Dual similarity solutions of MHD stagnation point flow of Casson fluid with effect of thermal radiation and viscous dissipation: stability analysis.

In this paper, the rate of heat transfer of the steady MHD stagnation point flow of Casson fluid on the shrinking/stretching surface has been investigated with the effect of thermal radiation and viscous dissipation. The governing partial differential equations are first transformed into the ordinary (similarity) differential equations. The obtained system of equations is converted from boundary value problems (BVPs) to initial value problems (IVPs) with the help of the shooting method which then solved by the RK method with help of maple software. Furthermore, the three-stage Labatto III-A method is applied to perform stability analysis with the help of a bvp4c solver in MATLAB. Current outcomes contradict numerically with published results and found inastounding agreements. The results reveal that there exist dual solutions in both shrinking and stretching surfaces. Furthermore, the temperature increases when thermal radiation, Eckert number, and magnetic number are increased. Signs of the smallest eigenvalue reveal that only the first solution is stable and can be realizable physically.

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 ( δ ) 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.

Quadruple solutions of mixed convection flow of magnetohydrodynamic nanofluid over exponentially vertical shrinking and stretching surfaces: Stability analysis.

BACKGROUND AND OBJECTIVE: The last two and half decades are witnessed a great surge in the use convective fluids for enhancement of heat transfer of minerals ethylene glycol, oil and water due to their numerous applications in the industrial segments including chemical production, microelectronics, power generation, transportation, and air-conditioning. For this purpose, different procedures were applied to upgrade the thermal conductivity of common fluid but could not. Further, Choi and Eastman in 1995 introduced nanofluid which has good thermal properties as compared to common fluids. After that, it can be seen that researchers, mathematicians, and scientists tried to understand the principles of nanofluids and how to implicate them in many different practical applications. In this work, the Buongiorno model has been considered for nanofluid. One of the prime objectives is to consider all possible multiple solutions of the model because these solutions cannot be seen experimentally. METHODS: The governing equations of fluid flow have been transformed in the form of ordinary differential equations. These equations have been solved by two methods namely, shooting method and three-stage Lobatto IIIa formula. RESULTS: The effects of different parameters on temperature, velocity, concentration profiles, skin friction coefficient, Sherwood number, and reduced Nusselt number were obtained and presented graphically. It was noticed that four solutions existed at definite ranges of the parameters for high suction over both surfaces for the first time. The results of the stability analysis revealed that only the first solution is more stable and possess physical reliability compared to the remaining solutions. CONCLUSION: The graphs also indicated that the fluid velocity decreases as the thermophoresis parameter increases but the opposite behavior observed for both temperature and concentration profiles in the first solution. Furthermore, it was detected that the concentration profile declined at the higher values of the Brownian motion parameter.

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.