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.

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.

PurposeThis paper aims to study the natural convection from a heated T-open pipe of nanoencapsulated phase change material in a cavity. The impact of the presence of nanoencapsulated phase change materials (NEPCM) in water was studied on the thermal behavior of these novel nanoliquids in the presence of natural convection flows. The entropy generation for these nanoliquids was also investigated. Design/methodology/approachThe NEPCM is modeled as a lumped phase change nanoparticle with a phase change material core and a polymer shell. The governing equations for a uniform mixture of NEPCM-water are written based on the conservation of mass, energy and also fluid motion. The natural convection effects were also taken into account. The finite element method was used to solve the governing equations. The entropy generation was also computed and studied. FindingsIncreasing the aspect ratio (AR) from 0.05 to 0.2 enhanced the average Nusselt number by 9%, while total entropy generation rose by 13%, indicating improved convective heat transfer near the bottom wall due to increased surface area. Enhancing the NEPCM nano particles volume fraction from 0 to 0.05 led to a 15% increase in heat transfer efficiency and a 9% rise in entropy, with negligible change in flow patterns. Growing the NEPCM fusion temperature from 0.1 to 0.5 slightly improved the Nusselt number by 5% and increased entropy by 3%, showing minor thermal gains with limited hydrodynamic impact. Practical implicationsNEPCMs have demonstrated significant potential in heat and mass transfer for cooling systems and thermal energy storage. Encapsulation technology has been widely used to improve the stability, specificity and bioavailability of essential food ingredients, as well as the performance of NEPCM suspensions in cooling applications. Additionally, the NEPCM suspensions use the latent heat of nanoparticles and can effectively control surface temperatures. Originality/valueThe natural convection heat transfer and the entropy generation of NEPCM suspension are addressed in an enclosure with T-open heated walls for the first time.

Purpose This study aims to investigate magnetohydrodynamic (MHD) flow and heat transfer over a nonlinear permeable stretching/shrinking sheet, incorporating the effects of viscous dissipation and Joule heating. This work addresses the gaps in understanding dual solutions and stability in such systems, with implications for industrial and thermal management applications. Design/methodology/approach The governing partial differential equations are reduced to a system of ordinary differential equations through the application of similarity transformations. Analytical solutions for reduced skin friction and heat transfer coefficients are derived under specific parametric conditions. Physical insights are extracted through graphical and tabular representations of key parameters: suction strength, Prandtl number, magnetic field intensity, viscous dissipation, mass flux and sheet shrinking rate. Findings Dual solutions (upper and lower branches) emerge for the shrinking sheet case, with the upper branch extending as suction increases. Stability analysis confirms the lower branch’s instability. Parametric studies reveal that suction, viscous dissipation and Joule heating significantly influence temperature profiles and boundary layer thickness, whereas magnetic effects dominantly alter flow dynamics. Practical implications The findings are critical for industrial processes involving stretching/shrinking sheets, such as polymer extrusion, glass production and metal rolling. Unlike stretching flows, shrinking sheet flows exhibit backward-flow behavior, necessitating external forces (e.g. suction or imposed flow) to stabilize the boundary layer. This study provides actionable strategies for optimizing thermal regulation and flow control in such systems. Originality/value This work advances the understudied area of MHD flow with combined viscous and Joule heating effects on nonlinear permeable sheets. Novel contributions include the identification of critical thresholds for dual solutions, stability characterization and the application of asymptotic methods to resolve complex ordinary differential systems. The results offer a framework for enhancing efficiency in thermal-fluid systems reliant on conductive media.

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.

This study analyzes unsteady ternary hybrid nanofluid flow and heat transfer over a generalized stretching/shrinking wall using both analytical and numerical methods. By applying similarity transformations, the governing nonlinear partial differential equations are reduced to a system of ordinary differential equations, which are numerically solved using the MATLAB bvp4c function. We find that the system exhibits two solution branches-an upper and a lower-within certain parameter ranges. A detailed stability analysis is conducted to determine the stability of these solutions. Additionally, the study presents analytical solutions for specific cases, which are relevant to heat exchangers in low-velocity environments. Next, MINITAB software is used to statistically model the interactions of the parameters and assess their impact on the heat transfer performance (measured through the local Nusselt number), identifying low, medium, or strong effects through regression analysis. Finally, a sensitivity analysis is performed on the regression function obtained in MINITAB, focusing on key input parameters. To the best of our knowledge, this study is novel, as no previous work has explored this problem, making both the analytical and numerical results original.

The purpose of this paper is to describe the stead MHD mixed convection flow over a permeable vertical flat plate embedded in a Darcy–Forchheimer porous medium. Using appropriate similarity variables, the partial differential equations are transformed into ordinary (similar) differential equations, which are numerically solved using the bvp4c function in MATLAB. The numerical results are used to present graphically and in tables, illustrations of the reduced skin friction, reduced Nusselt number, velocity, and temperature profiles. Dual (upper and lower branch) solutions are discovered in this exciting analysis. Although numerous studies on the mixed convection past a vertical plate embedded in a fluid-saturated porous medium exist, none of the researchers have focused on the Darcy–Forchheimer flow with asymptotic solutions. The behavior of the flow and heat transfer has been thoroughly analyzed with the variations in governing parameters, such as Darcy–Forchheimer G, suction/injection S , MHD M, and mixed convection λ parameters.

Purpose This paper aims to study numerically the steady natural convective heat transfer of a hybrid nanosuspension (Ag-MgO/H2O) within a partially heated/cooled trapezoidal region with linear temperature profiles at inclined walls under an effect of uniform Lorentz force. This investigation is useful for researchers studying in the area of cavity flows to know features of the flow structures and nature of hybrid nanofluid characteristics. In addition, a detailed entropy generation analysis has been performed to highlight possible regimes with minimal entropy generation rates. Design/methodology/approach The governing equations formulated using the Oberbeck–Boussinesq approach and single-phase nanoliquid model are transformed to a non-dimensional form by using non-dimensional variables. The obtained equations with appropriate boundary conditions are resolved by the finite difference technique. The developed code has been validated comprehensively. Analysis has been performed for a wide range of governing parameters, including Rayleigh number (Ra = 105), Prandtl number (Pr = 6.82), Hartmann number (Ha = 0–100), magnetic field inclination angle ( φ = 0–?/2) and nanoparticles volume fraction ( φ hnf = 0 and 2%). Findings It has been shown that inclined magnetic field can be used to manage the energy transport performance. An inclusion of nanoparticles without Lorentz force influence allows forming more stable convective regime with descending heat plume in the central zone, while such a regime was performed for clear fluid only for moderate and high Hartmann numbers. Moreover, the average overall entropy generation can be decreased with a growth of the Hartmann number, while an addition of hybrid nanoparticles allows reducing this parameter for Ha = 30 and 50. The average Nusselt number can be increased with a growth of the nanoparticles concentration for low values of the magnetic field intensity. Originality/value Governing equations written using the conservation laws and dimensionless non-primitive variables have been resolved by the finite difference approach. The created numerical code has been verified by applying the grid independence test and computational outcomes of other researchers. The comprehensive analysis for various key parameters has been performed.

Purpose This study aims to investigate the dual solutions for axisymmetric flow and heat transfer due to a permeable radially shrinking disk in copper oxide (CuO) and silver (Ag) hybrid nanofluids with radiation effect. Design/methodology/approach The partial differential equations that governed the problem will undergo a transformation into a set of similarity equations. Following this transformation, a numerical solution will be obtained using the boundary value problem solver, bvp4c, built in the MATLAB software. Later, analysis and discussion are conducted to specifically examine how various physical parameters affect both the flow characteristics and the thermal properties of the hybrid nanofluid. Findings Dual solutions are discovered to occur for the case of shrinking disk ( λ < 0). Stronger suction triggers the critical values’ expansion and delays the boundary layer separation. Through stability analysis, it is determined that one of the solutions is stable, whereas the other solution exhibits instability, over time. Moreover, volume fraction upsurge enhances skin friction and heat transfer in hybrid nanofluid. The hybrid nanofluid’s heat transfer also heightened with the influence of radiation. Originality/value Flow over a shrinking disk has received limited research focus, in contrast to the extensively studied axisymmetric flow problem over a diverse set of geometries such as flat surfaces, curved surfaces and cylinder. Hence, this study highlights the axisymmetric flow due to a shrinking disk under radiation influence, using hybrid nanofluids containing CuO and Ag. Upon additional analysis, it is evidently shows that only one of the solutions exhibits stability, making it a physically dependable choice in practical applications. The authors are very confident that the findings of this study are novel, with several practical uses of hybrid nanofluids in modern industry.

The bidirectional flow and thermal transfer of magnetohydrodynamics (MHD) and radiative nanofluid (magnetite-vacuum pump oil) due to a nonlinear shrinking surface in a three dimensional system is studied. The boundary layer model is first transformed into a set of ordinary differential equations using the similarity transformations, and then solved using the bvp4c solver. The accuracy of the present model is justified by comparing present data with the numerical values from the published findings. The effect of factors (magnetic parameter, radiation parameter and nanoparticles volumetric concentration) on the development of responses (skin friction coefficient and thermal rate) and critical value (separation value from laminar to turbulent flow) is observed through the graphical presentation. In addition, two solutions are attained where the first solution is affirmed as the reliable solution through stability analysis. Conclusively, the suction effect is necessary in generating the solutions under the phenomenon of opposing shrinking flow. The addition of magnetic parameter and nanoparticles concentration can enhance both responses as well as the critical value while the radiation parameter tends to reduce the heat transfer coefficient. The types of stretching/shrinking velocity (linear/nonlinear) also affect the heat transfer rate. The critical value can be extended by using the linear velocity, but, for thermal enhancement, the nonlinear form of velocity can significantly develop the thermal rate better than the linear shrinking surface.

Purpose This paper aims to present the steady dual solutions on three-dimensional flow and heat transfer of nanofluid over a permeable non-linearly shrinking surface with two-order velocity slips conditions. Boundary layer assumption is considered in the mathematical modelling. Authors comprehend from previous studies and papers that the shrinking surfaces are extremely important in current engineering and environmental systems. Design/methodology/approach Using appropriate similarity variables, the full partial differential equations (PDF) are modified into a specific set of ordinary (similar) differential equations (ODE). The resulting non-linear ordinary differential system is then solved both analytically for some particular cases and numerically for the general case using the function bvp4c from MATLAB for characteristic values of the parameters which govern the equations. The transformed mathematical model is analysed using the bvp4c procedure. Based on the given assumptions, this study is able to produce multiple solutions of the problem. Findings The ordinary (similarity) differential equations have two branches solutions, upper and lower branch solutions, given some interval of shrinking and velocity slip parameters. The authors consider here a temporal stability analysis, as they want to establish which of the solutions are stable and which are not. In a distinct paragraph, the authors discuss in detail and present in a graphical manner the effects of shrinking and second-order slip flow model on the skin friction coefficient, surface wall heat flux and dimensionless velocity and temperature profiles. The analysis reveals that the second order slip has a big influence on the flow and heat transfer characteristics. Originality/value The present discoveries are unique and truly new for the research of the three-dimensional stretching/shrinking forced convection flow and heat transfer nanofluids. The nanofluid is a water-based nanofluid (H 2 O), which contains one type of nanoparticles, namely, copper (Cu). Of course, the analysis can be further extended considering other types of nanoparticles such as alumina (Al 2 O 3 ). The authors assume that the thermal equilibrium is reached for the base fluid together with the suspended nanoparticles and that the nanoparticles are uniform in dimension and form.

The present paper gets together problems about unsteady boundary layers with industrial applications. We can enumerate some examples of applications as the flow over a flying helicopter, the fluid flow over the surface of turbines' blades and compressors, the fluid flow over the aerodynamic surfaces of flying vehicles, etc. The unsteadiness in the flow field is due to the external flux (or the surface of the body) or by external factors of stream (or the body surface). When the fluid movement close to a surface is created nonuniform, the inviscid flow in the region of that surface appear instantaneously but the viscous layer near the body is slowly created and, in time, it reaches fully-developed steady-state. This review paper deals with the unsteady boundary layer flow of nanofluids, hybrid nanofluids, micropolar fluids, and porous media. It is shown how the set of the governing partial differential models of the considered problems are formulated into nonlinear coupled ordinary differential equations using suitable similarity transformations. The problems are fully specified in terms of characterizing parameters, known as fluid nanofluids and hybrid particle interaction parameters, mass transfer parameters (suction and injection), micropolar parameters, mixed convection parameters, local Nusselt number, etc. The effects of these parameters on the velocity and temperature fields, the skin friction coefficient, and the local Nusselt number are presented using figures and tables, respectively. We emphasize that such a review of unsteady boundary layer flow has not presented before. It is well known that nanofluids have a determinant role in heat transfer problems, getting an augmentation of thermal conductivity by controlling their characteristics as per requirements. Nanofluids and hybrid nanofluids possess unique characteristics that have attracted many researchers over the past several decades to design new thermal systems for different engineering applications. Therefore, this review synthesizes the actual investigations on unsteady boundary layer papers, their thermo-physical properties, thermodynamic and hydrodynamic behaviours published by researcher.

This study is concerned with the three-dimensional (3D) stagnation-point for the mixed convection flow past a vertical surface considering the first-order and second-order velocity slips. To the authors' knowledge, this is the first study presenting this very interesting analysis. Nonlinear partial differential equations for the flow problem are transformed into nonlinear ordinary differential equations (ODEs) by using appropriate similarity transformation. These ODEs with the corresponding boundary conditions are numerically solved by utilizing the bvp4c solver in MATLAB programming language. The effects of the governing parameters on the non-dimensional velocity profiles, temperature profiles, skin friction coefficients, and the local Nusselt number are presented in detail through a series of graphs and tables. Interestingly, it is reported that the reduced skin friction coefficient decreases for the assisting flow situation and increases for the opposing flow situation. The numerical computations of the present work are compared with those from other research available in specific situations, and an excellent consensus is observed. Another exciting feature for this work is the existence of dual solutions. An important remark is that the dual solutions exist for both assisting and opposing flows. A linear stability analysis is performed showing that one solution is stable and the other solution is not stable. We notice that the mixed convection and velocity slip parameters have strong effects on the flow characteristics. These effects are depicted in graphs and discussed in this paper. The obtained results show that the first-order and second-order slip parameters have a considerable effect on the flow, as well as on the heat transfer characteristics.

For the first time, the laminar jet in a free-stream flow around a corner with/without a moving wall condition (as it assists the flow) is studied theoretically. The corner may contract the streamlines or expand them depending on the angle of inclination. In the context of incompressibility, it is presented a composite non-similar transformation of the Navier-Stokes (N-S) equations in Cartesian space to approximately decipher the nature of the boundary layer flow over the entire physical domain; i.e. 0 <= x, y < infinity. The transformed Partial Differential Equations (PDEs) are then solved by an in-house MATLAB code, which employs an implicit algorithm of a tridiagonal form with quasi-linearization technique. For a stationary wall, as the negative angle of the corner meets Falkner-Skan (F-S) limitation and expands beyond, the predictive model starts to show a critical location, which reveals directly the flow separation point. This critical location moves toward the jet origin as the angle of inclination becomes more negative. Here, these locations are extracted as a function of the inclination parameter in order to manifest the suitability of a jet to delay boundary layer separation. A similar scenario also applies to the moving wall case. In this case, it was recorded that a moving wall condition delays separation for a considerable distance. It should be pointed out that up to date, there is no theoretical investigation on this specific flow geometry despite the many associated applications; and the limited available reports on an analogous flow geometry are those from the Computational Fluid Dynamics (CFD) and on a case by case basis. More specifically, the present geometry is quite similar to the flow passing the way down to the trailing edge of an airfoil, with a jet generator mechanism such as Dielectric Barrier Discharge (DBD) plasma actuator to delay flow separation on the backside of the airfoil. Here, by the use of the theoretical non-similarity concept empowered by some recent advancements (see Jafarimoghaddam, 2020 and Jafarimoghaddam, 2021), the fundamental problem of a jet discharged in the lower boundary of a freestream flow over a wedge is successfully solved. This theoretical breakthrough suggests promising avenues for the future design of airfoils with moving parts/flaps as an innovative step forward to enhance maneuverability of the flying objects. (C) 2022 Published by Elsevier Inc.

Purpose The purpose of this paper is to numerically study the problem of mixed convection flow of a hybrid nanofluid past a vertical wedge with thermal radiation effect. Design/methodology/approach The governing nonlinear partial differential equations are transformed into a system of ordinary differential equations by a similarity transformation, which is then solved numerically through the function bvp4c from MATLAB for different values of the governing parameters. The solutions contain a mixed convection parameter λ that has a considerable impact on the flow fields. Findings It is found that the solutions of the ordinary (similarity) differential equations have two branches, upper and lower branch solutions, in a certain range of the mixed convection and several other parameters. To establish which of these solutions are stable and which are not, a stability analysis has been performed. The effects of the governing parameters on the fluid flow and heat transfer characteristics are illustrated in tables and figures. It is found that dual (upper and lower branch) solutions exist for both the cases of assisting and opposing flow situations. A stability analysis has also been conducted to determine the physical meaning and stability of the dual solutions. Practical implications This theoretical study is significantly relevant to the applications of the heat exchangers placed in a low-velocity environment and electronic devices cooled by fans. Originality/value The case of mixed convection flow of a hybrid nanofluid past a vertical wedge with thermal radiation effects has not been studied before, and hence all generated numerical results are claimed to be original and novel.

Purpose The purpose of this paper is to study the effects of thermal radiation and homogeneous-heterogeneous reactions in the three-dimensional hybrid nanofluid flow past a permeable stretching/shrinking sheet. Design/methodology/approach The combination of aluminum oxide (Al 2 O 3 ) and copper (Cu) nanoparticles with total volumetric concentration is numerically analyzed using the existing correlations of hybrid nanofluid. With the consideration that both homogeneous and heterogeneous reactions are isothermal while the diffusion coefficients of both autocatalyst and reactant are same, the governing model is simplified into a set of differential (similarity) equations. Findings Using the bvp4c solver, dual solutions are presented, and the stability analysis certifies the physical/real solution. The findings show that the suction parameter is requisite to induce the steady solution for shrinking parameter. Besides, the fluid concentration owing to the shrinking sheet is diminished with the addition of surface reaction. Originality/value The present findings are novel and can be a reference point to other researchers to further analyze the heat transfer performance and stability of the working fluids.

The flow of the hybrid nanofluid (copper–titanium dioxide/water) over a nonlinearly stretching surface was studied with suction and radiation effect. The governing partial differential equations were then converted into non-linear ordinary differential equations by using proper similarity transformations. Therefore, these equations were solved by applying a numerical technique, namely Chebyshev pseudo spectral differentiation matrix. The results of the flow field, temperature distribution, reduced skin friction coefficient and reduced Nusselt number were deduced. It was found that the rising of the mass flux parameter slows down the velocity and, hence, decreases the temperature. Further, on enlarging the stretching parameter, the velocity and temperature increases and decreases, respectively. In addition, it was mentioned that the radiation parameter can effectively control the thermal boundary layer. Finally, the temperature decreases when the values of the temperature parameter increases.

In the present research, it is solved in an innovative manner the essentially Non-Similar nanofluids jet discharged over a generalized non-linearly stretching wall. The Non-Similar set-up is interpreted as a combination of wall jet flow at the leading edge and non-linearly stretching sheet flow far away from it. Nanofluids are simulated employing Buongiorno's two-phase model together with a modified boundary condition for nanoparticles transport equation. The compositely-transformed partial differential equations (PDE) of Non-Similar forms were solved numerically after employing a new mapping function to abridge the horizontal coordinate of an infinite length into a finite one. The numerical algorithm contains quasi-linearization technique together with an implicit algorithm of tridiagonal form. It should be pointed out that such a new Non-Similar setup is brought into account for the first time in the literature. In this respect, we summarize the outstanding novel findings of the present work as: I. Accounting a modified boundary condition for nanoparticles concentration at the wall, the socalled and widely-used Brownian motion parameter, namely Nb, will have absolutely NO impact on Nusselt number or more precisely, on Energy Equation. In this respect, it is presented a new way of defining dimensionless variables. II. As the stretching ratio increases, it is observed a discrepancy in boundary layer thickness; it expands close to the jet area, shrinks in the transition region and adapts the far-field condition, either by expanding or shrinking, depending on the stretching ratio. III. As the stretching ratio increases, Nusselt number decreases slightly shortly after the jet origin and then increases dramatically to match the stretching region. IV. As the stretching ratio increases, nanoparticles at the wall become more concentrated at the downstream and hence, close to the jet region as well as the transition region, the concentration is sparse. V. As thermophoresis parameter, namely N-t, increases, convective heat transfer coefficient drops non-uniformly, from the leading edge to downstream, as well as nanoparticles concentration at the wall. VI. As Schmidt number increases, convective heat transfer drops; however, nanoparticles concentration at the wall increases over the entire spatial domain. ARTICLE INFO Article history: Received 6 May 2020 Received in revised form 16 October 2020 Accepted 19 November 2020 Available online 21 November 2020 Keywords: The non-linearly stretching wall jet Two-phase nanofluids Heat transfer and nanoparticles concentration analysis Numerical solution Non-similar flow (C) 2020 Elsevier Masson SAS. All rights reserved.

In the present work, Blasius problem subject to a moving and permeable wall (as a universal scheme) is tackled analytically and numerically in a comprehensive manner. In the analytic part, it is initially employed perturbation technique to develop some new asymptotic solutions; then, a modified scheme of Adomian Decomposition Method (namely, Duan–Rach ADM) combined with Jafarimoghaddam contraction mapping theorem, 2019 is brought into account to provide some new insights to the nonlinearity. Particularly, this combination led to an accurate analytic estimation of the critical points within the nonlinearity as well as an excellent improvement of the series solution presumably for the 1st time in the state of art. In the numerical part, the nonlinearity underwent Runge–Kutta–Fehlberg (RKF45) algorithm and the dual-nature solutions were confirmed. As a new finding in this part, it is mentioned to a critical injection rate of Scr.≈−0.873 for the existence of duality in the solution. In other words, for Scr.<−0.873 dual-nature solutions transform into single-nature ones.

This paper studies the boundary layer flow and heat transfer characteristics past a permeable isothermal stretching/shrinking surface using both nanofluid and hybrid nanofluid flows (called modified Buongiorno nonliquid model). Using appropriate similarity variables, the PDEs are transformed into ODEs to be solved numerically using the function bvp4c from MATLAB. It was found that the solutions of the resulting system have two branches, upper and lower branch solutions, in a certain range of the suction, stretching/shrinking and hybrid nanofluids parameters. Both the analytic and numerical results are obtained for the skin friction coefficient, local Nusselt number, and velocity and temperature distributions, for several values of the governing parameters. It results in the governing parameters considerably affecting the flow and heat transfer characteristics.