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The use of hybrid nanofluids in practical applications is pivotal for enhanced heat transfer efficiency especially for electronics cooling, and manufacturing processes. This study delves into numerically investigating the unsteady water-based (alumina+copper+titanium dioxide) ternary hybrid nanofluid flow over a permeable biaxial shrinking sheet, considering the influence of thermal radiation. The model, initially formulated as partial differential equations (PDEs), is adeptly transformed into ordinary differential equations (ODEs) via established similarity transformations. Subsequently, a numerical solution employing the finite difference scheme in bvp4c MATLAB unravels the behaviors of crucial physical quantities—across various parameter configurations. Remarkably, this study reveals the presence of two potential solutions, among which only one exhibits physical stability. Notably, the findings underscore the efficacy of enlarging the boundary suction parameter and diminishing thermal radiation for augmenting heat transfer within the specified conditions of ternary hybrid nanofluid. A noteworthy finding of this study reveals that an increase in the boundary suction parameter by 4% leads to a remarkable 9% delay in the boundary layer separation of the ternary hybrid nanofluid, thus maintaining the laminar phase flow. This study offers crucial guidance and insights for researchers and practitioners delving into the mathematical or experimental aspects of ternary hybrid nanofluid dynamics.

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.

The exclusive behaviour of hybrid nanofluid has been actively emphasized due to the determination of improved thermal efficiency. Therefore, the aim of this study is to highlight the stagnation point Al2O3-Cu/H2O hybrid nanofluid flow with the influence of Arrhenius kinetics and thermal radiation over a stretching/shrinking sheet. This particular work is distinctive because it presents a novel hybrid nanofluid mathematical model that takes into account the highlighted issue with a combination of multiple consequences that have not yet been addressed in prior literature. The bvp4c package embedded in MATLAB software is used to address the formulated ordinary differential equations and specified boundary conditions based on similarity solutions. The flow is assumed to be incompressible and laminar, and the hybrid nanofluid is made up of two different types of nanoparticles. The findings demonstrate the viability of dual solutions within the defined ranges of the physical parameters. As predicted, the hybrid nanofluid flow has been convincingly proved to enhance the skin friction coefficient and the heat transfer performance as opposed to viscous flow and nanofluid flow. The heat of reaction and radiation parameters also act as contributing factors in the progress of thermal enhancement. On the other hand, the reaction rate parameter unexpectedly displays a decreasing trend in the heat transfer rate of the current study. It is anticipated that this study will benefit future research into this potential heat transfer fluid, particularly in the areas of thermal systems and boundary layer analysis.

In many applications, hybrid nanofluids showed superior heat transfer outcomes; nevertheless, further study is needed to expand the range of applications for hybrid nanofluids. Therefore, in this study, the radiative magnetohydrodynamic (MHD) mixed convective alumina-copper/water hybrid nanofluid flow past an inclined shrinking plate is analyzed. By incorporating similarity transformations, the PDEs of the flow model is converted to ODEs. The boundary value problem of the fourth-order accuracy code (bvp4c) is implemented to solve the mathematical model numerically. When preliminary assumptions are appropriate, the above-proposed method may provide non-unique outcomes. Due to the plate’s shrinking motion, two solutions are possible. The first solution is stable based on a stability study. Therefore, we only rely on the first solution for effective practical uses. The findings reported that using less copper concentration (1 % volume fraction instead of 2 %) and applying more thermal radiation and MHD effect, the heat transfer rate might increase significantly when the plate is inclined considerably (at 70 degrees). It is possible to avoid the flow separation by upsurging the copper volume fraction from 1 % to 2 % at a higher MHD effect. The boundary layer separation is not affected by the employment of various inclination angles, variable thermal radiation, and mixed convection. This study offers valuable insight into fundamental transport phenomena such as the transmission of heat, momentum, or mass. Thus, it provides valuable information on the gradients of essential factors to control the boundary layer flow pattern.

The aim of this paper is to highlight the output of the investigation on the MHD and radiative flow and thermal characteristics of a non-Newtonian Reiner–Philippoff nanofluid with Brownian and thermophoresis diffusion effects. The model studied is embedded in the Buongiorno theory. This unique model is designed to observe both shear thickening and shear thinning properties on that particular fluid with embedded Brownian and thermophoresis diffusion implications. The proposed model consists of continuity, momentum, energy, and concentration equations constructed using the theoretical assumptions and are reduced to a set of ordinary differential equations (ODEs) before solving it using the bvp4c function in MATLAB software. Two solutions are observed, and their physical significance is justified using the temporal stability analysis. From the standpoint of the Reiner–Philippoff fluid parameter, the skin friction coefficient as well as the heat and mass transfer rates are at maximum for the shear-thickening fluid followed by the Newtonian and shear-thinning fluids. The thermophoresis parameter is noticed to decline the heat and mass transfer rate whereas the Brownian motion parameter boosts the mass transfer rate but decreases the heat transfer rate.

Heat transfer is an important process in many engineering, industrial, residential, and commercial buildings. Thus, this study aims to analyse the effect of MHD and viscous dissipation on radiative heat transfer of Reiner–Philippoff fluid flow over a nonlinearly shrinking sheet. By adopting appropriate similarity transformations, the partial derivatives of multivariable differential equations are transformed into the similarity equations of a particular form. The resulting mathematical model is elucidated in MATLAB software using the bvp4c technique. To determine the impact of physical parameters supplied into the problem, the results are shown in the form of tables and graphs. The findings reveal that the heat transfer rate reduces as the Eckert number and radiation parameter are introduced in the operating fluid. However, increasing the magnetic parameter raises both the skin friction coefficient and the local Nusselt number, which impulsively improves the heat transfer performance. The suction effect has a noticeable influence on the Reiner–Philippoff fluid, since increasing the suction parameter's value is seen to enhance the skin friction coefficient and the heat transfer performance. The dual solutions are established, leading to the stability analysis that supports the first solution’s validity.

The field of magnetohydrodynamics (MHD) encompasses a wide range of physical objects due to their stabilising effects. Thus, this study concerns the numerical investigation of the radiative non-Newtonian fluid flow past a shrinking sheet in the presence of an aligned magnetic field. By adopting proper similarity transformations, the governing partial derivatives of multivariable differential equations are converted to similarity equations of a particular form. The numerical results are obtained by using the bvp4c technique. According to the findings, increases in the suction parameter resulted in higher values of the skin friction and heat transfer rate. The same pattern emerges as the aligned angle and magnetic parameter are considered. On the other hand, the inclusion of the Bingham number, the Reiner–Philippoff fluid, and the thermal radiation parameters deteriorate the heat transfer performance, evidently. The dual solutions are established, which results in a stability analysis that upholds the validity of the first solution.

The magnetohydrodynamic (MHD) radiative flow of a hybrid alumina-copper/water nanofluid past a permeable vertical plate with mixed convection is the focal interest in this present work. Dissimilar to the traditional nanofluid model that considers only one type of nanoparticles, we consider the hybridization of two types of nanoparticles in this work which are alumina and copper. The governing flow and heat transfer equations are simplified to the ordinary differential equations (ODEs) with the adaptation of conventional similarity transformations which are then evaluated by the bvp4c solver (MATLAB) to generate the numerical solutions. The solutions are generated and illustrated in the form of graph to be easily observed. Although dual solutions are obtained in this study, only one solution is determined to be stable. By reducing the concentration volume of copper and increasing the magnetic and radiation parameters, the boundary layer separation can be hindered. With the occurrence of opposing flow due to the mixed convection parameter, the heat transfer can be enhanced when the concentration volume of copper is being reduced and when the magnetic and radiation parameters are being proliferated.

The proficiency of hybrid nanoparticles in augmenting the heat transfer has fascinated many researchers to further analysing the working fluid. The present paper is focused on the MHD hybrid nanofluid flow with heat transfer on a moving plate with Joule heating. The combination of metal (Cu) and metal oxide (Al2O3) nanoparticles with water (H2O) as the base fluid is used for the analysis. Similarity transformation reduces the complexity of the PDEs into a system of ODEs, which is then solved numerically using the function bvp4c from MATLAB for different values of the governing parameters. Two solutions are obtained when the plate is moved oppositely from the free stream flow. Analysis of flow stability unveils the first solution as the real physical solution, which is realizable in practice. From physical perspective, the real solution must be available for all cases of λ which affirms the finding from stability analysis. An upsurge of suction’s strength and magnetic parameter enhances the heat transfer operation and extends the critical value λc. Meanwhile, there is no change on the critical value when the Eckert number is added. This study is important in determining the thermal behavior of Cu-Al2O3/H2O when the physical parameters like magnetic field and Joule heating are embedded. The results are new and original with many practical applications in the modern industry.

New formulations are produced for problems associated with multiple cracks in the upper part of thermoelectric-bonded materials subjected to remote stress using hypersingular integral equations (HSIEs). The modified complex stress potential function method with the continuity conditions of the resultant electric force and displacement electric function, and temperature and resultant heat flux being continuous across the bonded materials’ interface, is used to develop these HSIEs. The unknown crack opening displacement function, electric current density, and energy flux load are mapped into the square root singularity function using the curved length coordinate method. The new HSIEs for multiple cracks in the upper part of thermoelectric-bonded materials can be obtained by applying the superposition principle. The appropriate quadrature formulas are then used to find stress intensity factors, with the traction along the crack as the right-hand term with the help of the curved length coordinate method. The general solutions of HSIEs for crack problems in thermoelectric-bonded materials are demonstrated with two substitutions and it is strictly confirmed with rigorous proof that: (i) the general solutions of HSIEs reduce to infinite materials if G1=G2, K1=K2, and E1=E2, and the values of the electric parts are α1=α2=0 and λ1=λ2=0; (ii) the general solutions of HSIEs reduce to half-plane materials if G2=0, and the values of α1=α2=0, λ1=λ2=0 and κ2=0. These substitutions also partially validate the general solution derived from this study.

The boundary layer flows exhibit symmetrical characteristics. In such cases, the flow patterns and variables are symmetrical with respect to a particular axis or plane. This symmetry simplifies the analysis and enables the use of symmetry-based boundary conditions or simplifications in mathematical models. Therefore, by using these concepts, the governing equations of the radiative flow of a hybrid nanofluid past a stretched and shrunken surface with the effect of a magnetic dipole are examined in this paper. Here, we consider copper (Cu) and alumina (Al2O3) as hybrid nanoparticles and use water as a base fluid. The heat transfer rate is enhanced in the presence of hybrid nanoparticles. It is observed that the heat transfer rate is increased by 10.92% for the nanofluid, while it has a 15.13% increment for the hybrid nanofluid compared to the base fluid. Also, the results reveal that the non-uniqueness of the solutions exists for a certain suction and shrinking strength. Additionally, the ferrohydrodynamic interaction has the tendency to reduce the skin friction and the heat transfer coefficients for both solution branches. For the upper branch solutions, the heat transfer rate increased over a stretching sheet but decreased for the shrinking sheet in the presence of the radiation. It is confirmed by the temporal stability analysis that one of the solutions is stable and acceptable as time evolves.

Abstract The proficiency of hybrid nanofluid from Cu-Al2O3/water formation as the heat transfer coolant is numerically analyzed using the powerful and user-friendly interface bvp4c in the Matlab software. For that purpose, the Cu-Al2O3/water nanofluid flow between two parallel plates is examined where the lower plate can be deformed while the upper plate moves towards/away from the lower plate. Other considerable factors are the wall mass suction/injection and the magnetic field that applied on the lower plate. The reduced ordinary (similarity) differential equations are solved using the bvp4c application. The validation of this novel model is conducted by comparing a few of numerical values for the reduced case of viscous fluid. The results imply the potency of this heat transfer fluid which can enhance the heat transfer performance for both upper and lower plates approximately by 7.10% and 4.11%, respectively. An increase of squeezing parameter deteriorates the heat transfer coefficient by 4.28% (upper) and 5.35% (lower), accordingly. The rise of suction strength inflates the heat transfer at the lower plate while the presence of the magnetic field shows a reverse result.

The objectives of the present study are (i) to observe the duality of solutions, and (ii) to investigate the flow and heat transfer of the hybrid nanofluid past a shrinking cylinder in the appearance of Joule heating. The single phase nanofluid model with modified thermophysical properties are used for the mathematical model. The similarity transformation simplifies the model (PDEs) into similarity (ordinary) differential equations. bvp4c solver is used to compute the reduced equations. For the validation part, the analytical solution is developed using an exact analytical method and compared with the numerical values for several cases. First and second solutions are observable for the shrinking cylinder case only if suction parameter is applied. Meanwhile, only the first solution is found to be stable from the stability analysis. The application of high suction strength make the reduced heat transfer rate is lower for hybrid nanofluid (Cu-Al2O3/water) than alumina-water nanofluid but, opposite result is found for the skin friction coefficient. The addition of curvature parameter (flat plate to cylinder) can quicken the separation process of boundary layer. This results are conclusive to the pair of alumina and copper only.

This study concerns the three-dimensional hybrid nanofluid flow and heat transfer due to a deformable (stretching/shrinking) plate with power-law velocity and orthogonal surface shear. The flow due to the shrinking sheet is maintained with the imposition of wall mass suction. The effect of adding Cu and Al2O3 nanoparticles are represented by a homogeneous mixture model with the modified thermophysical properties. Two types of thermophysical properties for hybrid nanofluids are discussed and compared in this interesting work. The three-dimensional model is then, reduced into a relevant set of ordinary differential equations using similarity transformation. The results are generated using the bvp4c solver and presented in the tables and graphs. Duality of solutions are observed in both stretching and shrinking regions, however, only the first solution is proved to be stable and realistic. Surprisingly, the heat transfer rate augments when the power law velocity is used. The hybrid nanofluid with an upsurge of copper volume fraction also reduces the rate of heat transfer.

The main objective of this work is to analyze and compare the numerical solutions of an unsteady separated stagnation point flow due to a Riga plate using copper–alumina/water and graphene–alumina/water hybrid nanofluids. The Riga plate generates electro-magnetohydrodynamics (EMHD) which is expected to delay the boundary layer separation. The flow and energy equations are mathematically developed based on the boundary layer assumptions. These equations are then simplified with the aid of the similarity variables. The numerical results are generated by the bvp4c function and then presented in graphs and tables. The limitation of this model is the use of a Riga plate as the testing surface and water as the base fluid. The results may differ if another wall surfaces or base fluids are considered. Another limitation is the Takabi and Salehi’s correlation of hybrid nanofluid is used for the computational part. The findings reveal that dual solutions exist where the first solution is stable using the validation from stability analysis. Graphene–alumina/water has the maximum skin friction coefficient while copper–alumina/water has the maximum thermal coefficient for larger acceleration parameter. Besides, the single nanofluids (copper–water, graphene–water and alumina–water) are also tested and compared with the hybrid nanofluids. Surprisingly, graphene–water has the maximum skin friction coefficient while alumina–water has the maximum heat transfer rate. The findings are only conclusive and limited to the comparison between graphene–alumina and copper–alumina with water base fluid. The result may differ if another base fluid is used. Hence, future study is necessary to investigate the thermal progress of these hybrid nanofluids.

The mixed convection flow with thermal characteristics of a water-based Cu-Al2O3 hybrid nanofluid towards a vertical and permeable wedge was numerically and statistically analyzed in this study. The governing model was constructed using physical and theoretical assumptions, which were then reduced to a set of ordinary differential equations (ODEs) using similarity transformation. The steady flow solutions were computed using the Matlab software bvp4c. All possible solutions were presented in the graphs of skin friction coefficient and thermal rate. The numerical results show that the flow and thermal progresses are developed by enhancing the controlling parameters (wedge parameter, volumetric concentration of nanoparticles, and suction parameter). Moreover, the response surface methodology (RSM) with analysis of variance (ANOVA) was employed for the statistical evaluation and conducted using the fit general linear model in the Minitab software. From the standpoint of statistical analysis, the wedge parameter and volumetric nanoparticle concentration have a considerable impact on all responses; however, the suction parameter effect is only substantial for a single response.

This paper examines the unsteady separated stagnation point (USSP) flow and thermal progress of Fe3O4–CoFe2O4/H2O on a moving plate subject to the heat generation and MHD effects. The model of the flow includes the boundary layer and energy equations. These equations are then simplified with the aid of similarity variables. The numerical results are generated by the bvp4c function and then presented in graphs and tables. The magnetic and acceleration (strength of the stagnation point flow) parameters are the contributing factors in the augmentation of the skin friction and heat transfer coefficients. However, the enhancement of heat generation parameter up to 10% shows a reduction trend in the thermal rate distribution of Fe3O4–CoFe2O4/H2O. This finding reveals the effectiveness of heat absorption as compared to the heat generation in the thermal flow process. From the stability analysis, the first solution is the physical solution. The streamline for the first solution acts as a normal stagnation point flow, whereas the second solution splits into two regions, proving the occurrence of reverse flow.

This paper aims to analyze the effects of different nanoparticles shape factor, EMHD and radiation parameters for the Cu–Al2O3/H2O nanofluid flow towards a stretching/shrinking Riga plate. The model is simplified into a set of ordinary (similarity) differential equations using the similarity transformation while the existing correlations are used to estimate the thermophysical properties for Cu–Al2O3/H2O. The comparison with previous results is in a good agreement with 0% error. Second solution is found and only exist in certain value of the shrinking parameter which reflects the unstableness of the solution. From the streamlines plot, the second solution dislocates the stagnation line far away from the wall surface to the reverse flow region as compared to the first solution. An increase of 1% EMHD parameter extends the separation value by 0.4%. This reflects the potential of EMHD parameter in delaying the separation process. Further, the heat transfer rate slightly increases with the rise of EMHD, radiation and shape factor parameters. The maximum heat transfer rate is acquirable for the ascendance nanoparticle concentration using blade-shape while the sphere-shape produce the lowest thermal rate. These findings are important in long run where we can plan for the heat transfer optimization of the cooling/heating applications.

The efficient performance of heat transfer fluid in terms of thermal conductivity plays a significant role in thermal engineering activities. In this work, the flow of boundary layer and heat transfer of hybrid nanofluid induced by an exponentially permeable stretching/shrinking curved surface is modelled and scrutinized numerically. Facilitated by bvp4c function, the ordinary differential equations are being solved. The implications of some intended parameters towards the physical quantities are plotted, while the comparison of results for the validation purpose is also tabulated. We found out that the boundary layer separation is deferred as copper volume fraction and curvature parameters increased. The stability analysis for the flow is conducted as the dual solutions are visible. The first (second) solution of the hybrid nanofluid flow is observed to be stable (unstable). For the physical solution, the increment in copper volume fraction leads to the accretion of the skin friction coefficient at the shrinking surface but has reduced the local Nusselt number. The presence of hybrid nanofluid has enhanced the velocity and temperature profiles. Generally, this study provides the initial prediction for the scientist and engineers in controlling the parameters to achieve the optimum desires for the related practical applications.

The widespread application of hybrid nanofluid in real applications has been accompanied by a large increase in computational and experimental research. Due to the unique characteristics of hybrid nanofluid, this study aspires to examine the steady two-dimensional mixed convection stagnation point flow of a hybrid nanofluid past a vertical plate with radiation, Dufour, and Soret effects, numerically. The formulations of the specific flow model are presented in this study. The model of fluid flow that is expressed in the form of partial differential equations is simplified into ordinary differential equations via the transformation of similarity, and then solved numerically by using the boundary value problem solver known as bvp4c in MATLAB, which implements the finite difference scheme with the Lobatto IIIa formula. Two possible numerical solutions can be executed, but only the first solution is stable and meaningful from a physical perspective when being evaluated via a stability analysis. According to the findings, it is sufficient to prevent the boundary layer separation by using 2% copper nanoparticles and considering the lesser amount of Dufour and Soret effects. The heat transfer rate was effectively upgraded by minimizing the volume fraction of copper and diminishing the Dufour effect. Stronger mixed convection would lead to maximum skin friction, mass transfer, and heat transfer rates. This important preliminary research will give engineers and scientists the insight to properly control the flow of fluids in optimizing the related complicated systems.

In this work, the mixed convection flow of non-Newtonian Eyring–Powell fluid with the effects of temperature dependent viscosity (TDV) were studied together with the interaction of dust particles under the influence of Newtonian Heating (NH) boundary condition, which assume to move over a vertical stretching sheet. Alternatively, the dusty fluid model was categorized as a two-phase flow that consists of phases of fluid and dust. Through the use of similarity transformations, governing equations of fluid and dust phases are reduced into ordinary differential equations (ODE), then solved by efficient numerical Keller–box method. Numerical solution and asymptotic results for limiting cases will be presented to investigate how the flow develops at the leading edge and its end behaviour. Comparison with the published outputs in literature evidence verified the precision of the present results. Graphical diagrams presenting velocity and temperature profiles (fluid and dust) were conversed for different influential parameters. The effects of skin friction and heat transfer rate were also evaluated. The discovery indicates that the presence of the dust particles have an effect on the fluid motion, which led to a deceleration in the fluid transference. The present flow model can match to the single phase fluid cases if the fluid particle interaction parameter is ignored. The fluid velocity and temperature distributions are always higher than dust particles, besides, the opposite trend between both phases is noticed with β. Meanwhile, both phases share the similar trend in conjunction with the rest factors. Almost all of the temperature profiles are not showing a significant change, since the viscosity of fluid is high, which can be perceived in the figures. Furthermore, the present study extends some theoretical knowledge of two-phase flow.

The flow of fluids over the boundaries of a rotating disc has many practical uses, including boundary-layer control and separation. Therefore, the aim of this study is to discuss the impact of unsteady magnetohydrodynamics (MHD) hybrid ferrofluid flow over a stretching/shrinking rotating disk. The time-dependent mathematical model is transformed into a set of ordinary differential equations (ODE’s) by using similarity variables. The bvp4c method in the MATLAB platform is utilised in order to solve the present model. Since the occurrence of more than one solution is presentable, an analysis of solution stabilities is conducted. Both solutions were surprisingly found to be stable. Meanwhile, the skin friction coefficient, heat transfer rate—in cooperation with velocity—and temperature profile distributions are examined for the progressing parameters. The findings reveal that the unsteadiness parameter causes the boundary layer thickness of the velocity and temperature distribution profile to decrease. A higher value of magnetic and mass flux parameter lowers the skin friction coefficient. In contrast, the addition of the unsteadiness parameter yields a supportive effect on the heat transfer rate. An increment of the magnetic parameter up to 30% reduces the skin friction coefficient by 15.98% and enhances the heat transfer rate approximately up to 1.88%, significantly. In contrast, the heat transfer is rapidly enhanced by improving the mass flux parameter by almost 20%.

This study examines the Blasius flow with Cu-Al2O3 hybrid nanoparticles over a moving plate. Additionally, the effects of viscous dissipation and radiation are considered. Similarity transformation is employed to convert the respective model into similarity equations. The results are generated by using bvp4c in MATLAB. Findings reveal that two solutions are attained when both the free stream and the plate move in opposite directions. Moreover, the domains of the velocity ratio parameter are extended when suction is available. Besides, the upsurge of radiation and hybrid nanoparticles lead to the heat transfer enhancement. The rise in radiation heat energy incorporated in radiation parameter leads to the development of fluid temperature as well as the thermal boundary layer. Meanwhile, hybrid nanoparticles offer good thermal characteristics because of synergistic effects. However, the effects reduce with the rise in Eckert number. The first solution is stable and acceptable based on the temporal stability analysis. Furthermore, the critical/separation values of the physical parameters are also reported. With these findings, the optimized productivity will be achieved as well as the processes on certain products can be planned according to the desire output. This significant preliminary study provides future insight to the engineers and scientist on the real applications.

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.

An investigation has been done on the hybrid nanofluid slip flow in the existence of heat generation over an exponentially stretching/shrinking permeable sheet. Hybridization of alumina and copper with water as the base fluid is considered. The mathematical model is simplified through the similarity transformation. A numerical solver named bvp4c in Matlab software is utilized to facilitate the problem-solving process and dual solutions are attained. The influences of several pertinent parameters on the main physical quantities of interest and the profiles are scrutinized and presented in the form of graphs. Through the stability analysis, only the first solution is considered as the physical solution. As such, the findings conclude that the upsurges of volume fraction on the copper nanoparticle could enhance the skin friction coefficient and the local Nusselt number.

This numerical study was devoted to examining the occurrence of non-unique solutions in boundary layer flow due to deformable surfaces (cylinder and flat plate) with the imposition of prescribed surface heat flux. The hybrid Al2O3-Cu/water nanofluid was formulated using the single phase model with respective correlations of hybrid nanofluids. The governing model was simplified by adopting a similarity transformation. The transformed differential equations were then numerically computed using the efficient bvp4c solver with the ranges of the control parameters 0.5%≤ϕ1,ϕ2≤1.5% (Al2O3 and Cu volumetric concentration), 0≤K≤0.2 (curvature parameter), 2.60) and a flat plate (K=0) with the inclusion of only the suction (transpiration) parameter. The real and stable solutions were mathematically validated through the stability analysis. The Al2O3-Cu/water nanofluid with ϕ1=0.5% (alumina) and ϕ2=1.5% (copper) has the highest skin friction coefficient and heat transfer rate, followed by the hybrid nanofluids with volumetric concentrations (ϕ1=1%,ϕ2=1%) and (ϕ1=1.5%,ϕ2=0.5%), respectively. Surprisingly, the flat plate surface abates the separation of boundary layer while it enhances the heat transfer process.

The present work highlights the stagnation point flow with mixed convection induced by a Riga plate using a Cu-Al 2 O 3 /water hybrid nanofluid. The electromagnetohydrodynamic (EMHD) force generated from the Riga plate was influential in the heat transfer performance and applicable to delay the boundary layer separation. Similarity transformation was used to reduce the complexity of the governing model. MATLAB software, through the bvp4c function, was used to compute the resulting nonlinear ODEs. Pure forced convective flow has a distinctive solution, whereas two similarity solutions were attainable for the buoyancy assisting and opposing flows. The first solution was validated as the physical solution through the analysis of flow stability. The accretion of copper volumetric concentration inflated the heat transfer rate for the aiding and opposing flows. The heat transfer rate increased approximately up to an average of 10.216% when the copper volumetric concentration increased from 0.005 ( 0.5 % ) to 0.03 ( 3 % ) .

The present study emphasizes the combined effects of double stratification and buoyancy forces on nanofluid flow past a shrinking/stretching surface. A permeable sheet is used to give way for possible wall fluid suction while the magnetic field is imposed normal to the sheet. The governing boundary layer with non-Fourier energy equations (partial differential equations (PDEs)) are converted into a set of nonlinear ordinary differential equations (ODEs) using similarity transformations. The approximate relative error between present results (using the boundary value problem with fourth order accuracy (bvp4c) function) and previous studies in few limiting cases is sufficiently small (0% to 0.3694%). Numerical solutions are graphically displayed for several physical parameters namely suction, magnetic, thermal relaxation, thermal and solutal stratifications on the velocity, temperature and nanoparticles volume fraction profiles. The non-Fourier energy equation gives a different estimation of heat and mass transfer rates as compared to the classical energy equation. The heat transfer rate approximately elevates 5.83% to 12.13% when the thermal relaxation parameter is added for both shrinking and stretching cases. Adversely, the mass transfer rate declines within the range of 1.02% to 2.42%. It is also evident in the present work that the augmentation of suitable wall mass suction will generate dual solutions. The existence of two solutions (first and second) are noticed in all the profiles as well as the local skin friction, Nusselt number and Sherwood number graphs within the considerable range of parameters. The implementation of stability analysis asserts that the first solution is the real solution.

The numerical study of nanofluid stagnation point flow coupled with heat and mass transfer on a moving sheet with bi-directional slip velocities is emphasized. A magnetic field is considered normal to the moving sheet. Buongiorno’s model is utilized to assimilate the mixed effects of thermophoresis and Brownian motion due to the nanoparticles. Zero nanoparticles’ flux condition at the surface is employed, which indicates that the nanoparticles’ fraction are passively controlled. This condition makes the model more practical for certain engineering applications. The continuity, momentum, energy and concentration equations are transformed into a set of nonlinear ordinary (similarity) differential equations. Using bvp4c code in MATLAB software, the similarity solutions are graphically demonstrated for considerable parameters such as thermophoresis, Brownian motion and slips on the velocity, nanoparticles volume fraction and temperature profiles. The rate of heat transfer is reduced with the intensification of the anisotropic slip (difference of two-directional slip velocities) and the thermophoresis parameter, while the opposite result is obtained for the mass transfer rate. The study also revealed the existence of non-unique solutions on all the profiles, but, surprisingly, dual solutions exist boundlessly for any positive value of the control parameters. A stability analysis is implemented to assert the reliability and acceptability of the first solution as the physical solution.