Carbon nanotubes (CNTs) spark interest due to their inimitable characteristics, leading to a multitude applications across various sectors. Thus, a mathematical model is developed for hybrid carbon nanotubes flow towards stagnation zone on an exponentially permeable cylinder. The flow is unsteady under stretching conditions. Nanoparticle geometry and thermal source are the physical manifestations of thermal energy. The basic formulation that defines the mathematical description of unsteady flow is recast into highly nonlinear differential equations through a new self-similarity variable. To produce observational data, a numerical tool (bvp4c) in Matlab utilized. The responses to flow factors are physically depicted through graphical illustration. The calculation yielded a non-unique nature in both (elongation/contraction) zones, with clearly spotted the respective critical points. The key results indicate that an exalts exponential parameter lead to prolong the onset of turbulent flow. The presence of hybrid carbon nanotubes reduced the range of solutions. Further, noticeable linear drop found in thermal source and shape factor with the strength of hybrid carbon nanotubes. It is also found that the temperature profile of platelet nanoparticle exhibits the highest recorded values among all considered cases.

Researchers are motivated to understand the behavior and properties of hybrid nanofluids due to their wide range of applications. For example, unsteady hybrid nanofluid flow can occur in marine propellers, hydrofoil flutters, rotor blades, and turbomachines. This study examines the unsteady mixed convection flow of a hybrid nanofluid over a radially shrinking disk. The time-dependent governing partial differential equations and associated boundary conditions are formulated and transformed into a system of non-linear ordinary differential equations using similarity transformations. These equations are solved numerically using MATLAB’s bvp4c function. Two solutions are obtained, and a stability analysis confirms that only the first solution is stable. In this flow problem, increasing both the Biot number and the mixed convection parameter increases the local Nusselt number and local skin friction coefficient. Increasing the mixed convection parameter from its lowest to highest considered values leads to increases of 113 α =-0.7, Bi=0.7, and λ =1.5 ). Meanwhile, the local skin friction coefficient is minimized when these parameters are at their lowest levels (i.e., α =-0.7, Bi=0.3, and λ =0.5 ). At these optimal conditions, the local sensitivity analysis suggests that the local Nusselt number is most sensitive to the Biot number, whereas the local skin friction coefficient is most sensitive to the mixed convection parameter.

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.

Carbon nanotubes (CNTs) have proven their value in diverse multidisciplinary applications. For this purpose, the current study sheds light on time-reliant properties of electrically conducting flow of hybrid carbon nanotubes, scenario involving joule dissipation at a permeable cylinder that can expand and shrink. To get a precise insight into numerical outcome, the unsteady governing momentum and energy equations in cylindrical coordinates rendered into pertinent ODEs via incorporating the rescaling technique, Thereafter, the rendered equations cracked numerically via a built-in function in MATLAB (BVP4C) package. Notably, the sundry parameters yield two distinct solutions in both assisting and opposing zones, so the flow separation is identified. The governing physical factors are well explored through various graphical forms with physical explanations. Graphical observations declare, that heightening the value of curvature and volume fraction parameters contributes to speed up the onset of turbulence flow, augmentation in skin friction rate is noted through unsteadiness, magnetic field, and curvature parameters. Additionally, the stability assessment clearly specifies the mathematical robustness of the first branch as time passes. This study stands out for it is an inimitable configuration that holds significant addition in optimization of modern heat transfer applications.

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 ).

Fluid flow may strike a surface at an angle due to the physical limitations of the nozzle or contouring of the surface. The heat transfer optimization for the Al2O3-Cu/water hybrid nanofluid flow impinging obliquely on a permeable shrinking sheet is analyzed in this study. Flow over a shrinking sheet may occur during polymer and metal sheet extraction, wire drawing, and glass-fiber production. The first step in this study involves reducing the governing partial differential equations and boundary conditions into non-linear ordinary differential equations via similarity transformation. Subsequently, these equations are solved using built-in finite difference code in MATLAB bvp4c solver. It is found that the increment of the suction parameter enhances the heat transfer rate represented by the physical quantity of interest called the local Nusselt number. However, the opposite occurs when the nanoparticle volume fraction of Cu and the magnitude of the shrinking parameter increase. Meanwhile, the normal and shear components of skin friction are augmented by the rise in the suction parameter and nanoparticle volume fraction of Cu. Then, the statistical analysis and optimization done using the response surface methodology (RSM) revealed that the local Nusselt number is highly impacted by the suction parameter, followed by the shrinking parameter and nanoparticle volume fraction of Cu. The maximum value of local Nusselt number is approximated to be 13.30539 when the magnitude of suction is at the highest, while shrinking parameter and nanoparticle volume fraction of Cu are at the lowest.

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 existence of more than one diffusive component in fluid mixtures is observed in these situations: underground water flow, the mechanism of acid rain, the existence of contaminant in some certain mixture, etc. These diffusive components are occurred with the single temperature gradient (since all of the elements are dissolved into the same mixture) and 2 types of concentration gradients (since the dual diffusive components are dissolved in the same mixture). Besides, many industrial and engineering processes are utilizing the concept of convective fluid flow especially over a shrinking sheet. Therefore, a mathematical model for triple-diffusive flow over a nonlinear compressing sheet has been developed in this paper, and subjected to the Soret-Dufour effects. The model comprises of five initial equations namely continuity, momentum, energy, concentration of component 1 and concentration of component 2 equations, together with boundary conditions. These initial equations are expressed as partial differential equations. However, the finalized equations are in the form of ordinary differential equations. Later, the bvp4c programme provided by the Matlab Software is used to solve the ordinary differential equations and the boundary conditions. Three distinct values of each governing parameter are fixed into the bvp4c function, to observe the behaviour of the physical parameters, namely as local Nusselt number and local Sherwood number. The main finding of the dual numerical solutions varies for increasing governing parameters until they intersect at the critical points. In conclusion, the governing parameters affects the heat and mass transfer of the fluid flow model model.

This study aims to investigate the thermal boundary layer over a vertical plate in porous medium with a convective surface boundary condition. The governing systems of partial differential equations subject to the boundary conditions are transformed into the system of ordinary differential equations by employing the similarity transformation. The bvp4c method in Matlab software is used to numerically solve the equations. With the use of graphical and tabular data, the velocity and temperature profiles for various parameter values is obtained, analyzed, and discussed. The effects of the governing parameters involved including permeability parameter K and the buoyancy parameter lambda are examined and discussed. The local Nusselt number and skin friction coefficient increase as the permeability parameter's K value rises. The results suggest that dual solutions for opposing flow and the solution is unique for assisting flow. The results also shown that by increasing K , the permeability parameter, will increase the range of solutions. Lastly, the permeability parameter, K , has the consequence of expanding the range values of buoyancy parameter, lambda for which solutions are discovered.

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.

Carbon nanotube (CNTs) has an extraordinary interest due to its escalating applications in industry and technological processes. Thus, the analysis of unsteady stagnation-point of carbon nanotubes through exponential permeable (extending/shrinking) sheet with slip effects is inspected in this work. Comprising SWCNTs/MWCNTs which suspended into a carrier fluid (Water (H2O) or kerosene (oil)). The similarity transformation technique is adapted to attain the dimensionless system of differential equations from the envisioned mathematical model in (pde). Later, the finite difference method is utilized in MATLAB software to attain the approximate solutions. The outcomes of quantities of practical interest versus crucial parameters are discussed graphically and interpreted physically. According to the finding, the double solution discovered for both extending and shrinking, the addition of SWCNTs volume fraction will speed up the separation of flow at the critical point, stability analysis performed the rational of first solution as time evolves. Finally, this analysis supports the adoption of controlling parameters in abroad range of distinct industries including processing and biomedical.

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.

This study is to analyse the problem of slip flow via exponentially stretching/shrinking sheet in carbon nanotubes (CNTs) with heat generation effects. The governing partial differential equations are transformed into nonlinear ordinary differential equations via transformation of similarity. The bvp4c solver in Matlab is then used to resolve them numerically. Water is used as the base fluid together with single wall and multi wall CNTs. The flow parameters effect is investigated, shown in the graphs form, and physically evaluated for the dimensionless velocity, temperature, skin friction, and Nusselt numbers. The results show that there are unique solutions for stretching sheets and non-unique solutions for shrinking sheets. In addition, compared to the case of a linearly stretching/shrinking sheet, the region of the stretch/shrink parameter where the similarity solution exists for the case of exponential stretching/shrinking sheet is greater.

The two-dimensional mathematical model ofwater-based hybrid nanofluid, where the nanoparticles of the model are alumina (Al2O3) and copper (Cu) is analyzed in this article. It describes the heat and mass transfer which are induced by concentration and temperature differences, respectively. The current mathematical model extended the works by implementing both directions of moving sheet in the boundary conditions: stretching and shrinking, and use the exponential variations of the sheet velocity, temperature, and concentration of the hybrid nanofluid at the sheet. The final numerical solutions can be obtained by implementing Matlab bvp4c, which involves the step of choosing the most reliable solution in an actual fluid situation. This selection technique on numerical solutions is known as stability analysis and only needs to apply when more than one numerical solution appears in the Matlab bvp4c program. Finally, the controlling parameters such as nanoparticle solid volume fraction, suction, shrinking/stretching, Soret and Dufour cause an increment or decrement in the flow, heat and mass transfer in the hybrid nanofluid. For the stable solution, fluid velocity becomes slower whereas temperature and concentration of the fluid increase when the percentage of Cu, as well as Al2O3, rises into the water. Moreover, in case of local Nusselt number and local Sherwood number it is proved that Soret effect is the opposite phenomenon of Dufour effect.