This study analyzes the steady hybrid nanofluid flow over a permeable, non-isothermal cone and wedge. The heat transfer analysis considers the effects of thermal radiation and convective boundary conditions. Non-linear ordinary differential equations, derived through similarity transformation of the governing partial differential equations and boundary conditions, are solved numerically using the bvp4c solver. The resulting triple solutions are then subjected to a stability analysis. It is confirmed that only the first solution is stable and physically meaningful, while the other solutions are unstable. The physical quantities of interest, namely the local skin friction coefficient and local Nusselt number, are found to be higher for assisting mixed convection flow than for opposing flow. Compared to the wedge geometry, hybrid nanofluid flow over the cone exhibits a lower local skin friction coefficient but a higher local Nusselt number. Furthermore, optimization results from the response surface methodology (RSM) indicate that the maximum local Nusselt number, corresponding to the highest heat transfer rate at the cone/wedge surface, can be achieved at high values of the Biot number, radiation parameter, and wall temperature parameter.

Hybrid nanofluid flow over a Riga plate has broad potential applications in biomedical, chemical, and engineering fields. This study analyzes the mixed convection stagnation-point flow of a hybrid nanofluid over a Riga plate. The effects of thermal radiation, suction, and convective boundary condition are considered by imposing related terms into the governing partial differential equations and boundary conditions. These equations are then reduced into non-linear ordinary differential equations using similarity transformation, and the bvp4c solver in Matlab is used to compute the numerical results. Dual solutions are presented, but only the stable first solution is analyzed and discussed. The presence of suction is found to enhance the magnitude of the local skin friction coefficient, local Nusselt number, and velocity profile of the hybrid nanofluid. However, increasing suction causes the temperature profile to drop. Meanwhile, increasing the nanoparticle volume fraction of Cu and Al_2O_3 in the Al_2O_3 -Cu/ H_2 O hybrid nanofluid raises the local skin friction coefficient but reduces the local Nusselt number. In addition, the response surface methodology (RSM) revealed that the suction parameter, Biot number, and radiation parameter favorably impact the local Nusselt number. With desirability of 99.88 S=0.7 , Bi=0.7 , and R=1.5 ).

For efficient heating and cooling applications, minimum wall shear stress and maximum heat transfer rate are desired. The current study optimized the local skin friction coefficient and Nusselt number in Al2O3-Cu/water hybrid nanofluid flow over a permeable shrinking rotating disk. First, the governing equations and boundary conditions are solved numerically using the bvp4c solver in MATLAB. Von Kármán’s transformations are used to reduce the partial differential equations into solvable non-linear ordinary differential equations. The augmentation of the mass transfer parameter is found to reduce the local skin friction coefficient and Nusselt number. Higher values of these physical quantities of interest are observed in the injection case than in the suction case. Meanwhile, the increase in the magnitude of the shrinking parameter improved and reduced the local skin friction coefficient and Nusselt number, respectively. Then, response surface methodology (RSM) is conducted to understand the interactive impacts of the controlling parameters in optimizing the physical quantities of interest. With a desirability of 66%, the local skin friction coefficient and Nusselt number are optimized at 1.528780016 and 0.888353037 when the shrinking parameter (λ) and mass transfer parameter (S) are −0.8 and −0.6, respectively.

Many real-world devices, such as heat exchangers, geothermal reservoirs, and cooling systems, utilize the concept of boundary layer flow across a cone geometry. The current study presents and analyses the mathematical formulation for the mixed convection flow of a hybrid nanofluid over a permeable stationary cone. The heat transfer analysis considers the effects of thermal radiation and convective boundary condition. Numerical and statistical analyses of this flow problem yield new, physically significant results. The numerical analysis is carried out using the bvp4c solver in Matlab. Similarity transformations are performed to obtain a system of nonlinear ordinary differential equations from the governing partial differential equations and boundary conditions. In both assisting and opposing flows, spherical- and platelet-shaped nanoparticles are observed to produce the lowest and highest local skin friction coefficient, respectively. The spherical- and blade-shaped nanoparticles also offer the highest and lowest local Nusselt number, respectively, with a difference of 6.4% (assisting) and 6.03% (opposing). Meanwhile, the increase in the mixed convection parameter raised the velocity profile but diminished the temperature profile of the hybrid nanofluid. Then, the relationship of the Biot number (Bi)$( {Bi} )$, suction (S), and thermal radiation (R) parameters with the local Nusselt number is investigated through the response surface methodology (RSM). The local Nusselt number for the current flow problem is estimated to be maximized at 0.814323 (assisting) and 0.814629 (opposing) when these parameters are at the highest range of S=2.0$S\ = \ 2.0$, R=1.0$R\ = \ 1.0$, and Bi=0.5$Bi\ = \ 0.5$. Several researchers had presented experimental studies conducted at different temperatures (15, 25, 35 degrees C), mass flow rates (ranging from 0.00076 to 0.041 kg/s), and nanoparticle concentrations (0.387, 0.992, 3.12, 4.71 mass%).

The fundamental goal of this study is to scrutinize the flow and the heat transfer performance of Reiner-Philippoff hybrid ferrofluid (magnetic nanofluid) magnetite-cobalt ferrite/water, magnetite/water, and cobalt ferrite/water with radiation effect past a permeable stretching/shrinking sheet. Different variations of Bingham number, Reiner-Philippoff parameter, and radiation effect are examined to estimate the thermal progress across the fluid at the boundary layer when the sheet is shrunk and stretched. The governing fluid flow model is remodeled into a nonlinear set of ordinary differential equations via transformation of similarity. The new transformed system is numerically solved aided by MATLAB's bvp4c solver. Within a certain set of physical parameters, two distinct solutions can be generated. Hybrid ferrofluid has been shown to have an improved heat transfer performance when the sheet is shrunk compared to mono ferrofluids. The laminar state of the flow can be maintained efficiently when a hybrid ferrofluid is considered. The radiation parameter can be used to enhance the thermal progress except when the sheet is shrunk. The first solution has been shown to be stable by stability analysis.

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.

Rotating machinery, gas turbine rotators, and air cleaning equipment are some industrial and electronic applications of hybrid nanofluids as heat transfer fluids. Considering these potential applications, the axisymmetric flow of a hybrid nanofluid towards a permeable rotating disk with a uniform shrinking rate is analysed in the current study. Nonlinear ordinary differential equations and boundary conditions are generated, using Von Kármán’s transformations, from the governing partial differential equations and boundary conditions. Then, a sophisticated bvp4c solver containing finite difference code is utilized for solving the boundary value problem numerically. Following the discovery of dual solutions, stability analysis is performed, and only the first solution is stable. Besides that, the magnitude of the local skin friction coefficient is found to increase with the rise of shrinking and injection parameters. However, the augmentation of the shrinking and injection parameters reduces and enhances the local Nusselt number. Meanwhile, the enhancement of injection parameter is observed to reduce the hybrid nanofluid’s momentum and thermal boundary layer thickness.

Hybrid nanofluid flow past a stretching/shrinking sheet has various applications in industrial and engineering processes, e.g. in glass blowing, the extrusion of polymer sheets, and paper production. Motivated by these numerous uses of hybrid nanofluid in diverse geometries and conditions, the present study analyzes the solutions for MHD flow of Ag-CuO/water hybrid nanofluid past a shrinking sheet. The governing equations and boundary conditions are formulated together with the effects of Brownian motion, double stratification, porous medium, suction, slips, and thermophoresis. Then, similarity transformations are employed to form non-linear ordinary differential equations and boundary conditions for numerical computation in Matlab using the bvp4c solver. A significant finding of triple solutions in the shrinking sheet case prompted a stability analysis to be carried out, and the results show that only the first solution is stable. The effects of controlling parameters on the physical quantities of interest, velocity, temperature, and concentration profiles are analyzed and discussed. The heat and mass transfer rates are noted to improve, with an average of 7.91% and 258.36%, by increasing the Darcy number related to the permeability of the porous medium. Meanwhile, augmenting the nanoparticle volume fraction of Ag from 0.03 to 0.05 enhances the skin friction, heat transfer, and mass transfer rates by 9.4%, 7.36%, and 150.31%, respectively. However, the heat and mass transfer performances of the hybrid nanofluid are inhibited by the double stratification parameters.

The flow between bounded surfaces is known as internal flow. The internal flow between disks has many significant applications, such as gas turbine rotors, rotating machinery, food processing technology, and air cleaning machines. In the current study, the nanofluid flow between two disks, nonpermeable and stationary, and the other permeable, rotating and shrinking, is analysed. The governing partial differential equations and boundary conditions are proposed with the inclusion of radiation and heat generation effects. Then, similarity transformations are utilised in deriving the nonlinear ordinary differential equations and boundary conditions for computation using the bvp4c solver. Multiple solutions are obtained, and only the first solution is stable. The combination Mn-ZnFe2O4/C2H6O2 nanofluid is found to produce the lowest magnitude of skin friction coefficient and the highest heat transfer rate.

Including slip boundary conditions in the study involving the flow of foams, emulsions, polymer solutions, and suspensions over moving surfaces are crucial for real-life applications. The current study analysed the multiple slips effects on the magnetohydrodynamics (MHD) flow of Ag-CuO/water hybrid nanofluid past a permeable stretching/shrinking sheet embedded in a porous medium. Appropriate similarity variables are introduced for transforming the governing equations and boundary conditions into ordinary differential equations before being solved using the bvp4c solver. Dual solutions are yielded from the numerical computation of flow over a shrinking sheet, and the first solution is identified as stable through a stability analysis. It is found that the imposition of velocity, thermal, and mass slips promotes the reduction of momentum, thermal, and concentration boundary layer thickness, respectively. The hybrid nanofluid around the sheet is observed to flow at a different velocity from the sheet due to the imposition of velocity slip. Thermal and mass slips, meanwhile, obstruct the flow's ability to transport heat and mass. An increased suction parameter, however, can aid in enhancing the rates of heat and mass transfers.

Real-world applications of hybrid nanofluid flow past a cone and wedge include the design of spacecraft, nuclear reactors, solar power collectors, and many others. A steady, hybrid nanofluid flow is explored over a porous, non-isothermal, non-isosolutal cone and wedge. The heat transfer analysis includes the effects of thermal radiation and convective boundary condition. Then, non-linear ordinary differential equations are derived from partial differential equations and boundary conditions of the stated flow problem. The bvp4c solver is used for numerical calculation, resulting in dual solutions. Stability analysis confirmed that only the first solution is stable and physically meaningful. The physical quantities of interest (i.e., local skin friction coefficient, local Nusselt number, and local Sherwood number) for assisting flow are found to be higher than the opposing flow. Compared to the wedge, the hybrid nanofluid flow over a cone has a lower local skin friction coefficient and greater local Nusselt and Sherwood numbers. Meanwhile, the optimization results from the response surface methodology (RSM) described that the maximum local Nusselt number corresponding to the heat transfer rate could be achieved at high values of the Biot number, radiation, and wall temperature parameters.

Purpose This paper aims to study the stagnation point flow of Al 2 O 3 –Cu/H 2 O hybrid nanofluid over a radially shrinking disk with the imposition of the magnetic field, viscous-Ohmic dissipation and convective boundary condition. Design/methodology/approach Similarity variables are introduced and used in reducing the governing partial differential equations into a system of ordinary differential equations. A built-in bvp4c solver in MATLAB is then used in the computation of the numerical solutions for equations (7) and (8) subject to the boundary conditions (9). Then, the behavior of the flow and thermal fields of the hybrid nanofluid, with various values of controlling parameters, are analyzed. Findings The steady flow problem resulted in multiple (dual) solutions. A stability analysis performed to identify the stable solution applicable in practice revealed that the first solution is stable while the second solution is unstable. The skin friction coefficient and Nusselt number of the hybrid nanofluid are found to be greater than the Al 2 O 3 –H 2 O nanofluid. Thus, the hybrid nanofluid has a better heat transfer performance than the nanofluid. Besides that, the presence of the magnetic field, suction, convective boundary condition and the enhancement of nanoparticle volume fraction of Cu augments the skin friction coefficient and Nusselt number of the hybrid nanofluid. Meanwhile, the presence of viscous-Ohmic dissipation reduces the heat transfer performance of the fluid. Originality/value To the best of the authors’ knowledge, the present results are original and new for the study of the flow and heat transfer of Al 2 O 3 –Cu/H 2 O hybrid nanofluid past a permeable radially shrinking disk. Considerable efforts have been directed toward the study of the boundary layer flow and heat transfer over stretching/shrinking surfaces and disks because of its numerous industrial applications, such as electronic, power, manufacturing, aerospace and transportation industries. Common heat transfer fluids such as water, alumina, cuprum and engine oil have limited heat transfer capabilities due to their low heat transfer properties. In contrast, metals have higher thermal conductivities than these fluids. Therefore, it is desirable to combine the two substances to produce a heat transfer medium that behaves like a fluid but has higher heat transfer properties.

The suction draws the ambient fluid towards the permeable surface. Generally, suction delays the detachment of a boundary layer from a surface and improves the heat transfer rate. The current study analyses the effects of suction on the oblique stagnation-point flow of hybrid nanofluid (i.e., Cu-Al2O3/H2O) over a shrinking surface. Similarity transformations are adopted to reduce the governing equations and boundary conditions into non-linear differential equations and boundary conditions. Then, numerical computation is carried out using the bvp4c package. Dual solutions are found, and the first solution is identified as the stable solution through a stability analysis. The increase in the suction parameter enhances the normal and shear components of the skin friction and the temperature gradient. Consequently, it reduces the momentum and thermal boundary layer thickness to slow down boundary layer separation and enhances the heat transfer rate at the shrinking surface.

Fluid flow over a biaxial stretching/shrinking surface may arise in fiber production and wrapping processes. The current study considered the three-dimensional flow of a hybrid nanofluid past a biaxial stretching/shrinking sheet with thermal radiation and suction. This flow problem is translated into nonlinear partial differential equations and boundary conditions. After similarity transformations, the numerical computations are conducted using the bvp4c solver. The calculation yielded dual solutions that prompted a stability analysis, demonstrating that only the first solution is stable and significant. Cu-Al2O3/H2O hybrid nanofluid produced the highest temperature profile compared to Cu/H2O and Al2O3/H2O nanofluids. As observed from this study, a further increase in the temperature profile of the hybrid nanofluid can be achieved by enhancing the shrinking and radiation parameters. Meanwhile, the magnitude of the skin friction coefficient and heat transfer rate rises with the suction parameter. At the same time, the suction parameter reduces the thickness of the momentum and thermal boundary layers. Then, response surface methodology (RSM) is used to develop a correlation between the response, Nusselt number, Re  1/2 x Nux, and governing parameters of the problem. The RSM suggested that the suction parameter positively affects the heat transfer rate. However, the opposite behavior is observed for the nanoparticle volume fraction of Cu and Al2O3. The heat transfer rate is estimated to be optimized at 6.02216 when & phi;Cu = & phi;Al2O3 = 0.02 and S = 3.0.

A mathematical model for the unsteady, two-dimensional mixed convection stagnation point flow over a Riga plate is presented in this study. Convective boundary conditions, time-dependent derivatives, mixed convection, radiation effects, and the Grinberg term were all incorporated into the formulation of the governing equations and boundary conditions. By incorporating similarity transformations, ordinary differential (similarity) equations (ODEs) are derived from the partial differential equations (PDEs) of the flow model. The boundary value problem of the fourth-order accuracy code (bvp4c) was implemented in MATLAB (2017b, The MathWorks, Inc., Natick, MA. USA, 2017) to solve the mathematical model numerically. Due to the plate's shrinking motion, two (dual) solutions are possible (first and second solutions). Based on the stability analysis, it was found that the first solution is stable and physically realizable in practice, while the second solution is not stable and not physically realizable in practice. It was found that the increase in the mixed convection parameter, modified Hartmann number, and unsteadiness parameter improved the hybrid nanofluid's temperature profile. In addition, increasing the unsteadiness parameter decreased the velocity profile and the skin friction coefficient. Thus, the numerical results suggested that the augmentation of the modified Hartmann number, mixed convection parameter, and unsteadiness parameter can enhance the heat transfer performance in this flow model. This study offers valuable insight into fundamental transport phenomena such as the transmission of momentum, heat, or mass. Hence, it provides valuable information on the gradients of essential factors to control the boundary layer flow pattern.

An engineered fluid, called nanofluid, is expected to have better thermal conductivity than conventional working fluids. The superior heat transfer performance and various possible applications promote the analysis of nanofluids in different flow geometries. This paper studies the flow of non-Newtonian Burgers' nanofluids over a permeable stretching/shrinking surface with a heat source/sink. In the current study, we highlight the use of the single-phase nanofluid model in studying the boundary layer flow. The basic partial differential equations are transformed into ordinary (similarity) differential equations. Then, the resulting equations and boundary conditions are solved numerically in MATLAB using the bvp4c package. Triple solutions are presented, and stability analysis certifies that the first solution is physically realizable in practice. It is found that the increment of the heat source parameter raised the temperature profile of the nanofluids. Al2O3/H2O and Cu/H2O nanofluids produced the highest skin friction coefficient in the flow over stretching and shrinking surfaces, respectively. Meanwhile, Cu/H2O nanofluid showed a better heat transfer performance when compared to Al2O3/H2O and TiO2/H2O nanofluids. The present study is novel and could serve as a reference to other researchers for further analysis of heat transfer performance and the rheological behavior of nanofluids.

The impact of nanoparticle shapes (i.e., blade, brick, cylindrical, platelet, and spherical) on the MHD flow of Ag-MgO/water hybrid nanofluid over a stretching/shrinking sheet is scrutinized in this study. Chemical reaction and activation energy are included in the governing partial differential equations of the flow problem. On the boundary, velocity slip and zero mass flux conditions are considered. The simplification of the governing equations and boundary conditions into non-linear ordinary differential equations is done through similarity transformation. Then, the bvp4c solver in Matlab is deployed for computation, with the results generated in the form of numerical solutions and graphs. It is found that the usage of spherical-shaped nanoparticles produces the lowest magnitude of skin friction coefficient, and the implementation of blade-shaped nanoparticles in the hybrid nanofluid provides the highest enhancement of heat transfer rate. The increment in activation energy slows down the chemical reaction that raises the concentration profile of the hybrid nanofluid. However, the concentration profile decreases as the reaction rate increases.

Purpose The purpose of this paper is to numerically analyze the stagnation point flow of Cu-Al2O3/water hybrid nanofluid with mixed convection past a flat plate and circular cylinder. Design/methodology/approach The similarity equations that reduced from the boundary layer and energy equations are solved using the bvp4c solver. The duality of solutions is observed within the specific range of the control parameters, namely, mixed convection parameter λ, curvature parameter γ and nanoparticles volumetric concentration ϕ1 for alumina, while for copper ϕ2. The stability analysis is also designed to justify the particular solutions’ stability. Additionally, the idea to obtain the solution for large value of λ and γ is also presented in this paper. Findings Two solutions exist in opposing and assisting flows up to a critical value λc where λc lies in the opposing region. An upsurge of the curvature parameter tends to extend the critical value (delay the separation process), whilst increase the heat transfer performance of the working fluid. Meanwhile, the application of hybrid Cu-Al2O3/water nanofluid also can decelerate the separation of laminar boundary layer flow and produce higher heat transfer rate than the Cu–water nanofluid and pure water. Originality/value The results are new and original. This study benefits to the other researchers, specifically in the observation of the fluid flow characteristics and heat transfer rate of the hybrid nanofluid. Also, this paper features with the mathematical formulation for the solution with large values of λ and γ.

To fill the existing literature gap, the numerical solutions for the oblique stagnation-point flow of Cu-Al2O3/H2O hybrid nanofluid past a shrinking surface are computed and analyzed. The computation, using similarity transformation and bvp4c solver, results in dual solutions. Stability analysis then shows that the first solution is stable with positive smallest eigenvalues. Besides that, the addition of Al2O3 nanoparticles into the Cu-H2O nanofluid is found to reduce the skin friction coefficient by 37.753% while enhances the local Nusselt number by 4.798%. The increase in the shrinking parameter reduces the velocity profile but increases the temperature profile of the hybrid nanofluid. Meanwhile, the increase in the free parameter related to the shear flow reduces the oblique flow skin friction.

Hybrid nanofluid has been widely used in various heat transfer applications especially as the heat exchanger due to the great thermal conductivity compared to the conventional fluid. However, numerous investigations should still be carried out to properly understand its properties. Hence, in this study, a three-dimensional radiative flow of hybrid Cu-Al2O3/water nanofluid past a permeable shrinking plate is numerically analyzed. The boundary layer including the energy equations are reduced to a system of ordinary differential equations using the similarity transformations and are then solved numerically by using the bvp4c solver in MATLAB. The application of suction through the permeable plate is necessary in aiding the fluid motion past the shrinking surface. Dual solutions are also observable, hence the stability analysis is conducted to mathematically validate the real solution. The enhancement of copper volumetric concentration in the hybrid nanofluid is capable in decelerating the boundary layer separation.