MHD STAGNATION POINT FLOW WITH THERMAL RADIATION AND SLIP EFFECT OVER A LINEAR STRETCHING SHEET

This research investigates the flow of stagnation point magnetohydrodynamic (MHD) and heat transfer along the stretched sheet in the existence of radiation and slip effects. With the help of similarity variables, the governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs). The BVP4C technique in Matlab function has been used to simplify the governing ODEs. The numerical outcomes for temperature and velocity profiles, coefficient of skin friction and Nusselt Number have been achieved and matched with the findings in literature. The findings are compared to previously reported results. In addition, the impacts of numerous related parameters on the profiles of velocity and temperature are shown, and the results of every related parameter are presented using graphs. The velocity profile decreases as the magnetic force, suction, and permeability parameters rise

Analysis of dual solution for MHD flow of Williamson fluid with slippage

This study investigates the numerical solutions of MHD boundary layer and heat transfer of the Williamson fluid flow on the exponentially vertical shrinking sheet, having variable thickness and thermal conductivity under effects of the velocity and thermal slip parameters. It is also assumed that shrinking/stretching velocity, as well as the wall temperature, has the exponential function form. In this study, the continuity, momentum and energy equations with buoyancy parameter and Hartmann number are incorporated especially in the Williamson fluid flow case. Similarity transformation variables have been employed to formulate the ordinary differential equations (ODEs) from partial differential equations (PDEs). The resultant ODEs are solved by shooting method with Runge Kutta of fourth order method in Maple software. The effects of the different applied non-dimensional physical parameters on the boundary layer and heat transfer flow problems are presented in graphs. The effects of Williamson parameter, Prandtl number, and slip parameters on velocity and temperature profiles have been thoroughly demonstrated and discussed. The numerical results show that the buoyancy force and the slip parameters contribute to the occurrence of the dual solutions on the boundary layer and heat transfer flow problems. Furthermore, the stability analysis suggests that the first solution is stable and physically possible

Mathematical analysis of magnetohydrodynamic (MHD) flow of micropolar nanofluid under buoyancy effects past a vertical shrinking surface: dual solutions

In this paper, we explore dual solutions of MHD flow, heat and mass transfer of micropolar nanofluid over a linear vertical shrinking surface with buoyancy effects, which was not considered in the previous works. The governing fluid flow equations of this problem are transformed into nonlinear boundary value problems (BVPs) of ordinary differential equations (ODEs) by applying similarity variables. The resultant BVPs are converted into initial value problems (IVPs) by using shooting method which then resolved by employing Runge Kutta of order four. The impacts of the governing parameters, such as suction parameter, material parameter, Richardson number, magnetic parameter, Prandtl number, thermophoresis and Brownian motion parameters on velocity, angular velocity, temperature, and concentration are illustrated graphically. The results indicate that the existence of a range of dual solutions and no-solutions. When Richardson number (delta) is increased, the reduction of the velocity of micropolar nanofluid has occurred in the second solution. The stability analysis on dual solutions, however, reveals that only the first solution is stable

Multiple solutions of Cu-C6H9NaO7 and Ag-C6H9NaO7 nanofluids flow over nonlinear shrinking surface

Model of Casson nanofluid flow over a nonlinear shrinking surface is considered. Model of Tiwari and Das is applied to nanofluid comprising of sodium alginate with copper and silver. The governing nonlinear equations incorporating the effects of the viscous dissipation are transformed into boundary value problems (BVPs) of ordinary differential equations (ODEs) by using appropriate similarity transformations. The resulting equations are converted into initial value problems (IVPs) using the shooting method which are then solved by Runge-Kutta method of fourth order. In order to determine the stability of the dual solutions obtained, stability analysis is performed and discovered that the first (second) solution is stable (unstable) and physically realizable (unrealizable). Both the thickness of the thermal boundary layer as well as temperature increase when the Casson parameter (β) is increased in the second solution

Triple solutions of micropolar nanofluid in the presence of radiation over an exponentially preamble shrinking surface: Convective boundary condition

In this study, we attempt to obtain all probable multiple solutions of the magnetohydrodynamic (MHD) steady flow of micropolar nanofluid on an exponentially shrinking surface by the consideration of concentration slip, thermal radiation, and convective boundary condition with help of the revised model of Buongiorno. The significance of the mass suction on the existence of multiple solutions is integrated. The suitable pseudo‐exponential similarity variables have been adopted to transfer the system of nonlinear partial differential equations into a system of nonlinear quasi‐ordinary ordinary differential equations. The resultant system has been solved by employing the Runge–Kutta fourth‐order method along with the shooting method. Three different ranges of solutions are noticed, namely triple solutions and single solution. When ranges of the suction parameter ar

Dual solutions and stability analysis of a hybrid nanofluid over a stretching/shrinking sheet executing MHD flow

In this paper, the unsteady magnetohydrodynamic (MHD) flow of hybrid nanofluid (HNF) composed of Cu−Al2O3 /water in the presence of a thermal radiation effect over the stretching/shrinking sheet is investigated. Using similarity transformation, the governing partial differential equations (PDEs) are transformed into a system of ordinary differential equations (ODEs), which are then solved by using a shooting method. In order to validate the obtained numerical results, the comparison of the results with the published literature is made numerically as well as graphically and is found in good agreements. In addition, the effects of many emerging physical governing parameters on the profiles of velocity, temperature, skin friction coefficient, and heat transfer rate are demonstrated graphically and are elucidated theoretically. Based on the numerical results, dual solutions exist in a specific range of magnetic, suction, and unsteadiness parameters. It was also found that the values of f′′(0) rise in the first solution and reduce in the second solution when the solid volume fraction ϕCu is increased. Finally, the temporal stability analysis of the solutions is conducted, and it is concluded that only the first solution is stable

Rotating 3D flow of hybrid nanofluid on exponentially shrinking sheet: Symmetrical solution and duality

This article aims to study numerically the rotating, steady, and three-dimensional (3D) flow of a hybrid nanofluid over an exponentially shrinking sheet with the suction effect. We considered water as base fluid and alumina (Al2O3), and copper (Cu) as solid nanoparticles. The system of governing partial differential equations (PDEs) was transformed by an exponential similarity variable into the equivalent system of ordinary differential equations (ODEs). By applying a three-stage Labatto III-A method that is available in bvp4c solver in the Matlab software, the resultant system of ODEs was solved numerically. In the case of the hybrid nanofluid, the heat transfer rate improves relative to the viscous fluid and regular nanofluid. Two branches were obtained in certain ranges of the involved parameters. The results of the stability analysis revealed that the upper branch is stable. Moreover, the results also indicated that the equations of the hybrid nanofluid have a symmetrical solution for different values of the rotation parameter (Ω)

Effects of stefan blowing and slip conditions on unsteady MHD casson nanofluid flow over an unsteady shrinking sheet: Dual solutions

In this article, the magnetohydrodynamic (MHD) flow of Casson nanofluid with thermal radiation over an unsteady shrinking surface is investigated. The equation of momentum is derived from the Navier–Stokes model for non-Newtonian fluid where components of the viscous terms are symmetric. The effect of Stefan blowing with partial slip conditions of velocity, concentration, and temperature on the velocity, concentration, and temperature distributions is also taken into account. The modeled equations of partial differential equations (PDEs) are transformed into the equivalent boundary value problems (BVPs) of ordinary differential equations (ODEs) by employing similarity transformations. These similarity transformations can be obtained by using symmetry analysis. The resultant BVPs are reduced into initial value problems (IVPs) by using the shooting method and then solved by using the fourth-order Runge–Kutta (RK) technique. The numerical results reveal that dual solutions exist in some ranges of different physical parameters such as unsteadiness and suction/injection parameters. The thickness of the velocity boundary layer is enhanced in the second solution by increasing the magnetic and velocity slip factor effect in the boundary layer. Increment in the Prandtl number and Brownian motion parameter is caused by a reduction of the thickness of the thermal boundary layer and temperature. Moreover, stability analysis performed by employing the three-stage Lobatto IIIA formula in the BVP4C solver with the help of MATLAB software reveals that only the first solution is stable and physically realizable

Numerical investigation of multiple solutions for caputo fractional-order-two dimensional magnetohydrodynamic unsteady flow of generalized viscous fluid over a shrinking sheet using the Adams-type predictor-corrector method

In this paper, magnetohydrodynamic (MHD) flow over a shrinking sheet and heat transfer with viscous dissipation has been studied. The governing equations of the considered problem are transformed into ordinary differential equations using similarity transformation. The resultant equations are converted into a system of fractional differential boundary layer equations by employing a Caputo derivative which is then solved numerically using the Adams-type predictor-corrector method (APCM). The results show the existence of two ranges of solutions, namely, dual solutions and no solution. Moreover, the results indicate that dual solutions exist for a certain range of specific parameters which are in line with the results of some previously published work. It is also observed that the velocity boundary layer decreases as the suction and magnetic parameters increase

Stability analysis and dual solutions of micropolar nanofluid over the inclined stretching/shrinking surface with convective boundary condition

The present study accentuates the heat transfer characteristics of a convective condition of micropolar nanofluid on a permeable shrinking/stretching inclined surface. Brownian and thermophoresis effects are also involved to incorporate energy and concentration equations. Moreover, linear similarity transformation has been used to transform the system of governing partial differential equations (PDEs) into a set of nonlinear ordinary differential equations (ODEs). The numerical comparison has been done with the previously published results and found in good agreement graphically and tabular form by using the shooting method in MAPLE software. Dual solutions have been found in the specific range of shrinking/stretching surface parameters and the mass suction parameter for the opposing flow case. Moreover, the skin friction coefficient, the heat transfer coefficient, the couple stress coefficient, and the concentration transfer rate decelerate in both solutions against the mass suction parameter for the augmentation of the micropolar parameter respectively. The first (second) solution is the stable (unstable) solution and can (not) be considered as a real solution as the values of the smallest eigenvalues are positive (negative)