Magnetohydrodynamic MHD flow and heat transfer for

Magnetohydrodynamic (MHD) flow and heat transfer for a fluid has enormous applications in many engineering problems such as MHD power generators, petroleum industries, plasma studies, geothermal ML355 extractions, the boundary layer control in the field of aerodynamics and many other applications [7–27]. The MHD flow along a stretching cylinder has been considered by the following authors [28–31]. The non-adherence of the fluid to a solid boundary, known as velocity slip, is a phenomenon that has been observed under certain circumstances. When fluid is encountered in microelectromechanical systems, the no-slip condition at the solid-fluid interface is abandoned in favor of a slip flow model which represents more accurately the non-equilibrium region near the interface. In all of the above-mentioned investigations, the no-slip condition at the boundary was assumed. Even in literature, the slip flow over a flat plate has not been studied sufficiently. Zheng et al. [32] investigated the flow and radiation heat transfer of a nanofluid over a stretching sheet with velocity slip and temperature jump in porous medium. Mukhopadhyay [33] analyzed the slip effects on MHD flow along a horizontal stretching cylinder. Very recently, Abbas et al. [34] studied the slip effects on the flow over an unsteady stretching/shrinking cylinder in the presence of suction. Heat transfer characteristics of the stretching sheet problem have been restricted to two boundary conditions of either prescribed temperatures or heat flux on the wall in the published papers. Most recently, heat transfer problems for boundary layer flow concerning a convective boundary condition was investigated by Aziz for the Blasius flow [35]. Following Aziz, many researchers investigated the boundary layer flows with convective boundary condition [36–40].
Equations of motion Consider the steady axi-symmetric slip flow of an incompressible fluid along a vertical stretching cylinder in the presence of a uniform magnetic field. The x-axis is measured along the axis of the tube and the r-axis is measured in the radial direction. It is assumed that the uniform magnetic field of intensity acts in the radial direction. The magnetic Reynolds number is assumed to be small so that the induced magnetic field is negligible in comparison with the applied magnetic field. It is also assumed that the left side of the cylinder is heated by convection from a hot fluid at temperature which provides a heat transfer coefficient and T is the ambient fluid temperature (see Fig. 1). The continuity, momentum and energy equations governing such type of flow [33] are as follows: Where u and v are the components of velocity respectively in x and r directions, is the kinematic viscosity, ρ is the fluid density, μ is the dynamic viscosity of the fluid, g is the acceleration due to gravity, β is the volumetric coefficient of thermal expansion, σ is the electric conductivity of the medium, is the uniform magnetic field, α is the thermal diffusivity of the fluid and T is the fluid temperature.
Results and discussion For the verification of the present numerical results, we have compared F ″(0) values with those of Kandelousi [11] in the absence of slip and mixed convection parameters which are presented in Table 1. The comparison results are found to be good. In order to analyze the results, numerical computations are carried out for various values of curvature parameter C, mixed convection parameter λ, magnetic parameter M, Prandtl number Pr and surface convection parameter γ. For illustrations of the results, numerical values are plotted in Figs. 2–8. The values of the skin friction coefficient and the Nusselt number are calculated and presented in Table 2. The effects of C and A on the dimensionless horizontal velocity profile are illustrated in Fig. 2. It is clear that the curvature parameter decreases the velocity profile near the wall and increases it far away from the wall. When a slip occurs, the flow velocity near the stretching wall is no longer equal to the stretching velocity of the wall. With the increase in A such slip velocity increases and consequently the fluid velocity decreases because under the slip condition, the pulling of the stretching wall can only be partly transmitted to the fluid. Thus the increasing values of the slip parameter decelerate the dimensionless velocity profile.