This study discovers the mass and heat transfer characteristics of a magnetohydrodynamic GO-Ag-CuO-Al₂O₃/EG nanofluid streaming past a shrinking sheet under the force of various physical effects. The Powell-Eyring fluid model is engaged to account for the effect of a magnetic field, a stagnation point, viscous dissipation, radiation, Joule heating, and suction. Through similarity transformations, the governing partial differential equations are reduced to a system of ordinary differential equations, which are then resolved numerically using the bvp4c solver in MATLAB. The results affirmed that raising the heat transfer complements the thermal boundary layer, whereas reductions in the skin friction coefficient contribute to a reduction in drag force. Moreover, the velocity profile rises because of the shrinking effect, while the temperature profile decreases. The enhanced thermal conductivity provided by the quaternary nanoparticle suspension (GO-Ag-CuO-Al₂O₃) suggests that this fluid can maintain lower surface temperatures under high-heat flux conditions compared to mono or hybrid nanofluids. Consequently, these characteristics are particularly advantageous for electronic device cooling and heat exchangers in renewable energy systems where rapid heat dissipation is critical. Furthermore, the observed reduction in drag force under the influence of the magnetic field provides a theoretical basis for optimizing energy efficiency in magnetohydrodynamic pumps and metallurgical processing. These findings offer specific design parameters for boosting thermal management in industrial applications involving tetra-hybrid nanofluids.

Numerical simulation was utilised to evaluate entropy generation and the thermo-hydrodynamic behaviour of the unsteady magnetised Mg-Fe3O4 hybrid Powell-Eyring nanofluid flow over a porous, thermally stratified, curved surface, in relation to thermal engineering case studies. This study seeks to understand the interactions among flow stability, heat transfer, and entropy generation, considering the effects of magnetic field strength, viscous dissipation, Joule heating, the porous medium's resistance, and buoyancy forces from the surroundings. The Keller-Box Method was then applied to solve the coupled, nonlinear boundary-layer equations, thereby building confidence in the velocity, temperature, and concentration profiles, which were thoroughly validated against established benchmark studies. The findings showed that, together with the porous medium and magnetic field factors, these factors reduce the flow velocity and increase the temperature gradients and entropy generation, revealing essential consequences for the system’s energy efficiency. The hybrid nanoparticles’ superior thermal-conducting and flow-stabilising effects around curved biomedical and microfluidic surfaces were observed. The characteristically advanced, multifaceted integration of physical effects is a step forward in the development of magneto-thermofluidic devices. Discrete, validated numerical findings emphasise the control of resultant flow and heat transfer in magnetically aided cancer hyperthermia, targeted drug delivery, and progressive cooling as advanced cooling aids.

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

The present investigation analyses the Brownian diffusion and thermophoretic effects influence the motion and thermal-mass transport characteristics of an electromagnetic free, convective, radiative, and chemically reacting Williamson fluid flow over an inclined porous surface. The novelty of this investigation lies in considering the effects of chemical reaction, heat source/sink and Brownian motion on flow, heat and mass transfer with two boundary conditions: (i) Prescribed surface temperature and surface mass (PST & PSM), and (ii) Prescribed heat and mass flux (PHF & PMF). The complex nonlinear coupled partial differential equations are transformed into ordinary differential equations using similarity transformations and solved numerically through the BVP4C solver on MATLAB, with outcomes presented via tables and graphical illustrations, while Surface Response Methodology (SRM) is applied to analyze parameter sensitivity. Results reveal that Browinan motion, thermophorosis, radiation and heat source parameters enhance the temperature on both the PST and PHF cases but reverse phenomenon is observed with Weissenberg number, whereas as Biot number elevates temperature and concentration profiles. The statistical accuracy of skin friction prediction is high, with R2 values of 98.39 % and 98.92 % for Cases 1 and 2, respectively. Furthermore, the Nusselt number exhibits greater sensitivity to thermal radiation and the Weissenberg number compared to the heat source parameter.

Rice tungro disease is one of the most destructive viral diseases that affects rice production in Southeast Asia, especially in Malaysia, where the recurring outbreaks continue to pose a challenge to food security and sustainable agriculture. Green leafhoppers transmit the disease through co-infection of Rice Tungro Bacilliform Virus (RTBV) and Rice Tungro Spherical Virus (RTSV) and require effective and environment friendly management strategies. Roguing, the systematic removal of infected plants, is widely practised as it is simple and inexpensive. However, its effectiveness is dependent on dynamic factors such as disease prevalence, timing, crop growth stage and vector activity. Compartmental disease models are used to provide a mathematical view of dynamic roguing strategies in a soil-plant-crop system setting. It critically reviews the existing numerical methods to solve the dynamic roguing models and highlights the limitations of the conventional time-stepping techniques in terms of accuracy, stability and computational efficiency. The unexploited promise of spectral weighted residual methods as powerful global numerical schemes to improve the reliability of long-term simulation in plant virus epidemiology is given special attention. The study highlights important modelling and computational gaps and proposes a more robust mathematical framework for supporting sustainable rice disease management and evidence-based agricultural decision making.

In the present work, we study the numerical approximation of nonlinear elliptic equations using the mortar expanded mixed method. This framework is based on the mortar finite element method and non-overlapping domain decomposition, allowing independent discretization of each subdomain. An auxiliary interface variable is defined on the internal boundary of subdomains which plays the role of a Dirichlet boundary condition for subdomain problems. This mortar variable also enforces the weak continuity across the interface between subdomains. We combine the multiblock mortar approach with the expanded mixed method to provide accurate approximations for small coefficients. We prove the existence and uniqueness of the discrete problem by employing fixed point argument. Optimal-order convergence estimates are derived for subdomain approximations in the L2 norm. An error bound for the interface approximation is also presented. The computational results matching the theoretical findings are reported.

The stagnation point flow of a ternary hybrid nanofluid (THNF) over a sheet that stretches or shrinks exponentially is investigated in this study. The primary goal is to assess the implication of Joule heating, magnetohydrodynamics (MHD), thermal radiation, and boundary slips on the physical quantities and flow profiles. Besides, attention is also given to the occurrences of multiple solutions in this fluid flow situation. The continuity, momentum and energy equations that described the fluid flow problem are converted into a simpler form of ordinary differential equations (ODEs). This is achieved by applying a similarity transformation, which make the equations easier to solve. Solving the resulting equations using the bvp4c solver in MATLAB software yields results that are analyzed and illustrated through a combination of tables and graphical representations. The analysis reveals that, for a shrinking sheet, an increase in Joule heating reduces the heat transfer. Similarly, a higher thermal slip factor leads to a decreased heat transfer rate in this case. In contrast, parameters such as magnetic field strength, radiation, velocity slip and suction contribute to enhancing the heat transfer rate of THNF. Furthermore, the results indicate that the THNF used in this study exhibits a better heat transfer rate compared to nanofluid (NF) and hybrid nanofluid (HNF). Notably, a shrinking sheet is observed to exhibit multiple solutions when the shrinking parameter falls within a defined range, specifically when lambda > lambda(c). By scrutinizing the analysis of stability, the first solution (upper solution) was determined to be consistently stable and applicable in real-world settings. These findings offer insightful information into the optimization of heat transfer processes in nanofluid-based systems under complex flow conditions. However, these findings apply only to a THNF mixture of alumina, copper, and titania. THNFs may have different flow dynamics and thermal properties depending on the mixtures of nanoparticles.

This study investigates the Blasius flow and heat transfer characteristics of a copper-alumina-titania/water ternary hybrid nanofluid over a moving flat plate. A similarity transformation is applied to reduce the governing partial differential equations into a system of similarity equations, which are then solved numerically using the Matlab-based boundary value solver. The analysis reveals the existence of dual solutions under certain flow conditions, particularly when the plate and free stream move in opposite directions. An increment of 0.5% titania concentration led to the upsurge of skin friction and heat transfer coefficients which shows that the copper-alumina-titania/water exhibits superior flow and heat transfer performances as compared to the copper-alumina/water. These findings provide a basis for optimizing operational efficiency and designing processes tailored to specific outcomes.

This paper reports on the fluid flow characteristics as well as the heat transfer attribute of a Reiner-Philippoff (RP) fluid past a permeable shrinking wedge with a particular focus on the incorporation of the AA7075-AA7072/methanol hybrid nanofluid. Through the application of suitable transformations, the original model in partial differential equations (PDEs) is converted into ordinary differential equations (ODEs) of a specific form. The ODEs are then solved using the bvp4c solver in MATLAB software. The findings showed that when the magnitude of the magnetic parameter is increased, skin friction and heat transfer rate both are increased. Moreover, the inclusion of hybrid nanoparticles has a positive impact on the system, leading to a 6.16% increment in magnitude of skin friction while boosted about 24% improvement in thermal performance. The confirmation of dual solutions leads to a study of stability analysis to examine the reliability of the first solution. It is important to note that the current findings are novel and original for the study of RP hybrid nanofluid past a permeable shrinking wedge.

It is commonly known that fluids behave differently depending on the application. Couple stress, which is related to the fluid’s internal tensions, is one of its most important and distinguishing properties. This characteristic is essential for modeling fluid flow in microchannels, where elevated shear rates distinguish the flow. Hence, this study offers a numerical investigation of convective heat transfer in a couple stress alloys/methanol (AA7072-AA7075/CH3OH) hybrid nanofluid. Similarity transformations are applied to convert the governing partial differential equations into a system of ordinary differential equations and solved using MATLAB’s bvp4c function. The result shows that the presence of couple stress effects and hybrid nanoparticles has a significant impact on both velocity and temperature distributions. It is observed that larger values of the couple stress parameter decrease skin friction and increase the thickness of the thermal boundary layer. Moreover, the suction parameter is also crucial in enhancing the heat by reducing boundary layer thickness, thereby increasing the local Nusselt number. However, higher concentrations of the nanoparticles produce larger viscosity which may diminish the efficiency of convective heat transfer. The results offer valuable insights into optimizing hybrid nanofluid properties for complex thermal management systems, such as cooling systems and energy devices.

Hybrid nanofluids have emerged as advanced heat transfer materials for industrial applications. In this study, Alumina (Al₂O₃) and Copper (Cu) nanoparticles, dispersed in a Carboxymethyl Cellulose (CMC)-water-based fluid, are considered to form a non-Newtonian hybrid nanofluid with shear thinning behaviour, chosen for their superior thermophysical properties, stability, and practical applicability in advanced thermal management systems. By incorporating the non-Newtonian behavior of the Williamson fluid model together with the synergistic effects of hybrid nanoparticles, this study achieves a more accurate representation of practical fluid flows in industrial and engineering applications. Through appropriate transformations, the governing equations are reduced to similarity equations, which are then resolved using MATLAB’s bvp4c solver. Model accuracy is verified by comparing the results with an existing model, demonstrating good agreement. This study explores the effects of several fluid parameters, including mixed convection, suction, nanoparticle concentration, and the Williamson parameter, on fluid flow. The results reveal that increased mixed convection and suction enhance heat transfer performance, whereas higher Williamson parameter values and nanoparticle concentrations reduce heat transfer. Overall, the findings provide significant insights into the behaviour of hybrid nanofluids in non-Newtonian flows and offer a theoretical foundation for their application in heat transfer enhancement strategies across diverse engineering systems.

A heat source is frequently employed in the synthesis of new materials, the cooling of electronic devices, and the production of paper. Meanwhile, Joule heating is the process by which electrical energy is converted into thermal energy as a result of resistive losses, causing the generation of heat. This phenomenon is frequently employed in the development of electrical and electronic devices. This study attempts to emphasise these two parameters' impact on the Al2O3-Cu/H2O hybrid nanofluid as it passes through a shrinking sheet with a power law heat flux. This problem utilized the bvp4c to solve boundary value problems related to systems of ordinary differential equations. The study reveals that the flow progress is unaffected by both the Joule heating and the heat source parameters. Moreover, the results have conclusively proven that the presence of these two physical parameters slightly underperforms the efficiency of heat transfer in the hybrid nanofluid. At the same time, the boundary layer separation is not significantly impacted. As a result of the increased heat production in the flow system, it is remarked that the addition of nanoparticle volume fraction results in poorer heat transfer performance than claimed.

employed as the combination of particles, along with water as the base fluid over a vertical Riga plate is numerically addressed. The performance of heat transmission is influenced by the electromagnetohydrodynamic (EMHD) imposed produced from the Riga plate, and it could be used to postpone boundary layer separation. A model in the form of Partial Differential Equations (PDEs) is introduced to describe the physical behavior of the proposed problem. With the inclusion of relevant equation variables, this solver was employed to solve the respective equations. The characteristics of fluid velocity and temperature are investigated graphically. It is found the buoyancy assisting and opposing flows offered dual solutions whereas the purely forced convection flow the volumetric concentration of Cu increases the heat-transferring ability for assisting and opposing flows. The higher suction

In this study, we perform a mathematical analysis of couple stress fluid flow and heat characteristics over ashrinking sheet enhanced with hybrid nanoparticles and considering heat generation effects. Solid nanoparticles ofaluminium alloys (AA7072 and AA7075) are suspended in methanol to form the hybrid nanofluid. The similarityapproach is applied to transform the governing equations into similarity equations, which are subsequently solved using MATLAB's bvp4c function. The solutions for flow and temperature fields, skin friction coefficient, and Nusselt number are presented in both tabular and graphical formats. The findings indicate that hybrid nanofluids exhibit superior thermal conductivity, resulting in a significant enhancement of the heat transfer rate. Specifically, the addition of 2% hybrid nanoparticles increases the heat transfer rate by 0.78% compared to the base fluid. Furthermore, heat generation contributes to the thickening of the thermal boundary layer, thereby influencing the fluid temperature and ultimately reducing the heat transfer rate of the hybrid nanofluid.

Hybrid nanofluids are designed to improve conventional nanofluids' stability and other thermal properties. The present work investigates the flow of combined convective transport and the influence of radiation on the studied flow. A hybrid nanofluid ( Cu Al O  2 3 /water) flows through a vertically inclined stretching/shrinking sheet. To simplify the governing equations, the deterministic two-variable differential equations (PDEs) are systematically transformed into a system of one-variable differential equations by using appropriate similarity transformations. The bvp4c function of the MATLAB program is also used to solve the simplified mathematical model. The present study investigates and presents in tabular and graphical form the effects of stretching/shrinking surfaces, suction, and volume fraction of the nanoparticles on the velocity and temperature profiles as well as on the engineering quantities. The present results are first validated and confirmed as acceptable before the full calculations are performed. Overall, the results of this study show that the investigated parameters influence the flow characteristics, which can serve as a controlling factor for heat transfer.

This study introduces a mathematical model that addresses the mixed convection stagnation point flow of a non-Newtonian Reiner-Philippoff hybrid nanofluid over a shrinking sheet. Through the application of theoretical assumptions, the governing equations were formulated and subsequently simplified into a set of ordinary differential equations (ODEs). The model facilitates the computation of steady flow solutions utilizing the MATLAB software function bvp4c. The primary objectives of this research include an analysis of the effects of various parameters on flow dynamics and thermal behaviour. These parameters encompass the mixed convection parameter, the solid volume fraction of nanoparticles, and the mass flux parameter, all of which significantly influence flow characteristics. Numerical results have been obtained for critical metrics, including the skin friction coefficient, local Nusselt number, and the velocity and temperature profiles. The findings contribute to a deeper understanding of hybrid nanofluid behaviour in thermal management applications, offering valuable insights for future research endeavours in the domains of fluid dynamics and heat transfer.

This research focuses on solving hypersingular integral equations (HSIEs) numerically for thermoelectric bonded materials (TEBM) subjected to shear stress and weakened by two slanted cracks. The modified complex variable function (MCVF) method is applied, integrating continuity conditions (CC) for both the electric field effect (EFE) and the displacement electric function (DEF), which are used to formulate the governing HSIEs. Utilizing a curved length coordinate approach, the unknown functions associated with crack opening displacement (COD), as well as constants for current vector field and surface energy load, are expressed in terms of the fracture singularity basis function. These HSIEs are solved numerically using suitable quadrature techniques, with the crack traction as the right-hand term. The solutions for COD, current vector field, and surface energy load are then employed to calculate the dimensionless stress intensity factors (DSIFs), which offer insights into the stability of TEBM with two slanted cracks. Numerical simulations demonstrate that the computed DSIFs at the crack tips are consistent with previous research. Additionally, the results show that DSIFs are significantly affected by factors such as the ratio of elastic constants (ECR), crack geometry, and current vector field coefficients.

This study investigates the combined influence of magnetic field, suction effect and convective boundary condition on the flow and heat transfer characteristics of couple stress hybrid nanofluid (HNF) flow subjected to a stretching surface. The governing equations for the couple stress hybrid nanofluid flow are formulated and solved numerically using the efficient computational bvp4c solver embedded in the Matlab software. The hybrid nanofluid composed of a base fluid (methanol) and hybrid nanoparticles (AA7072 and AA7075-aluminium alloys) is characterized by considering the single phase nanofluid model. Validation is conducted with previously published results indicating that the presented model is correct. The analysis considers the presence of an external magnetic field, which has a significant influence on the flow and heat transfer behaviours (skin friction and heat transfer) as well as the velocity and temperature profiles. The findings contribute to a deeper understanding of the coupled effects of MHD and Biot number (generated by the convective boundary condition) on the hybrid nanofluid's flow and thermal progress. However, the results are applicable for the stretching plate only. Further, such insights are crucial for optimizing thermal management systems and enhancing heat transfer processes in advanced engineering applications.

This study is carried out to scrutinize the Hiemenz flow for ternary hybrid nanofluid flow across a stretching/shrinking sheet. This study aims to inspect the impacts of variations in the stretching/shrinking parameter and the volume fraction of nanoparticles on key aspects of the ternary hybrid nanofluid flow, specifically the skin friction, Nusselt number (which relates to heat transfer), velocity profiles, and the temperature profiles. The flow equations transform into a system of ordinary differential equations (ODEs) using a similarity transformation. Subsequently, the system is numerically solved using the MATLAB software’s 4th-order accuracy boundary value problem solver, known as “bvp4c”. Numeric findings reveal that skin friction values exhibit variations based on the magnitude of the stretching/shrinking parameter. Moreover, in the specific context of the flow problem being studied, the heat conduction efficiency of the hybrid (ternary) nanofluid surpasses that of the hybrid nanofluid. The system yields two distinct solutions within a specific shrinking/stretching parameter interval. Through an examination of the temporal stability of the solutions, it was determined that only one remained stable over an extended period. Remember that these current findings hold solely for the combination of copper, alumina, and titania.

Kesan nanozarah hibrid terhadap aliran bendalir Eyring-Powell pada permukaan mengecut dengan halaju hukum kuasa dikaji. Penjelmaan keserupaan yang sesuai digunakan untuk mengubah persamaan menakluk kepada persamaan keserupaan. Penyelesai masalah nilai sempadan bvp4c dalam perisian MATLAB digunakan untuk mendapatkan penyelesaian berangka. Hasil kajian mendapati bahawa nanozarah hibrid meningkatkan kedua-dua kecerunan halaju dan suhu, yang seterusnya meningkatkan geseran pada permukaan dan kadar pemindahan haba masing-masing pada 5.01% dan 0.59% berbanding bendalir asas. Namun, kuantiti fizikal tersebut menurun dan domain penyelesaiannya terjejas dengan kehadiran parameter bendalir Eyring-Powell. Daripada analisis kestabilan, hanya satu daripada dua penyelesaian tersebut stabil dalam jangka masa panjang.