The objective of this research is to acquire multiple branch solutions for the two-dimensional (2D) mixed convective flow of magnesium oxide (MgO)/water and titanium dioxide (TiO2)/water nanofluids over an exponentially vertical shrinking/stretching sheet with thermal radiation effect. Similarity transformations are applied to transform partial differential equations (PDEs) into ordinary differential equations (ODEs). For numerical findings, the shooting method is used via Maple software. Tremendously, four distinct branches of solutions are found in the case of opposing and assisting flows for each using influential parameters and nanoparticle volume fractions. In addition, stability analysis is also conducted that shows the first branch of the obtained solution in each case is physically stable (reliable) and acceptable, while the other three branches are unstable and considered as extraneous. Some of the key findings reveal that, for the stretching case, the skin friction reduces, but enhances the shrinking case of the surface when the rate of suction is increased. Moreover, the Nusselt number enhances for stretching case and reduces for shrinking case of sheet by increasing rate of the suction. The velocity profile increases with increasing rate of buoyancy or mixed convection parameter. Whereas, the profile of temperature decreases with larger values of the suction parameter, but enhances with higher nanoparticle volume fractions, the thermal radiation parameter, and the Biot number. However, TiO2 nanoparticles provide a greater rate of heat transfer and skin friction coefficient as compared to MgO nanoparticles. These findings provide novel physical insights for advanced thermal engineering applications.

This paper highlights the Prandtl nanofluid flow past a linearly stretchable surface in the existence of nanoparticles and gyrotactic microorganisms. The Prandtl fluid is a non-Newtonian behaviour which is represented by an extra stress tensor. Similarity variables are used to transform the PDEs into a coupled system of non-linear ordinary differential equations. An artificial neural network (ANN) method is also employed to estimate and validate the solution profiles. The results show that the activation energy and chemical reaction greatly affect mass transport, while the heat source/sink strongly affects heat transfer. Moreover, the gyrotactic microorganisms affect the bioconvective response of the nanofluid and improve the stability of the system in given circumstances. The comparison of the numerical and ANN-reliant solutions shows high correspondence, which proves the reliability and efficiency of the suggested ANN framework. The findings may support the design engineering of bio-nanofluid thermal and bioconvective transport systems.

The dynamics of peritoneal fluid has vital applications in various physiological functions, particularly in female reproductive physiology, which modulates the transport of gametes within the fallopian tubes. This research aims to study the interactive operating factors of peritoneal fluid flow, heat transfer, and mass transport under various physiological and pathological conditions. In this regard, the Jeffrey fluid model is applied to examine the non-Newtonian behavior of peritoneal fluid and its responses to thermal and magnetic influences. The study extends to consider thermophoresis and Brownian motion of nanoparticles in nanofluids with the purpose of enhancing thermal conductivity and optimizing fluid properties toward better reproductive health. Magnetic field effects on fluid dynamics, using magnetohydrodynamics, are also explored in relation to possible therapeutic interventions for endometriosis and tubal factor infertility. Numerical solutions and graphical interpretations are used to illustrate the impact of salient parameters such as Grashof numbers, Brownian motion, and shear-dependent viscosity on the fluid behavior. The results deepen the current understanding of the mechanics of peritoneal fluids and provide potential improvements in fertility treatments and biomedical applications. The proposed model can be applied to develop a non-invasive diagnostic technique for detection of endometriosis, to optimize infertility treatments and to design biomedical devices for peritoneal fluid manipulation in female reproductive therapy.

This research examines steady two-dimensional magnetohydrodynamic flow with integrated heat and mass transfer in a hybrid non-Newtonian Casson–Eyring–Powell nanofluid over a porous stretching sheet under convective boundary conditions. Using Lie symmetry analysis, the nonlinear governing equations, which include yield stress, viscoelasticity, magnetic effects, and nanoparticle transport, are turned into a simpler system of ordinary differential equations. We use MATLAB’s bvp4c solver to find the solution to the boundary value problem that comes from this. The results obtained illustrate that the skin friction coefficient, Nusselt number, and Sherwood number compare excellently with existing results, authenticating the validity of the modelled approach developed in the study. It also indicates that the synergistic integration of magnetic forces, non-Newtonian viscosity, and nanoparticle diffusion leads to an increase in the stability of the porous medium concerning its heat transfer performance. It was evident that the improved abilities related to thermal stability and efficiency, resulting from the individual and combined influences of magnetic fields, hybrid models, and nanoparticle motion, can be used as they are in industrial thermal processing and bioengineering heat transfer applications.

The purpose of the present investigation is to explore the implications of Cross fluid in a Darcy-Forchheimer porous medium due to the tri-hybrid nanofluid past a porous cylinder. Thermal radiation, heat generation, thermal convection, solutal convective and chemical reaction have been encountered in this analysis. Entropy generation has been accounted for under the fluidic friction, heat rate analysis, and porosity analysis. Three different nanoparticles of multiwall carbon nanotube (MWCNT), aluminum oxide (Al2O3), and silver (Ag) are utilized to illustrate the tri-hybrid nanofluid flow with Ethlene Glycol (EG) as the base fluid. The governance model, consisting of linked inadequate differential conditions, is transformed into an ordinary configuration of nonlinear coupled differential conditions by acceptable adjustments. The obtained outcomes in combination with the bvp4c approach are then used to resolve the generated ODEs. For discussion purposes, the impacts of the physical limitations on temperature profile, velocity, and concentration have also been illustrated. Numerical results have been obtained for the diffusion rate, heat transfer rate, drag force, and other factors. While the Forchheimer parameter and the inclination angle reduce the fluid flow’s velocity, the Biot number of heat and mass transfer influences the fluid’s temperature. According to the findings, hybrid nanofluid is the most effective way to improve heat transmission and may also be utilized for cooling. Three different kinds of nanofluids were used in a comparative examination to clarify the study’s conclusions. Changes in viscosity and porousness caused the nanofluids’ velocity to drop by 13.12% and 15.8%, respectively; however, trihybrid nanofluids with improved convection showed a 13.12% rise.

Many researchers have been drawn to the dynamics of non-Newtonian fluids due to their numerous uses across scientific and technological domains, including polymer extrusion, condensation mechanisms, and advanced cooling technologies. The uniqueness of this study lies in incorporating cross-diffusion effects on a nonlinear absorbent sheet. Additionally, the current work focused on the analysis of heat and mass transfer in a magnetohydrodynamic Casson nanofluid flow across a nonlinear absorbent sheet. Based on the laws of fluid motion, a physical problem for incompressible steady-state flow is formulated. The formulated model is converted into dimensionless ordinary differential equations by the application of a similarity transformation. Numerical simulations are computed by using the Finite Element Method (FEM) MATLAB built-in package. The study highlights several key outcomes that Casson's term improved the fluid's resistance to deformation, thereby thickening the velocity boundary layer. Brownian motion and Thermophoretic effects meaningfully augmented nanoparticle distribution, leading to an obvious rise in concentration distributions. The Dufour and Soret effects established a high connection between thermal and concentration fields, boosting both Nusselt and Sherwood numbers. Grid independence and validation have been conducted to validate the current model. These findings reveal new insights into combined thermal-solutal distribution in Casson nanofluids and give necessary information for the design of industrial heat-transfer systems, porous media technologies, energy devices, and advanced thermal systems applications.

The present study focuses on analysing and improving the heat -transfer performance of a rotating ternary hybrid nanofluid over a vertical flat surface under opposing mixed convection conditions. By employing suitable similarity transformations, the governing boundary-layer equations are reduced to nonlinear ordinary differential equations and subsequently solved using a MATLAB-based numerical approach. Furthermore, Response Surface Methodology (RSM) is used to investigate the combined influence of key parameters and to determine the conditions that maximise heat transfer efficiency. Increasing the concentration of nanoparticles, especially copper can greatly improve heat transfer efficiency, with copper nanoparticles showing the greatest enhancement, followed by aluminium oxide, (Al2O3) and titanium dioxide (TiO2) nanoparticles. Furthermore, desirability-based optimisation reveals that the heat-transfer rate attains a maximum value of 0.442152 with 99.93% desirability when the coded parameters A, B, and C (nanoparticle volume fractions) are at their maximum levels. Meanwhile, the decrement in skin friction along the x-direction is primarily influenced by the increase in volume fraction of copper nanoparticles, followed by titanium dioxide (TiO2) and aluminium oxide nanoparticles. The findings provide significant insights into optimising heat-transfer in complex fluid dynamics systems, with potential applications in diverse industrial and engineering domains.

This study uses the Backpropagation Bayesian Regularization scheme neural network (BBRSNN) technique to study the influence of slip conditions on the Ti₆Al₄V+TiO₂/Carboxymethyl Cellulose- water based hybrid nanofluid through an inclined rotating disc with Soret and Dufour effects. This model is useful for developing high-efficiency cooling systems, energy storage units, and chemical processing equipment because it helps precisely predict and optimize the behaviour of mass and heat transport in complicated fluid environments. While the Soret and Dufour effects are crucial for systems where coupled heat and mass diffusion occur, such as polymer processing, biomedical fluid transport, and membrane separation technologies, the inclusion of slip conditions enhances modeling accuracy in micro- and nano-scale flow devices, such as microchannel heat sinks. Engineers may create smarter, more effective industrial fluid-flow and thermal management systems thanks to the BR-NN optimization, which further improves model reliability. Following the validation of the approximate solution of multiple scenarios using the BBRSNN training and testing approach, the proposed model was given consideration for excellence. The proposed (BBRSNN) is confirmed using correlation analysis, mean squared error, and histogram of errors investigations. The accuracy level of the suggested technique ranges from 10−11 to 10−13.

Heat transfer is frequently employed in various industrial processes such as paper production, electronic device cooling, and the synthesis of new materials. Hence, this study aims to investigate the effect of Joule heating and magnetohydrodynamics (MHD) on the flow of a hybrid nanofluid with a power law heat flux past a shrinking sheet. The transformed governing equations are solved numerically using MATLAB’s bvp4c solver, and the results are validated against previously published data, showing excellent agreement (error < 0.01

In this research work, the 2D (two-dimensional) steady mixed convection MHD flow and heat transfer characteristics of nanofluids over an exponentially stretching/shrinking sheet are examined. In addition, suction/injection, heat source/sink, thermal radiation and slip parameter effects are considered. Initially, the problem is modelled in the form of PDEs, and then those PDEs equations are converted into ODEs using similarity transformations. Also, the solution of these equations is obtained by the shooting technique in Maple software. The three distinct branch solutions are found for each requisite posited influential parameter. Later, the stability analysis is performed to check that the first branch solution is stable and physically reliable. On the other hand, the second and third branch solutions are unstable. From the outcomes, it is seen that the skin friction increases for positive values of the stretching parameter and decreases for negative values of the shrinking parameter. The rate of heat transfer upsurges with the higher impact of the nanoparticles. The velocity profile escalates owing to the larger values of the stretching parameter, the nanoparticle volume fraction and the buoyancy parameter. In contrast, the suction and non-Newtonian parameter decelerate the velocity profile.

The fundamental mission of this study is to formulate and solve the mathematical model of the stagnation point flow of a hybrid nanofluid with the insertion of second-order velocity slips, magnetohydrodynamic (MHD), and radiation effects over a shrinking sheet, which are critical for enhancing thermal performance in industrial cooling and heat treatment processes. The model is transcribed into non-dimensional formulations using similarity variables and is solved numerically using the bvp4c solver in MATLAB. Dual solutions are executed, and the stable solution is validated via the stability analysis. In certain conditions, the comparison of current and prior findings demonstrates good agreement with nearly 0% relative error. The findings reported that the critical point is extended, and the bifurcation of the boundary layer is prevented by the boost in the magnitude of second-order velocity slips and copper volume fraction. The efficiency of heat transfer improves as the radiation effect and the copper volume fraction increase, particularly when the sheet is shrunk. The boost of copper volume fraction is also simulated to lessen the temperature and the thermal boundary layer thickness. Thus, the present model in this study has proven that the utilization of a hybrid nanofluid could increase the thermal performance of a system, and it could be used as a coolant for a heat treatment process.

It is well-known that blood is treated as a non-Newtonian fluid because its viscosity can vary with shear stress. The present study explores the effect of melting and stagnation-point micropolar fluid flow by involving blood-based copper (Cu)/copper-oxide (CuO) hybrid nanofluids and the features of heat transport across a moving sheet with inertial and microstructure characteristics. Initially, the problem is modelled in the form of partial differential equations and then changed into ordinary differential equations by using similarity variables. These equations are numerically solved using the fourth-order boundary value solver bvp4c and a neural network depending on the algorithm of the Levenberg-Marquardt back-propagation. The outcomes revealed that the suggested artificial neural network method could hold nonlinear data with minimum error and showed consistent performance across all phases, including training, validation, and testing. In addition, the friction factor and the micro-rotation coefficient increase significantly by up to 5.602% and 7.701%, respectively, with increasing nanoparticle volume fractions. In contrast, the heat transfer rate shows only very small variations with increasing nanoparticle volume fractions and changes in other influential parameters because of the imposed melting boundary conditions.

In this research, the impact of thermal effects, viscous energy loss, and magnetic-field interaction on a Williamson ternary hybrid nanofluid (Ag, SWCNT, MWCNT) that simulates blood flow (in 2D) over an elastically moving surface is assessed. An analytical technique is developed to provide a framework for enhancing convective heat transfer and reducing streamwise resistance in high-energy systems. Obtaining semi-analytical solutions to the governing nonlinear partial differential equations within the BVPh 1.0 and BVPh 2.0 packages for Mathematica involves transforming them into ordinary differential equations via special similarity variables, applying the homotopy analysis approximating method, and achieving residuals below 10(-5) in fewer than 20 steps. The analysis reveals that the resultant average heat transfer (Nu) is over 21% due to the surface cooling heat flux, the thickening of the opaque thermal layer from thermal effects, the extension of the Eckert number, the nanofluid volumetric concentration, and the magnetic Williamson number (slowing rates). And accurately including these ternary hybrid nanofluids in biomedical wearables, blood cooling, polymer extrusion cooling, and highly oriented micro- and electronic heat exchangers for efficient simultaneous temperature and shear stress, is revealing.

Developing bio-based lubricants from banana peel oil is emerging as a promising alternative to conventional lubricants. Infusing this biolubricant with graphene nanoparticles has the potential to enhance its tribological performance significantly by creating a tribofilm at the contact surface. This study aims to investigate the effectiveness of graphene-infused banana peel oil, with a focus on its tribological behaviour. Oil was extracted using Soxhlet extraction, and nanoparticle infusion was accomplished using ultrasonic homogenization. Tribological tests were conducted using the 4-ball tribometer to record the coefficient of friction (CoF) and wear scar diameter (WSD). The results showed that Sample 8 (0.5 vol%, 20 minutes, industrial graphene) obtained the best tribological performance, with the lowest CoF (0.0825) and WSD (137 & micro;m), while Sample 13 (0.3 vol%, 20 minutes, technical graphene) presented the highest COF (0.0939). Sample 14 (0.3 vol%, 20 minutes, technical graphene) exhibited the most severe wear in WSD value (170 & micro;m). In addition, Sample 8 demonstrated Graphene's crystal structure and its types having optimised composition to help reduce friction and wear by forming a protective layer that minimised direct surface contact. Furthermore, the developed bio-lubricants also demonstrated similar lubricant performance when compared to conventional engine oil.

This mathematical model investigates the complexity of buoyant convective flow and heat transfer phenomena interaction under the impact of Cattaneo-Christov theory. The uniform and unidirectional flow of Newtonian fluid across a heated bi-directional stretchable surface is considered here with influence of Lorentz force. Transportation of thermal energy is studied under influence of heat source and buoyancy force. Mathematical model in form of PDEs governs the flow and heat transport mechanism is transformed into similar ODEs. Moreover, the resulting outcomes of problem are graphically visualized and compute via bvp5c-MATLAB built-in command. These results prove that higher magnitude of magnetic force declines the flow velocity to zero in lateral directions. While buoyant motion of fluid enhances when Grashof number increases. Higher value of Prandtl number declines the lateral components of flow field. To verify these results the comparison tables for initial and final values of flow velocity with the previous study are included in limiting case.

This study examines magnetohydrodynamic (MHD) heat and mass transfer of a ternary hybrid nanofluid over a rotating sphere incorporating thermophoretic particle deposition, thermal radiation, activation energy and chemical reaction effects. The nanofluid consists of Cu – Fe_3O_4 – ZrO_2 nanoparticles dispersed in propylene glycol. The governing boundary layer equations are transformed into a system of nonlinear ordinary differential equations via similarity transformations, which are solved using the Gegenbauer wavelet method. Results indicate that increasing magnetic interaction suppresses velocity due to Lorentz force effects while enhancing thermal distribution. Higher nanoparticle volume fraction improves heat transfer but increases viscous resistance. Thermophoresis and activation energy significantly influence mass transfer characteristics. Comparative analysis reveals that the ternary hybrid nanofluid exhibits enhanced thermal performance relative to the corresponding hybrid nanofluid configuration. The findings provide theoretical insight into MHD-controlled rotating nanofluid systems.

This research investigates steady, incompressible, magnetized micropolar fluid flow between two parallel plates in a rotating frame with homogeneous-heterogeneous reactions and suction/injection effects. Thermal analysis includes thermal radiation with temperature-dependent viscosity and thermal conductivity. To investigate this complex scenario, the fundamental equations of flow are transformed into a coupled system of non-similar, dimensionless equations using a local non-similarity transformation, accommodating variable fluid properties and rotation effects. The coupled equations are solved numerically by using bvp4c in MATLAB. The study thoroughly examines the effect of various physical evolving parameters on the velocity, microrotation, temperature, and concentration profiles. Notably, the velocity profiles exhibit a unique dual behavior between the two plates, indicating a complex flow pattern. The effects of the variable viscosity parameter show a decreasing trend near the lower plate. As the fluid moves toward the upper plate, the velocity changes its behavior. Moreover, the influence of variable thermal conductivity is found to enhance the fluid temperature in the rotating system. The novelty lies in combining variable properties, chemical reactions, and rotation in a non-similar micropolar flow analysis, with applications in rotating microfluidic devices and MEMS.

This research studies the enhancement of heat transfer due to thermal radiation and viscous dissipation over a permeable and stretching surface of a two-dimensional porous medium, using a two-dimensional steady MHD hybrid nanofluid (HNF) flow model containing MWCNT-SWCNT-Cu-TiO2 nanoparticles in kerosene oil or blood. The single-nanofluid model limitations of the HNF model arise from its enhanced thermal conductivity and thermal stability. The governing partial differential equations are reduced to ordinary differential equations using the similarity method and solved semi-analytically with BVPh 2.0 due to improved convergence. Results show that the Lorentz force due to the magnetic field reduced velocities by 30-40 %. This also caused thermal boundary layer thickness to increase, while heat radiation (Rd=0-3) and Eckert's (Ec=0-1.5) dissipation increased temperatures by 25-35 % and Nusselt numbers to increase to 2.49. In a porous medium with a porosity parameter (epsilon=0.1-0.9), the Darcy drag effect reduces the skin-friction peak to 1.61 and improves the heat-momentum transfer rates with suction/injection. The nanofluid volume fraction phi= 0.01-0.1 increased the drag and transfer coefficients. The novelties are the dual-base, four-nanoparticle, HNF model, with the first and only known interaction among porosity-MHD-radiation, and the absence of research on steady stagnation or an impermeable sheet. The results are applicable to porous fins in nuclear coolers (velocity stabilisation), solar collectors (radiation utilisation), electronics (microchannels), and biomedical stents (blood flow control), yielding 20-50 % improvements over conventional fluids. BVPh results support real-time thermal systems engineering with a residual of < 1 % after 25 iterations.

This study introduces a novel integrated computational and statistical approach for analyzing steady boundary layer flow and heat transfer in a tetra hybrid nanofluid containing alumina, copper, silica and titania nanoparticles dispersed in water past a convectively heated moving plate with internal heat generation. The governing nonlinear equations are reduced via similarity transformations and solved using Matlab's bvp4c solver to obtain highly accurate velocity, temperature, skin friction, and Nusselt number distributions. Response Surface Methodology (RSM) and sensitivity analysis are employed to quantify and rank the influence of heat generation, Biot number, and suction parameter. The numerical results reveal that increasing the Biot number may enhance the heat transfer rate by approximately 42.6 for opposing flow case and 53.6% for assisting flow case, whereas stronger wall suction improves heat transfer by about 0.5-6.2%. However, higher heat generation slightly weakens the heat transfer rate up to 3.6% and 0.2%, for opposing and assisting flow cases, respectively. The response surface methodology and sensitivity analysis also reveal that the Biot number exerts the dominant influence on heat transfer, followed by suction strength and heat generation rate. The principal novelty lies in the exclusive integration of numerical simulation with statistical optimization for a tetra hybrid nanofluid under convective heating, an area rarely addressed in the literature. The proposed framework not only identifies the most influential parameters but also determines optimal ranges for maximizing thermal performance. These findings establish a benchmark for designing advanced thermal management systems in high-temperature industrial and energy applications.

Efficient heat transfer is a major challenge in systems like MHD pumps, electromagnetic cooling units and rotating heat exchangers especially under extreme conditions. The combined effects of magnetic fields, Joule heating and multiple nanoparticles create complex behaviors that are hard to solve using analytical methods. Therefore, reliable numerical methods and statistical optimization tools are needed to analyze and improve these systems. Hence, this work highlights ternary hybrid nanofluid flow over a permeable moving surface with the influence of magnetohydrodynamic (MHD), Joule heating and suction effects, integrating both computational and optimization techniques. The fluid comprises water-based ternary hybrid nanofluid containing aluminum oxide (Al2O3), copper (Cu) and titanium dioxide (TiO2) nanoparticles. The governing partial differential equations are transformed into a system of ordinary differential equations using similarity transformations and solved using the bvp4c solver (MATLAB). Validation of the numerical approach is carried out by comparing results with existing literature which showing excellent agreement for limiting cases. Response surface methodology (RSM) is applied to analyze interactions between the key parameters. Results indicate that both magnetic parameter and titania concentration significantly enhance the velocity profile due to the induced Lorentz force and altered thermal gradient. Analysis of variance (ANOVA) confirms that magnetic parameter and titania concentration are the most influential on thermal and flow responses. Sensitivity analysis further highlights strong linear and interaction effects.