A nonlinear permeable shrinking sheet refers to a boundary where the permeability changes nonlinearly, and the sheet itself is contracting or shrinking over time. Meanwhile, magnetohydrodynamics (MHD) models electrically conducting fluids as a single continuous medium. Hence, this research integrates magnetohydrodynamics (MHD) models electrically conducting fluids as a single continuous medium. The distribution of three nanoparticles—aluminum oxide (Al2O3), titanium dioxide (TiO2), and silver (Ag)—in water (H2O) is considered to represent the ternary nanofluid model. Two analyses are performed: (i) numerical analysis—to describe the mathematical model; and (ii) statistical analysis—to optimize the heat transfer rate using Response Surface Methodology (RSM). The numerical results are obtained through bvp4c scheme in MATLAB after applying similarity transformation to reduce the governing equations. The findings show dual solutions due to the shrinking parameter, while ternary nanoparticles enhance heat transfer more than mono- or hybrid nanofluids. The nonlinearity parameter increases temperature profiles, whereas heat generation decreases them. Further, Response Surface Methodology (RSM) is applied to optimize the highest heat transfer rate by identifying three optimal parameter values for nonlinearity, radiation, and heat generation. With a suggested desirability of 99.98

Purpose This study aims to investigate the behaviour of unsteady three-dimensional (3D) boundary layer flow and heat transfer generated by a non-axisymmetric stagnation point over a shrinking surface. The influence of thermal radiation is also included to represent realistic thermal environments. In particular, the work here focuses on how the shear-to-strain rate parameter and shrinking effects influence the flow structure, heat transfer rate, and stability of the solution. Design/methodology/approach The governing partial differential equations describing the mass, momentum, and energy transport are reduced into a system of ordinary differential equations using an appropriate similarity transformation for the non-axisymmetric stagnation point flow. The resulting boundary value problem is solved numerically using the bvp4c solver. In addition, stability analysis is carried out to distinguish physically meaningful solutions between dual solution branches. An asymptotic analysis is also developed for large values of the shear-to-strain rate parameter to provide analytical inputs into the limiting behaviour of the system. Findings The results show that the shear-to-strain rate parameter has a significant impact on the flow and thermal fields. An increase in this parameter leads to a reduction in both the reduced skin friction coefficients and the heat transfer rate at the surface. The presence of a shrinking surface generates dual solutions due to bifurcation behaviour, resulting in upper and lower solution branches. Stability analysis confirms that upper solution corresponds to a physically stable flow configuration. Furthermore, the asymptotic results indicate that both skin friction and heat transfer increase in proportion to the square root of the shear-to-strain rate parameter for large parameter values, which corresponds to thinner momentum and thermal boundary layers. Practical implications The findings provide information into controlling flow and heat transfer in systems involving stagnation point flow over shrinking or stretching surfaces. Understanding the stability characteristics helps in identifying physically realizable operating conditions, which is important in industrial processes such as aerodynamic surface design, cooling technologies, and material processing where non-uniform strain fields may occur. Originality/value This work extends the existing stagnation point flow studies by incorporating a non-axisymmetric 3D configuration with unsteady effects, thermal radiation, and shrinking surface conditions. The main originality of this work lies in the combined use of numerical simulation, stability analysis, and asymptotic analysis within a single non-axisymmetric stagnation point flow framework, which has not been simultaneously addressed in previous studies. The integration of these three approaches provides more understanding of the flow behaviour across different parameter regimes.

Achieving broadband plasmonic absorption across the entire solar spectrum remains a critical challenge for direct absorption solar collectors (DASCs), as single-morphology nanoparticles exhibit narrow spectral response limited to specific wavelength bands. This study introduces a quaternary morphological blending strategy that synergistically combines four distinct Ag nanoparticle geometries, nanocubes, nanodisks, nanoprisms, and nanorods to achieve continuous spectral coverage from 300 to 1500 nm. Finite element method (FEM) simulations optimized dimensional parameters for each morphology, while three-dimensional CFD modeling with the discrete ordinate radiation model (DORM) evaluated DASC thermal performance. The optimized blend (31% nanocubes, 23% nanodisks, 25% nanoprisms, 21% nanorods) achieved 99.4% spectral photothermal conversion efficiency at an ultra-low concentration of 0.0002 vol%, representing a 301.2% enhancement over pure water, while three-dimensional CFD modeling computationally predicted a DASC thermal efficiency of 83.3% at Re = 77, accounting for thermal losses and scattering effects. Sensitivity analysis confirmed system robustness, with all perturbed configurations maintaining efficiencies exceeding 97%. This quaternary approach establishes a new paradigm for nanofluid engineering, demonstrating that morphological diversity rather than increased particle loading governs optimal solar thermal conversion.

PurposeThis study aims to investigate a mixed convection stagnation-point flow and heat transfer of a ternary nanofluid over a vertical linearly stretching/shrinking sheet. The ternary nanofluid consists of aluminium oxide (Al2O3), copper oxide (CuO) and silver (Ag) nanoparticles dispersed in water (H2O). The effects of surface permeability and thermal radiation on flow and thermal characteristics are examined, with emphasis on heat transfer enhancement and parameter optimization.Design/methodology/approachThe governing partial differential equations describing momentum and energy transport are transformed into a system of ordinary differential equations using a similarity transformation approach. The resulting equations are solved numerically in MATLAB using the bvp4c solver. To further evaluate the parameter influence and optimize thermal performance, response surface methodology and normalized sensitivity analysis are employed. A predictive correlation relating key parameters to the heat transfer response is also developed.FindingsThe analysis shows that ternary nanofluid yields the highest heat transfer rate compared to hybrid and mono-nanofluids under identical operating conditions. Dual solutions are observed in the shrinking regime which indicates multiple flow states. Furthermore, statistical analysis reveals the relative influence of Al2O3, CuO and Ag volume fractions on the heat transfer rate, where Ag exhibits the strongest influence on the heat transfer rate due to its high thermal conductivity, and the optimized combination of these ternary nanoparticles leads to maximum heat transfer performance.Practical implicationsThe combined numerical and statistical framework offers a structured approach for improving thermal transport in nanofluid-based systems. The findings may assist in the design of enhanced cooling and thermal regulation applications where improved heat transfer efficiency is required, as well as contributes to energy efficiency efforts that aligned with sustainable development goals (SDG 7).Originality/valueThis work integrates similarity-based numerical modelling with statistical optimization and normalized sensitivity analysis for a ternary nanofluid system under mixed convection stagnation-point flow. The development of a predictive correlation and systematic parameter ranking also provides further information into multi-nanoparticle heat transfer behaviour.

In electromagnetism, a magnetic dipole is a tiny loop of electric current or a pair of magnetic poles. As the loop size decreases to zero while maintaining a constant magnetic moment, it forms a magnetic dipole. Composed by the magnetic particles, ferromagnetic fluids align with magnetic fields and when a magnetic dipole interacts with such fluids, the particles magnetize the fluid and influence the dipole's field. Hence, this study investigates the magnetic dipole and velocity slip on ternary hybrid ferrofluid flow past a shrinking surface. The model considers three magnetic nanoparticles - iron oxide (Fe3O4), cobalt ferrite (CoFe2O4), and copper (Cu) - dispersed in a base fluid. The similarity transformation technique is applied to derive mathematical models, which were solved numerically using bvp4c program in MATLAB. The analysis reveals that ferrohydrodynamic interaction reduces the skin friction coefficient and heat transfer rate but enhances velocity and temperature profiles. Additionally, the ternary hybrid ferrofluid is also shown to outperform both conventional ferrofluid and hybrid ferrofluid in fluid flow characteristics. Response Surface Methodology (RSM) is employed to identify the optimal combination of parameters, suggesting that the highest ferrohydrodynamic parameter and viscous dissipation, along with minimal Cu-nanoparticle concentration, maximize the heat transfer rate. Contour and surface plots illustrate these optimal conditions. This study highlights an innovative application of ternary ferrofluid with a magnetic dipole and employs RSM to optimize parameters for enhanced heat transfer performance, addressing a gap in existing literature and providing the way for further advancements in this field.

Purpose This study aims to investigate the behavior of a ternary nanofluid composed of alumina (Al2O3), copper (Cu) and titania (TiO2) nanoparticles dispersed in water (H2O) flowing over a permeable shrinking surface. Thi study applies the modified Buongiorno model, which accounts for Brownian motion, thermophoresis, gyrotactic microorganisms and radiation. The main aim is to analyze the thermal characteristics of current flow and to determine the optimal nanoparticle composition for maximizing heat transfer rates. Design/methodology/approach Two types of analysis are conducted. The numerical analysis is first involved by formulating the governing differential equations using a similarity transformation technique. The equations are then solved using the bvp4c solver in MATLAB, where detailed graphical results of velocity and temperature profiles are presented. At the same time, a statistical analysis is performed to optimize the heat transfer rate using response surface methodology (RSM) and the Taguchi method in Minitab software. Findings The numerical results reveal that the shrinking parameter produces dual solutions (first and second branches), where the first branch of the solution is proven to be stable by a temporal stability analysis. Notably, ternary nanofluid flow achieves the highest temperature distribution compared to single and hybrid nanofluids. The presence of radiation and thermophoresis parameters is also demonstrated to significantly increase the temperature profiles. From the optimization, the RSM suggests that the highest number of Al2O3, Cu and TiO2 nanoparticles predicts the maximum heat transfer rate at 33.6%, while the Taguchi method estimates it at 32.3%. Practical implications The current mathematical model and statistical approach used in this study can be applied in areas such as biomedical fluids or drug delivery systems, where improved heat transfer and fluid stability are important in achieving optimal outputs. The ability to control and enhance thermal properties through nanoparticle composition and surface conditions can benefit various engineering and medical applications. Originality/value The originality of this work lies in applying both RSM and the Taguchi method to a ternary nanofluid system modeled by the modified Buongiorno model. While the Buongiorno model has been widely used in nanofluid studies, its combination with statistical optimization techniques for a ternary nanoparticle mixture provides a new understanding of thermal performance concepts. This is especially true with gyrotactic microorganisms and radiation. The dual approach of optimization techniques allows for both theoretical and practical optimization of nanofluid-based systems.

Jeffrey model refers to a type of non-Newtonian fluids where it captures both viscous and elastic properties of the fluid. It builds on the Oldroyd-B fluid model by adding a feature called retardation time, which represents the delay of a fluid experiences in responding to stress. Therefore, we keen to study on the Jeffrey stagnation point ternary hybrid nanofluid flow over a permeable shrinking surface with heat generation, velocity slip, and thermal slip. The ternary hybrid nanofluid model is developed using three nanoparticles: alumina (Al2O3), copper (Cu), and titania (TiO2), with water (H2O) serving as the base fluid. The mathematical derivation involves applying a similarity transformation technique to a set of partial differential equations, reducing them to non-linear ordinary differential equations. These equations are then solved using MATLAB’s bvp4c function. In the meantime, two branches of solutions are obtained via the shrinking parameter. The analysis compares mono, hybrid, and ternary nanofluids, revealing that the three nanoparticles achieve the highest heat transfer rate. The findings also show that the Deborah number, velocity slip, and heat generation parameter increase the temperature distribution, while thermal slip reduces them. Additionally, Response Surface Methodology (RSM) was employed to optimize the volume fractions of the three nanoparticles, which confirming that their highest values maximize the heat transfer rate. The findings from this study suggest promising applications for ternary hybrid nanofluids that work with optimization processes such as high-performance solar thermal collectors, geothermal energy systems, or medical thermal therapies.

Purpose Magnetohydrodynamics (MHD) in nanofluids is crucial in boundary layer flow as it enables the manipulation of fluid motion through magnetic fields, which leads to improved stability and efficiency. This study aims to introduce a model and solutions for the boundary layer flow of a ternary hybrid nanofluid past a permeable shrinking sheet, integrating both magnetohydrodynamic and slip effects. Design/methodology/approach The model is firstly expressed as partial differential equations and subsequently converted into ordinary differential equations (ODEs) through a similarity transformation technique. A finite difference scheme with the Lobatto IIIa formula in MATLAB is applied to numerically solve the ODEs, where the respective outcomes provide insights into the skin friction coefficient, Nusselt number, velocity profiles and temperature profiles. Findings The results highlight the significance of enhancing magnetohydrodynamic effects and first-order velocity slip to reduce skin friction, improve heat transfer, delay boundary layer separation, increase flow velocity and lower fluid temperature. In addition, the stable numerical solution is scrutinized using response surface methodology (RSM) to validate and optimize flow control. The RSM optimization confirms that higher suction, magnetohydrodynamic effects and first-order slip levels are essential for minimizing skin friction and maximizing heat transfer simultaneously. Originality/value The presented model together with the numerical and statistical results can be used as a guidance to control the flow and heat transfer that occur within a related practical application, especially in engineering and industrial activities such as cooling technologies, energy harvesting or fluid transport in nanotechnology, where precise control of heat transfer and fluid dynamics is essential for optimizing performance and reducing energy consumption.

This study investigates the boundary layer flow of penta-hybrid nanofluid with suction and velocity slip over a stretching/shrinking sheet. The exploration of penta-hybrid nanofluid, which consist of five different nanoparticles, is motivated by the synergistic effects that enhance heat transfer performance. A mathematical model is formulated in the form of partial differential equations based on the fluid flow configuration, which are then reduced to ordinary differential equations and solved numerically using the finite difference scheme in MATLAB. Dual solutions are obtained, and the impacts of various parameters are analyzed. This study demonstrates that penta-hybrid nanofluid with a high concentration of titanium dioxide significantly enhances the heat transfer rate compared to those with lower concentrations, particularly when operating under conditions of lower suction and velocity slip at the boundary, with the sheet in a stretching condition. Additionally, the increase in suction and velocity slip allows the sheet to shrink more extensively, which helps delay boundary layer separation and prevents the turbulent flow. Furthermore, Supervised 2 Satisfiability Reverse Analysis (S2SATRA) is employed to understand parameter interactions and optimize heat transfer performance. The analysis identifies the best-induced logic for accurately predicting the heat transfer rate, which is expressed as: Re_x^ - 1/2Nu_x = ( λ∨ϕ_ 5) ∧( ϕ_ 2∨ϕ_ 1) ∧( ϕ_ 3∨ ϕ_ 4) and Re_x^ - 1/2Nu_x = ( S ∨ϕ_ 4) ∧( λ∨ ϕ_ 3) ∧( A ∨ϕ_ 5) .

Purpose - This paper aims to explore dual solutions for the fl ow of a hybrid nanofluid over a permeable melting stretching/shrinking sheet with nanoparticle shape factor, second-order velocity slip conditions and viscous dissipation. The hybrid nanofluid is formulated by dispersing alumina (Al2O3) and copper (Cu) nanoparticles into water (H2O). Design/methodology/approach - The governing partial differential equations (PDEs) are fi rst reduced to a system of ordinary differential equations (ODEs) using a mathematical method of similarity transformation technique. These ODEs are then numerically solved through MATLAB's bvp4c solver. Findings - Key parameters such as slip parameter, melting parameter, suction parameter, shrinking parameter and Eckert number are examined. The results reveal the existence of two distinct solutions (upper and lower branches) for the transformed ODEs when considering the shrinking parameter. Increasing value of Cu-volume fraction and the second-order velocity slip enhances boundary layer thicknesses, whereas the heat transfer rate diminishes with rising melting and suction parameters. These numerical results are illustrated through various fi gures and tables. Additionally, a stability analysis is performed and confirms the upper branch is stable and practical, while the lower branch is unstable. Practical implications - The analysis of hybrid nanofluid fl ow over a shrinking surface has practical significance with applications in processes such as solar thermal management systems, automotive cooling systems, sedimentation, microelectronic cooling or centrifugal separation of particles. Both steady and unsteady hybrid nanofluid fl ows are relevant in these contexts. Originality/value - While the study of hybrid nanofluid fl ow is well-documented, research focusing on the shrinking fl ow case with specific parameters in our study is still relatively scarce. This paper contributes to obtaining dual solutions specifically for the shrinking case, which has been less frequently addressed.

A nanofluid refers to a suspension of nanoparticles in a conventional fluid, which finds unique applications in diverse sectors, including engineering, technology, and medicine. When multiple nanoparticles are suspended, it creates a hybrid nanofluid. In this study, we aim to investigate an unsteady flow of hybrid nanofluid over a permeable shrinking inclined rotating disk subjected to heat radiation, magnetohydrodynamics and slip effects. The chosen nanoparticles for this study are alumina (Al2O3) and copper (Cu), incorporated into a base fluid of water (H2O) to create the hybrid nanofluid. An appropriate method of similarity transformation is executed along a set of partial differential equations that were reduced to a system of nonlinear ordinary differential equations, where numerical outcomes were then obtained via bvp4c in MATLAB software, with the influence of various parameters such as unsteadiness parameter, nanoparticle volume fraction, shrinking, radiation, magnetic and velocity slip parameters, shown in tables and figures. Multiple solutions (including dual, upper, and lower branch solutions) are identified for the governing similarity equations. Through the conducted stability analysis, it is determined that the upper branch solutions exhibit stability and physically realizable in practice, while the lower branch solutions are unstable. Our numerical findings showed that dual solutions exist when ε_c≤ε≤ - 1 , where ε_c < 0 is the critical value of ε for which the boundary value problem poses physical solutions applicable in practice. Yet, the boundary value problem lacks a similarity solution for ε≤ε_c≤ 0 , and the complete set of partial differential equations needs to be solved numerically. Improvements in heat transfer rate are observed concerning the radiation parameter, nanoparticle fraction, and shrinking parameter. Furthermore, azimuthal velocity profiles show an increase influenced by velocity slip and magnetic parameters. The non-dimensional physical parameters, including stretching/shrinking, suction, slip, and unsteadiness, are also considered and their effects are presented in figures and tables.

A nanofluid refers to the dispersion of nanoparticles in a regular fluid and has a unique application in various sectors, including medicine, engineering, and technology. When multiple nanoparticles are suspended in a regular fluid, it creates a hybrid nanofluid. In this study, we aim to investigate homogenous–heterogenous reactions in Bödewadt hybrid nanofluid flow over a permeable rotating disk with radiation. The base fluid chosen for this study is water (H2O), while the nanoparticles iron oxide (Fe3O4) and cobalt ferrite (CoFe2O4) are utilized to create the hybrid nanofluid. An appropriate method of similarity transformation is executed along a set of partial differential equations (PDEs) that were reduced to a system of nonlinear ordinary differential equations (ODEs). Numerical outcomes were then obtained via bvp4c in MATLAB software, with the influence of various parameters such as nanoparticle volume fraction, homogenous/heterogenous reaction strength parameters, suction, shrinking/stretching parameters, and radiation parameter. Additionally, asymptotic analysis was conducted to show that the concentration boundary layer on the disk can be performed subject to a large number of suctions. The present findings reveal that a rise in the volume fraction of nanoparticles results in a reduction in radial velocity profiles, temperature profiles, and tangential fields. As thermal radiation levels rise, a notable reduction in the local Nusselt number is evident. Moreover, there is an observed linear escalation in wall surface concentration when the heterogeneous strength parameter attains higher values. The presented results demonstrate that all flow fields are significantly affected by the participating parameters.

This research aimed to develop a numerical solution to analyze the effects of solar radiation and nanoparticle shape factors on the flow of a hybrid nanofluid past a shrinking Darcy-Forchheimer porous medium. The base fluid chosen for this study is water (H2O), and the hybrid nanofluid consists of nanoparticles of silver (Ag) and titanium dioxide (TiO2) in four different shapes: bricks, cylinders, platelets, and blades. To account for solar radiation, the energy model incorporated a radiative heat flux, while the momentum problem considers the influence of a magnetic field. The application of an appropriate similarity transformation method converts the partial differential equations (PDEs) model into a system of nonlinear ordinary differential equations (ODEs). The mathematical model is solved using the shooting technique method and the bvp4c solver. The obtained results, along with the effects of the nanoparticle shape factor, solar radiation parameter, shrinking parameter, Darcy-Forchheimer number, and nanofluid volume fraction, are visually presented through figures and tables. It is worth noting that, in our numerical results, we observed the presence of dual solutions when λ < 0. Our findings indicate that the thermal transmittance increases with an increase in the nanoparticle shape factor and solar radiative parameter. Additionally, we observed an escalation in the velocity distribution in relation to the shrinking parameter and nanofluid volume fraction. Before reaching the two solutions, a flow stability analysis revealed that the first branch appears to be the most stable. Overall, these findings provide valuable insights into the behaviour of hybrid nanofluid flow in the presence of solar radiation and porous media.

In this work, we investigate the unsteady flow of gyrotactic microorganisms over a shrinking surface in the presence of hybrid Ag-TiO2/H2O nanofluid. The hybrid nanofluid considered in our analysis comprises nanoparticles with distinct properties, improving the fluid’s thermal and transport characteristics. Additionally, we incorporate higher-order slip conditions to capture the complex interfacial dynamics. The mathematical model governing the flow is formulated using the Buongiorno-Tiwari-Das nanofluid framework or modified Buongiorno’s nanofluid model, which accounts for the impact of Brownian motion and thermophoresis on the nanoparticle distribution. The resulting nonlinear ordinary differential equations, which were derived from a set of partial differential equations by a similarity transformation technique, are solved via bvp4c method in MATLAB. We present a detailed parametric study to elucidate the influence of various physical parameters on the flow and microorganisms’ behavior. Our analysis reveals two distinct solutions when shrinking parameter λ<0, as well as the intricate interplay between gyrotaxis microorganisms, nanoparticle migration, unsteadiness flow, and slip effects on the current model. The presence of the Brownian motion constant was observed to enhance the heat transfer rate, nanofluid concentration and the mobility of microorganisms near the wall. The shrinking parameter was also found to increase the heat transfer rate, while the second-order slip parameter had a diminishing effect. Further, suction parameter and nanoparticle volume fraction positively influenced the velocity profiles. Prior to identifying dual solutions, a temporal stability analysis is performed, justifying the stability of the first solution.

We investigated the mixed convection boundary layer flow over a permeable surface embedded in a porous medium, filled with a nanofluid and subjected to thermal radiation, magnetohydrodynamics (MHD) and internal heat generation. The nanofluid consists of water (H2O) as the base fluid and nanoparticles such as copper (Cu), aluminium oxide (Al2O3) and titanium dioxide (TiO2). The governing system nonlinear partial differential equations is transformed into a set of ordinary differential equations using a similarity transformation, which are then solved numerically for various parameter values. The numerical solutions are obtained using the shooting technique method and bvp4c method, via MAPLE and MATLAB, respectively. Our findings revealed that the velocity distribution decreases with the shrinking parameter, while the presence of nanoparticles enhances the respective profiles. The velocity profiles were also observed to exhibit mixed patterns influenced by magnetic, radiation, and suction parameters. Further, the solutions bifurcated into two branches prior to the shrinking parameter. A stability analysis is performed to determine the stability of the solutions between two branches. We thoroughly discussed the characteristics of the respective solutions and their stability in detail.

The objective of this study is to investigate the effects of mixed convection flow dispersed with a hybrid nanofluid over a permeable shrinking surface past a stagnation-point region considering the influence of second-order velocity slip and variable viscosity on the flow behaviour. For the hybrid nanofluid, water (H2O) is chosen to be the base fluid, while silver (Ag) decorated copper oxide (CuO) nanoparticles are employed as the hybrid component. To achieve the mathematical model, a suitable method of similarity transformation is applied to convert the partial differential equations (PDEs) model into a system of non-linear ordinary differential equations (ODEs). The shooting technique method and bvp4c solver in MAPLE and MATLAB are employed to obtain the analytical solutions of the mathematical model. The obtained results, including the impacts of variable viscosity, second-order velocity slip, mixed convection parameter, suction, shrinking parameter, and nanofluid volume fraction, are presented through tables and figures. The study reveals the existence of dual solutions (upper and lower branches) prior to shrinking sheet . Furthermore, the thermal distribution exhibits mixed behaviours with respect to the variable viscosity number and second-order slip parameter, while demonstrating an increase with the presence of Ag- . The velocity distribution experiences an enhancement with both concentration and variable viscosity number. Stability analysis is then employed and shows that the first branch is stable, whereas the second branch exhibits an opposite outcome.

The objective of this research is to study the significance of second-order velocity slip in a Darcy-Forchheimer porous medium with the hybrid nanofluid flow toward a permeable shrinking surface. Heat generation and radiative heat flux are introduced in the energy model. Two distinct nanoparticles of aluminum oxide (Al2O3) and copper (Cu) are used to represent the hybrid nanofluid flow with water (H2O) as the base fluid. An appropriate method of similarity transformation is applied to reduce a PDE system into a model of non-linear ODEs. With the aid of a bvp4c solver in Matlab, the respective findings are graphically presented for the profiles of velocity and temperature, skin friction coefficient, and Nusselt numbers with physical parameters, such as suction, porous medium permeability, Darcy-Forchheimer number, shrinking, radiation, and nanoparticle volume fraction. The hybrid nanofluid presented a higher estimation of heat and mass transfer rates than the classic mono-nanofluid. Moreover, the parameters of Darcy-Forchheimer number and second-order velocity slip significantly expand the fluid flow. It is found that the occurrence of opposing flow (lambda < 0) will generate two solutions, where the implementation of stability analysis perceived the first solution to be the most realizable.

The major objective of this study is to identify the behaviour of flow and heat transfer on mixed convection with the appearance of hybrid nanofluid in a porous medium, heat generation, suction/injection, thermal radiation and magnetohydrodynamics (MHD). Two distinct fluids of Ag (silver) and TiO2 (titanium dioxide) are used in this model to represent the hybrid condition and a model of non-linear ODEs is obtained by conducting a technique of similarity transformation. The findings and solutions are gained with the aid of the shooting method in Maple and the bvp4c solver in Matlab. The outcomes of dimensionless velocity and temperature profiles are exemplified quantitively through respective figures with performing parameters such as suction, mixed convection, thermal radiation and volume fraction of nanoparticle. Through our comparison analysis, the application of added hybrid nanoparticles revealed a better performance of thermal transmittance compared to the traditional mono-nanoparticle. We also identified two different branches of solutions in the occurrence of λ < 0 (opposing flow). In addition, a stability analysis was conducted throughout this research due to the existence of two levels of solutions, and the first branch is perceived to be the most solid and recognizable solution.

The study of hybrid nanofluid and its thermophysical properties is emerging since the early of 2000s and the purpose of this paper is to investigate the flow of hybrid nanofluid over a permeable Darcy porous medium with slip, radiation and shrinking sheet. Here, the hybrid nanofluid consists of Cu/water as the base nanofluid and Al2O3–Cu/water works as the two distinct fluids. The governing ordinary differential equations (ODEs) obtained in this study are converted from a series of partial differential equations (PDEs) by the appropriate use of similarity transformation. Two methods of shooting and bvp4c function are applied to solve the involving physical parameters over the hybrid nanofluid flow. From this study, we conclude that the non-uniqueness of solutions exists through a range of the shrinking parameter, which produces the problem of finding a bigger solution than any other between the upper and lower branches. From the analysis, one can observe the increment of heat transfer rate in hybrid nanofluid versus the traditional nanofluid. The results obtained by the stability of solutions prove that the upper solution (first branch) is stable and the lower solution (second branch) is not stable.

Communication poses several problems, particularly to English second language (ESL) learners.In overcoming these problems, some ESL learners employ communicative strategies.This study reports the efforts undertaken by Malaysian ESL learners in enhancing their communication with the native speakers during collaborative cross-cultural online activities through Astronomy Online Lab (AOL).This AOL module was utilized to achieve twofold objectives namely enhancing students' communication skills and promoting knowledge exchange during the live discussions.29 undergraduate students from a Malaysian public university and a UK university explored the Astronomy topics online and experienced the simulation during the two live sessions conducted for one month.Data were collected from a face-to-face interview with all the Malaysian students.The students' expectations before the live sessions, difficulties confronted when communicating with the native speakers, the communicative strategies and experience during the interactive activities were explored.The findings revealed that conversational issues such as the native's accent and the challenge in formulating strategies for solving communication problems during the conversations posed difficulties to the students.A general pattern of indirect and interactional communicative strategies as promoted by Dörnyei and Scott (1997) was evident during the interactions.