Stagnation Point Boundary Layer Research Articles

Chemically reactive aspects of stagnation-point boundary layer flow of second-grade nanofluid over an exponentially stretching surface

Assessing the stagnation-point flow of a second-grade nanofluid with an induced magnetic field and the effects of Joule heating toward an exponentially extending surface is the leading goal of the current effort. The modeling of the thermal and solute energy equations has taken into account nonlinear thermal radiation as well as the impact of activation energy. The boundary of the sheet is subjected to the zero-mass flux and thermal slip boundary conditions. The simulated flow equations are converted into coupled nonlinear ODEs (ordinary differential equations) using similarity variables. The BVP4C MATLAB technique is used to numerically resolve these ODEs. Graphs and tabular data are used to perform the physical debate. The importance of a greater estimation of the magnetic Prandtl number and magnetic parameter is emphasized in order to obtain a better induced magnetic field profile. Additionally, the greater estimations of the second-order fluid parameter increase the rates of friction and heat transmission.

Dual solutions for general three-dimensional MHD boundary layer stagnation-point flow of hybrid nanofluid and heat transfer

PurposeThe evaluation of high thermal efficiency has actively highlighted the unique behaviour of hybrid nanofluid. Thus, the purpose of this paper is to emphasize the hybrid nanofluid’s stagnation point in three-dimensional flow with magnetic field.Design/methodology/approachThe defined ordinary differential equations systems are addressed using the bvp4c solver.FindingsThe results indicate that using dual solutions is possible as long as the physical parameters remain within their specified ranges. Hybrid nanofluid flow has been recognised for its superior heat transfer capabilities in comparison to both viscous flow and nanofluid flow. Furthermore, it has been demonstrated in the current study that augmenting the volume concentration of nanoparticles leads to a corresponding enhancement in the rate of heat transfer. When the velocity gradients ratio is augmented, there is a corresponding reduction in the thermal performance. The separation value grows as the magnetic parameter rises, which signifies the expansion of the boundary layer.Originality/valueThe originality of the paper highlights the general mathematical hybrid model of the three-dimensional problem with the magnetohydrodynamics (MHD) effect in the stagnation point flow. The comprehensive examination of the suggested model has not yet been thoroughly addressed in prior research.

Information of stagnation-point flow of Maxwell fluid past symmetrically exponential stretching/shrinking cylinder with prescribed heat flux

This paper deals with the numerical simulation of the stagnation-point boundary-layer flow of Maxwell fluid comprising heat transfer effects past a horizontal cylinder having exponential stretching/shrinking velocity. All geometrical aspects and physical conditions portrayed under the considered model are stated by a system of governing equations in cylindrical coordinates, for which similarity transformation is carried out to get non-linear ordinary differential equations. The secant shooting method is applied to the modeled equations to glean the graphical solution for the skin friction coefficient, local Nusselt number, longitudinal velocity, temperature dispersion, and stream function. Then further discussion was carried out in light of the governing physical parameters, and the nature of dual solutions has been discussed. The results indicate that dual solutions vanish by enhancing stretching and exist only for a limited belt of the shrinking parameter. Surface heat transfer varies inversely for an increasing stretching/shrinking parameter in case of a second branch solution. Curve relocations for axial velocity and temperature have also been observed. The symmetry of the first and second solution for the flow is found to be unaltered for Newtonian and non-Newtonian fluid.

A comparative study of unsteady MHD Falkner–Skan wedge flow for non-Newtonian nanofluids considering thermal radiation and activation energy

Boundary layer flow is very important in many engineering applications, such as thermal padding, oil bed retrieval, percolation, heat exchangers, and geothermal analysis. In this study, we explore unsteady magnetohydrodynamic bioconvective stagnation-point boundary layer flow over a wedge. The heat and mass transfer in Jeffrey and Oldroyd-B nanofluids in the presence of chemical reactions and under radiation effects are considered to develop a system of equations. The governing equations are transformed to a system of ordinary differential equations by employing suitable transformations, and the reduced system of equations is solved using the finite element technique (FEM). The effect of various parameters on the velocity, microrotation, temperature, microbe compactness, and concentration of nanoparticles are determined. The Nusselt number increases with the Brownian motion parameter (Nb) and thermophoresis parameter (Nt). The Nusselt number is higher for elongating wedges than shrinking wedges. The Sherwood number increases with the reaction rate (Ω) and activation energy parameter. A detailed parametric study is carried out, and the results are presented in graphical and tabular forms. The code performance and the exactness of the numerical scheme are verified by comparing our results with those in the literature.

Slip Effect on Stagnation Point Flow and Heat Transfer Over a Shrinking/Stretching Sheet in A Porous Medium with Suction/Injection

The effects of thermal and velocity slip on Newtonian fluid steady boundary layer stagnation-point flow and heat transfer through a porous shrinking sheet with suction effect are investigated in the present research. By using a similarity transformation, the partial differential equation (PDE) is converted to an ordinary differential equation (ODE) and solved using the shooting technique to get numerical results. The flow characteristics for various values of the physical parameters are discussed, taking into consideration the effects of suction/injection as well as velocity and thermal slip. The local Nusselt number and skin friction coefficient, as well as the temperature and the velocity profiles, are presented as results. It is noticed that when the suction and velocity slip increase, the skin friction coefficient and local Nusselt number rise, as does the range of solutions that exist.

Heat Transfer Analysis over the Boundary Layer Stagnation-Point Flow of Couple Stress Fluid over an Exponentially Stretching Sheet

The present study provides a comprehensive discussion for the problem of boundary layer flow and heat transfer analysis of stagnation point flow of couple stress fluid over an exponentially stretching surface. The governing equations of couple stress fluid model are assumed under boundary layer approach. The nonlinear partial differential equations are simplified by using similar similarity transformations. The analytical solutions of abridged equations are computed with the help of homotopy analysis method (HAM). The convergence of the HAM solutions have been deliberated by plotting <i>ћ </i>- curves and also through homotopy pade approximation. The physical features of pertinent parameters have been discussed through graphs.

RETRACTED: Aqueous aluminium–copper hybrid nanofluid flow past a sinusoidal cylinder considering three-dimensional magnetic field and slip boundary condition

We analyzed the problem of the steady general three-dimensional magnetohydrodynamics stagnation-point boundary layer flow past an impermeable wavy circular cylinder considering aluminium–copper/water hybrid nanofluid as the working fluid and velocity slip as well as temperature jump boundary conditions. The induced magnetic field effect was also taken into account. The analytical procedure is based on the model that implements the nanoparticles and base fluid masses to formulate the equivalent volume fraction, equivalent density, and equivalent specific heat at constant pressure which is then substituted in the chosen single-phase thermophysical properties. Then, the foregoing relations were used in basic governing PDEs (partial differential equations), according to Tiwari–Das nanofluid scheme. It is worth mentioning that the bvp4c code from MATLAB software that is a famous finite-difference method has been exploited for solving the final similarity ODEs (ordinary differential equations). Results demonstrate that the developed mass-based model can be successfully employed with great confidence to study the flow and heat transfer of hybrid nanofluid in other similar problems. Moreover, it is proved that the nodal stagnation points possess higher values of skin friction coefficients and local Nusselt numbers relative to those for the saddle stagnation points. Besides, enhancing the second nanoparticle's mass leads to increase in all parameters of engineering interest including skin friction coefficients along x and y directions as well as local Nusselt number.

Dual solutions of time-dependent magnetohydrodynamic stagnation point boundary layer micropolar nanofluid flow over shrinking/stretching surface

Dual solutions of time-dependent magnetohydrodynamic stagnation point boundary layer micropolar nanofluid flow over shrinking/stretching surface

Melting heat transfer of MHD stagnation point flow of non-Newtonian fluid due to an elastic sheet

This paper sheds light on the numerical investigation of 2D flow and melting heat transport of an electrically conducting boundary layer stagnation-point flow of Casson fluid in the direction of stretching sheet. Suitable changes are made in rehabilitating the flow governing partial differential equations into dimensionless ordinary differential equations and afterwards solved these ordinary differential equations by the fourth order Runge-kutta numerical method by utilising the most efficient shooting technique. The influence of melting heat transfer, magnetic field, Prandtl number, Rayleigh number and Schmidt number on velocity, energy and concentration distributions, are studied through graphically drawn results.

Diffusion-thermo Effects in Stagnation Point Flow of Second Grade Fluid past a Stretching Plate

Transmission of heat and mass in boundary layer flows over stretching surfaces play a significant role in metallurgy and polymer industry. In Current article the assisting and opposing flow of a second grade fluid towards a stretching sheet is analyzed to examine the heat and mass transfer in stagnation point boundary layer flow. Different flow parameters such as concentration, surface temperature and stretching velocity are supposed to variate linearly. The basic transport equations are transformed into non-linear ordinary differential equations by means of boundary layer approximation and similarity transmutations, which are then solved by employing nonlinear shooting (NLS) and Keller-box methods (KBM). These techniques are very useful for solving boundary-layer problems and are applicable to other general situations than that presented current study. The outcomes of velocity, temperature, concentration profile, skin-friction coefficient, heat and mass transfer coefficients are analyzed briefly in graphical and tabular formats. The mass transmission rate was found to be in direct relation with Schmidt number. Moreover, we predict that a rise in Prandtl number leads to a decline in temperature and thermal layer of boundary thickness for both supporting and contrasting flows. The outcomes of this article are important for the analysts in the field of second grade fluids. We believe that the article is very well prepared and the results are original and useful from both theoretical and application point of views.

Nodal/Saddle Stagnation Point Slip Flow of an Aqueous Convectional Magnesium Oxide–Gold Hybrid Nanofluid with Viscous Dissipation

In this analysis, convective heat transfer characteristics of a hybrid nanofluid mixture containing magnesium oxide (MgO) and gold (Au) nanoparticles are numerically studied. The impact of slip effects on nodal/saddle stagnation point boundary layer flow with viscous dissipation effect is mathematically modeled. The behavior of nanofluids is studied by employing Tiwari–Das nanofluid model. Pure water is the base fluid in this analysis. The governing partial differential equations with many independent variables are reduced to ordinary differential equations with one independent variable and then numerically solved by the Runge–Kutta–Fehlberg method with the desired accuracy. The outputs showed that MgO–Au/water hybrid nanofluid sharply raises the base fluid's thermal behavior. Results reveal that in the nodal and saddle point areas, the impact of higher slip effects significantly increases the local heat transfer rate.

A Stability Analysis of Boundary Layer Stagnation-Point Slip Flow and Heat Transfer towards a Shrinking/Stretching Cylinder over a Permeable Surface

A stability analysis of dual solution for the problem of stagnation-point slip flow over a stretching or shrinking cylinder is studied. The partial differential equations governing will be transformed to a set of coupled nonlinear nonsimilar equations via similarity transformations. The transformed governing equations are solved numerically using the bvp4c function in MATLAB software. Numerical calculations exhibit the existence of dual solution and the implementation of stability analysis proved that the first solution is stable and physically realizable.

The Boundary Layers of an Unsteady Stagnation Points Flow in Carbon Nanotubes

The unsteady stagnation-point boundary layer flow of carbon nanotubes is studied. Single-wall and multi-wall of carbon nanotubes are considered with ????2???? as a carrier fluid. The compatible transformations are utilized to change the governing equations of mathematical model which is in the form of partial differential equations (????????????????) into a set of a non-linear ordinary differential equations (???????????????? ). Solution of the problem obtained numerically through a powerful function (????????????4????) in MATLAB software. The impact of solid volume fraction and unsteadiness parameters on dimensionless velocity and temperature flow fields along with the magnitude of skin friction coefficient and local Nusselt number are discussed graphically and interpreted physically. The outcome indicates that the dual solutions observed when unsteadiness parameter is negative. Further, single-wall carbon nanotube has higher skin friction, as well as heat transfer rate, compare with multi-wall carbon nanotube at the surface.

Unsteady stagnation-point boundary layer flows of power-law fluids over a porous flat plate

The problem of unsteady stagnation-point flows of power-law fluids over a porous flat plate with mass transfer is considered with a view to examine the rheological behaviors of the fluids. This study is completely based on the four physical parameters, namely, flow strength parameter a, mass transfer parameter d, unsteadiness parameter $$\beta$$ and non-Newtonian power-law index n. For $$d=0$$ , the numerical results of this analysis reveal the existence of two types of solutions - one is attached flow solution (AFS) and the other is reverse flow solution (RFS) in a definite range of n ( $$0< n \le 2$$ ) when (a, $$\beta )= (1, - 1)$$ . The present analysis confirms that the velocity profile for any dilatant fluid $$(n > 1)$$ matches smoothly with the free stream velocity for a suitable amount of blowing $$d(< 0)$$ depending upon the values of n. We will also discuss the asymptotic behaviors of the boundary layer flows for large values of d, i.e., for $$d \rightarrow \pm \infty$$ . The asymptotic analysis ensures the existence of the above two solutions for large values of suction $$d>0$$ , whereas the boundary layer solution is terminated after a certain value of blowing $$d<0$$ , dependent on the values of $$\beta (< 0)$$ and n. Below this critical value of blowing d, this unsteady flow problem also provides us with a solution which does not appear to have a boundary layer character.

Stability Analysis on Stagnation-Point Flow and Heat Transfer towards a Permeable Stretching/Shrinking Sheet with Heat Source in a Casson Fluid

This paper deals with a stagnation-point boundary layer flow and heat transfer of a Casson fluid towards a stretching/shrinking sheet. The main objective of the present study is to analyse the effects of the injection parameter and heat source on the velocity and temperature profiles as well as the skin friction coefficient and the Nusselt number. It is vital to study the heat transfer and fluid flow problems in the presence of injection and heat source effects due to a wide variety of applications in engineering and industry. The governing nonlinear partial differential equations are transformed into a system of nonlinear ordinary differential equations by using similarity transformation, before being solved numerically using the boundary value problem solver bvp4c routine in MATLAB. Dual solutions are found to exist for the shrinking sheet case, whereas the solution is unique for the stretching case. The stability analysis has been performed to determine the stable solution. It is shown that the first solutions are stable and physically reliable while the second solutions are not. Further, the present results have been compared with the previous published results for a particular case and the comparisons are found to be in good agreement. The local Nusselt number is decreases with an increase in heat source parameter. Rising values of the injection parameter has decreases both the skin friction coefficient and the local Nusselt number.

Exact solution of stagnation point flow of MHD Cu–H2O nanofluid induced by an exponential stretching sheet with thermal conductivity

This study presents the stagnation point boundary layer flow of copper–water based nanofluid subject to the applied magnetic field. The flow is caused by the presence of an exponentially stretching porous wall. Heat transfer analysis is also taken into account along with the thermal radiation and Joule heating effects. Mathematical modeling is performed to convert the physical system into a set of mathematical equations which are further simplified by using suitable variables. Exact solutions for the velocity and temperature profiles are computed and interpreted for various physical parameters of interest. It is observed that by increasing the magnitude of the solid volume fraction of nanoparticles, the velocity profile is accelerated. The results also indicate an increment in the Eckert number increases the temperature and the thermal boundary layer thickness.

Mixed convection and stability analysis of stagnation-point boundary layer flow and heat transfer of hybrid nanofluids over a vertical plate

Purpose This study aims to study the mixed convection flow and heat transfer of Al2O3-Cu/water hybrid nanofluid over a vertical plate. Governing equations for conservation of mass, momentum and energy for the hybrid nanofluid over a vertical flat plate are introduced. Design/methodology/approach The similarity transformation approach is used to transform the set of partial differential equations into a set of non-dimensional ordinary differential equations. Finite-deference with collocation method is used to integrate the governing equations for the velocity and temperature profiles. Findings The results show that dual solutions exist for the case of opposing flow over the plate. Linear stability analysis was performed to identify a stable solution. The stability analysis shows that the lower branch of the solution is always unstable, while the upper branch of the solution is always stable. The results of boundary layer analysis are reported for the various volume fractions of composite nanoparticles and mixed convection parameter. The outcomes show that the composition of nanoparticles can notably influence the boundary layer flow and heat transfer profiles. It is also found that the trend of the variation of surface skin friction and heat transfer for each of the dual solution branches can be different. The critical values of the mixed convection parameter, λ, where the dual solution branches joint together, are also under the influence of the composition of hybrid nanoparticles. For instance, assuming a total volume fraction of 5 per cent for the mixture of Al2O3 and Cu nanoparticles, the critical value of mixing parameter of λ changes from −3.1940 to −3.2561 by changing the composition of nanofluids from Al2O3 (5 per cent) + Cu (0%) to Al2O3 (2.5%) + Cu (2.5 per cent). Originality/value The mixed convection stability analysis and heat transfer study of hybrid nanofluids for a stagnation-point boundary layer flow are addressed for the first time. The introduced hybrid nanofluid model and similarity solution are new and of interest in both mathematical and physical points of view.

Non-uniqueness of solutions to a MHD stagnation-point flow over an exponentially permeable stretching/shrinking sheet with velocity slip

In this study, a magnetohydrodynamic (MHD) stagnation-point boundary layer flow over an exponentially stretching/shrinking sheet with suction/injection and velocity slip effects is presented. The governing equations were first transformed into a non-linear ordinary differential equations with the help of similarity transformations. Following from there, the equations obtained were solved numerically utilizing the bvp4c programme in MATLAB. It was determined that the non-linear ordinary differential equations possessed non-uniqueness of solutions (dual solutions). Stability analysis was performed to verify the stability of the dual solutions obtained. The results revealed that the first solution was stable while the second solution was unstable. Apart from that, it was also found that as the magnetic, suction and velocity slip parameters increased, the range of the non-uniqueness of solutions increased correspondingly. Hence, it is proposed that the existence of magnetic, suction and velocity slip effects delayed the boundary layer separation.

MHD stagnation-point flow and heat transfer past a stretching/shrinking sheet in a hybrid nanofluid with induced magnetic field

PurposeThe purpose of this paper is to investigate the steady magnetohydrodynamics (MHD) boundary layer stagnation-point flow of an incompressible, viscous and electrically conducting fluid past a stretching/shrinking sheet with the effect of induced magnetic field.Design/methodology/approachThe governing nonlinear partial differential equations are transformed into a system of nonlinear ordinary differential equations via the similarity transformations before they are solved numerically using the “bvp4c” function in MATLAB.FindingsIt is found that there exist non-unique solutions, namely, dual solutions for a certain range of the stretching/shrinking parameters. The results from the stability analysis showed that the first solution (upper branch) is stable and valid physically, while the second solution (lower branch) is unstable.Practical implicationsThis problem is important in the heat transfer field such as electronic cooling, engine cooling, generator cooling, welding, nuclear system cooling, lubrication, thermal storage, solar heating, cooling and heating in buildings, biomedical, drug reduction, heat pipe, space aircrafts and ships with better efficiency than that of nanofluids applicability. The results obtained are very useful for researchers to determine which solution is physically stable, whereby, mathematically more than one solution exist.Originality/valueThe present results are new and original for the problem of MHD stagnation-point flow over a stretching/shrinking sheet in a hybrid nanofluid, with the effect of induced magnetic field.

MHD mixed convection boundary layer stagnation-point flow on a vertical surface with induced magnetic field

Purpose This study aims to perform a stability analysis on a steady magnetohydrodynamic (MHD) mixed convection boundary-layer stagnation-point flow of an incompressible, viscous and electrically conducting fluid over a vertical flat plate. The effect of induced magnetic field is also considered. Design/methodology/approach The governing boundary layer equations are transformed into a system of ordinary differential equations using the similarity transformations. The system is then solved numerically using the “bvp4c” function in MATLAB. Findings Dual solutions are found to exist for a certain range of the buoyancy parameter for both the assisting and opposing flows. The results from the stability analysis showed that the first solution (upper branch) is stable and valid physically, while the second solution (lower branch) is unstable. Practical implications This problem is important in many metallurgical processes, namely, drawing, annealing and tinning of copper wires. The results obtained are very useful for researchers to determine which solution is physically stable, whereby mathematically more than one solution exists for the skin friction coefficient and the heat transfer characteristics. Originality/value The present results of the stability analysis are original and new for the problem of MHD mixed convection stagnation-point flow of viscous conducting fluid over a vertical flat plate, with the effect of induced magnetic field.