In this paper, the problem of steady laminar boundary layer flow of an incompressible viscous fluid over a moving thin needle is considered. The governing boundary layer equations were first transformed into non-dimensional forms. These non-dimensional equations were then transformed into similarity equations using the similarity variables, which were solved numerically using an implicit finite-difference scheme known as the Keller-box method. The solutions were obtained for a blunt-nosed needle. Numerical computations were carried out for various values of the dimensionless parameters of the problem which included the Prandtl number Pr and the parameter a representing the needle size. It was found that the heat transfer characteristics were significantly
influenced by these parameters. However, the Prandtl number had no effect on the flow characteristics due to the decoupled boundary layer equations.
Linear stability analysis was used to investigate the onset of Marangoni convection in a two-layer system. The system comprised a saturated porous layer over which was a layer of the same fluid. The fluid was heated from below and the upper free surface was deformable. At the interface between the fluid and the porous layer, the Beavers-Joseph slip condition was used and in the porous medium the Darcy law was employed to describe the flow. Predictions for the onset of convection were obtained from the analysis by the perturbation technique. The effect of surface deformation and depth ratio, z (which is equal to the depth of the fluid layer/depth of the porous layer) on the onset of fluid motion was studied in detail.
This paper examines the stagnation point flow towards a stretching/shrinking cylinder in a hybrid nanofluid. Here, copper (Cu) and alumina (Al2O3) are considered as the hybrid nanoparticles while water as the base fluid. The governing equations are reduced to the similarity equations using a similarity transformation. The resulting equations are solved numerically using the boundary value problem solver, bvp4c, available in the Matlab software. It is found that the heat transfer rate is greater for the hybrid nanofluid compared to the regular nanofluid as well as the regular fluid. Besides, the non-uniqueness of the solutions is observed for certain physical parameters. It is also noticed that the bifurcation of the solutions occurs in the shrinking regions. In addition, the heat transfer rate and the skin friction coefficients increase in the presence of nanoparticles and for larger Reynolds number. It is found that between the two solutions, only one of them is stable as time evolves.
The steady two-dimensional stagnation-point flow and heat transfer past a permeable stretching/shrinking sheet with effects of viscous dissipation, Joule heating and partial velocity slip in the presence of a magnetic field is investigated. The partial differential equations are reduced to nonlinear ordinary differential equations by using a similarity transformation, before being solved numerically by shooting technique. Results indicate that the skin friction coefficient and the local Nusselt number increase as magnetic parameter increases. It is found that for the stretching sheet the solution is unique while for the shrinking sheet there exist nonunique solutions (dual solutions) in certain range of parameters. The stability analysis shows that the upper branch solution is stable while the lower branch solution is unstable.
The steady two-dimensional flow and heat transfer over a stretching/shrinking sheet in a nanofluid is investigated using Buongiorno's nanofluid model. Different from the previously published papers, in the present study we consider the case when the nanofluid particle fraction on the boundary is passively rather than actively controlled, which make the model more physically realistic. The governing partial differential equations are transformed into nonlinear ordinary differential equations by a similarity transformation, before being solved numerically by a shooting method. The effects of some governing parameters on the fluid flow and heat transfer characteristics are graphically presented and discussed. Dual solutions are found to exist in a certain range of the suction and stretching/shrinking parameters. Results also indicate that both the skin friction coefficient and the local Nusselt number increase with increasing values of the suction parameter.
The magnetohydrodynamic (MHD) stagnation point flow of a nanofluid over a permeable stretching/shrinking sheet is studied. Numerical results are obtained using boundary value problem solver bvp4c in MATLAB for several values of parameters. The numerical results show that dual solutions exist for the shrinking case, while for the stretching case, the solution is unique. A stability analysis is performed to determine the stability of the dual solutions. For the stable solution, the skin friction is higher in the presence of magnetic field and increases when the suction effect is increased. It is also found that increasing the Brownian motion parameter and the thermophoresis parameter reduces the heat transfer rate at the surface.
The steady boundary layer flow of a viscous and incompressible fluid over a moving vertical flat plate in an external moving fluid with viscous dissipation is theoretically investigated. Using appropriate similarity variables, the governing system of partial differential equations is transformed into a system of ordinary (similarity) differential equations, which is then solved numerically using a Maple software. Results for the skin friction or shear stress coefficient, local Nusselt number, velocity and temperature profiles are presented for different values of the governing parameters. It is found that the set of the similarity equations has unique solutions, dual solutions or no solutions, depending on the values of the mixed convection parameter, the velocity ratio parameter and the Eckert number. The Eckert number significantly affects the surface shear stress as well as the heat transfer rate at the surface.
An analysis is carried out to study the steady two-dimensional stagnation-point flow of a nanofluid over a stretching/shrinking sheet in its own plane. The stretching/shrinking velocity and the ambient fluid velocity are assumed to vary linearly with the distance from the stagnation point. The similarity equations are solved numerically for three types of nanoparticles, namely copper, alumina, and titania in the water-based fluid with Prandtl number Pr = 6.2. The skin friction coefficient, Nusselt number, and the velocity and temperature profiles are presented graphically and discussed. Effects of the solid volume fraction φ on the fluid flow and heat transfer characteristics are thoroughly examined. Different from a stretching sheet, it is found that the solutions for a shrinking sheet are non-unique.
The paper deals with a stagnation-point boundary layer flow towards a permeable stretching/shrinking sheet in a nanofluid where the flow and the sheet are not aligned. We used the Buongiorno model that is based on the Brownian diffusion and thermophoresis to describe the nanofluid in this problem. The main purpose of the present paper is to examine whether the non-alignment function has the effect on the problem considered when the fluid suction and injection are imposed. It is interesting to note that the non-alignment function can ruin the symmetry of the flows and prominent in the shrinking sheet. The fluid suction will reduce the impact of the non-alignment function of the stagnation flow and the stretching/shrinking sheet but at the same time increasing the velocity profiles and the shear stress at the surface. Furthermore, the effects of the pertinent parameters such as the Brownian motion, thermophoresis, Lewis number and the suction/injection on the flow and heat transfer characteristics are also taken into consideration. The numerical results are shown in the tables and the figures. It is worth mentioning that dual solutions are found to exist for the shrinking sheet.
In this paper, the unsteady stagnation-point boundary layer flow and heat transfer of a special third grade fluid past a permeable stretching/shrinking sheet has been studied. Similarity transformation is used to transform the system of boundary layer equations which is in the form of partial differential equations into a system of ordinary differential equations. The system of similarity equations is then reduced to a system of first order differential equations and has been solved numerically by using the bvp4c function in Matlab. The numerical solutions for the skin friction coefficient and heat transfer coefficient as well as the velocity and temperature profiles are presented in the forms of tables and graphs. Dual solutions exist for both cases of stretching and shrinking sheet. Stability analysis is performed to determine which solution is stable and valid physically. Results from the stability analysis depict that the first solution (upper branch) is stable and physically realizable, while the second solution (lower branch) is unstable.
The steady two dimensional magnetohydrodynamic (MHD) boundary layer flow and heat transfer over a stretching/shrinking permeable wedge is numerically investigated. The partial differential equations governing the flow and heat transfer are transformed into a system of ordinary differential equations using a similarity transformation. These equations are then solved numerically using the boundary value problem solver, bvp4c in Matlab software. It is found that dual solutions exist for a certain range of the shrinking strength. A stability analysis is performed to identify which solution is stable and physically reliable.
The steady boundary layer flow and heat transfer of a nanofluid past a nonlinearly permeable stretching/shrinking sheet is numerically studied. The governing partial differential equations are reduced into a system of ordinary differential equations using a similarity transformation, which are then solved numerically using a shooting method. The local Nusselt number and the local Sherwood number and some samples of velocity, temperature and nanoparticle concentration profiles are graphically presented and discussed. Effects of the suction parameter, thermophoresis parameter, Brownian motion parameter and the stretching/shrinking parameter on the flow, concentration and heat transfer characteristics are thoroughly investigated. Dual solutions are found to exist in a certain range of the stretching/shrinking parameter for both shrinking and stretching cases. Results indicate that suction widens the range of the stretching/shrinking parameter for which the solution exists.
Hybrid ferrofluid is a unique heat transfer fluid because it can be magnetically controlled and ideal in various applications. Further exploration to unleash its potential through studying heat transfer and boundary layer flow is crucial, especially in solving the thermal efficiency problem. Hence, this research focuses on the numerical examination of flow behaviour and heat transfer attributes of magnetized hybrid ferrofluid Fe3O4-CoFe2O4/water across a permeable moving surface considering the mutual effects of magnetohydrodynamic (MHD), viscous dissipation, and suction/injection. The problem was represented by the Tiwari and Das model with duo magnetic nanoparticle hybridization; magnetite Fe3O4 and cobalt ferrite CoFe2O4 immersed in water. The governing equations were transformed into ordinary differential equations using appropriate similarity variables and solved with bvp4c MATLAB. A dual solution is obtained, and via stability analysis, the first solution is stable and physically reliable. The significant influence of governing effects on the temperature and velocity profiles, the local skin friction coefficient and the local Nusselt number are analyzed and visually shown. The surge-up value of suction and CoFe2O4 ferroparticle volume concentration enhances the local skin friction coefficient and heat transfer rate. Additionally, the magnetic parameter and Eckert number reduced the heat transfer. Using a 1% volume fraction of Fe3O4 and CoFe2O4; the hybrid ferrofluid's convective heat transfer rate was shown to be superior to mono-ferrofluid and water by enhancing 2.75% and 6.91%, respectively. This present study also suggests implying a greater volume concentration of CoFe2O4 and lessening the magnetic intensity to maintain the laminar flow phase.
This research intends to investigate the effect of the nonlinearity of the surface velocity on the hybrid nanofluid flow behavior. Here, the total composition of Al2O3 (alumina) as well as Cu (copper) volume fractions, are implemented in a one-to-one ratio and then dispersed in water. The similarity equations are gained employing a similarity transformation, which is programmed in MATLAB software. The dual solutions are attainable for certain ranges with respect to the mass flux parameter S and the power-law index n. Also, the turning point occurs in the region of S<0 and n>1. Besides, the rise of n led to reduce the skin friction as well as the heat transfer coefficients with 39.44 % and 11.71 % reduction, respectively. Moreover, 14.39 % reduction of the heat transfer rate is observed in the presence of viscous dissipation (Eckert number). It is found that only the first solution is stable as time progresses. Generally, this study gives scientists and engineers a starting point for predicting how to control the parameters to achieve the best results for relevant practical applications.
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 use of hybrid nanoparticles to increase heat transfer is a favorable area of research, and therefore, numerous scientists, researchers, and scholars have expressed their appreciation for and interest in this field. Determining the dynamic role of nanofluids in the cooling of microscopic electronic gadgets, such as microchips and related devices, is also one of the fundamental tasks. With such interesting and useful applications of hybrid nanofluids in mind, the main objective is to deal with the analysis of the unsteady flow towards a shrinking sheet in a water-based hybrid ferrite nanoparticle in porous media, with heat sink/source effects. Moreover, the impact of these parameters on heat and mass transfers is also reported. Numerical results are obtained using MATLAB software. Non-unique solutions are determined for a certain shrinking strength, in addition to the unsteadiness parameter. The mass transfer and friction factor increase for the first solution due to the hybrid nanoparticles, but the heat transfer rate shows the opposite effect.
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.
Cooling and heating are two critical processes in the transportation and manufacturing industries. Fluid solutions containing metal nanoparticles have higher thermal conductivity than conventional fluids, allowing for more effective cooling. Thus, the current paper is a comparative exploration of the time-independent buoyancy opposing and heat transfer flow of alumina nanoparticles scattered in water as a regular fluid induced via a vertical cylinder with mutual effect of stagnation-point and radiation. Based on some reasonable assumptions, the model of nonlinear equations is developed and then tackled numerically employing the built-in bvp4c MATLAB solver. The impacts of assorted control parameters on gradients are investigated. The outcomes divulge that the aspect of friction factor and heat transport upsurge by incorporating alumina nanoparticles. The involvement of the radiation parameter shows an increasing tendency in the heat transfer rate, resulting in an enhancement in thermal flow efficacy. In addition, the temperature distribution uplifts due to radiation and curvature parameters. It is discerned that the branch of dual outcomes exists in the opposing flow case. Moreover, for higher values of the nanoparticle volume fraction, the reduced shear stress and the reduced heat transfer rate increased respectively by almost 1.30% and 0.0031% for the solution of the first branch, while nearly 1.24%, and 3.13% for the lower branch solution.
Objective: hybrid nanofluids have superior thermal efficiency and physical durability in contrast to regular nanofluids. The stagnation point flow of MHD micropolar hybrid nanofluids over a deformable sheet with viscous dissipation is investigated. Methodology: the controlling partial differential equations are converted to nonlinear ordinary differential equations using the transmuted similarity, and are subsequently solved using the bvp4c solver in MATLAB. The hybrid nanofluids consist of aluminum and copper nanoparticles, dispersed in a base fluid of water. Results: multiple solutions are obtained in the given problem for the case of shrinking as well as for the stretching sheet due to the variation in several influential parameters. Non-unique solutions, generally, exist for the case of shrinking sheets. In addition, the first branch solution is physically stable and acceptable according to the stability analysis. The friction factor is higher for the branch of the first solution and lower in the second branch due to the higher magnetic parameters, while the opposite behavior is seen in the case of the local heat transfer rate. Originality: the novelty of this model is that it finds multiple solutions in the presence of Cu and Al2O3 nanoparticles and also performs the stability analysis. In general, non-unique solutions exist for the phenomenon of shrinking sheets.