomly sized spherical nanoparticles flowing in a porous medium. It explores the impact of porous medium properties on nanofluid thermophysical characteristics, emphasizing the significance of permeability and pore size. The inclusion of random nanoparticle sizes adds practical relevance. Contrasting trends are observed, where thermal conductivity, density, and viscosity increase with particle volume fraction, while specific heat decreases. These findings offer valuable insights for engineering applications, providing a deeper understanding of nanofluid behavior in porous environments and guiding the design of efficient systems in various industrial contexts.