Free convection between vertical concentric annuli with induced magnetic field when inner cylinder is electrically conducting

An analysis is made for the fully developed laminar free convective flow in an open ended vertical concentric annuli with constant heat flux and constant temperature on the inner and outer walls, in the presence of a radial magnetic field. The length of the cylinder is assumed to be infinite and radiation heat transfer from the hot surface is assumed to be negligible. The inner cylinder is taken to be magnetic conducting while the outer cylinder is non-conducting. Buoyancy effect is also considered along with Boussinesq approximation. The induced magnetic field is taken into account arising due to the motion of an electrically conducting fluid. The governing linear simultaneous ordinary differential equations are first obtained in the non dimensional form and solved analytically for the velocity, induced magnetic field, temperature field and then skin-friction and induced current density are obtained. The expressions for the fluid flux and induced current flux in non-dimensional form have been also obtained. The effects of governing physical parameters occurring in the model are shown on the graphs and tables.


INTRODUCTION
There has been widespread interest in the study of natural convection between two vertical concentric cylinders. In the recent years, the study of transport phenomenon involving the annular geometry has attracted many researchers due to its applicability not only in the field of engineering but also in the field of Geophysics. Some applications of it are the optimization of solidification processes of metals and metal alloys, the study of geothermal sources, the treatment of nuclear fluid debris, the control of underground spreading of chemical wastes and pollutants and the design of magnetohydrodynamic power generators. In 1961, Ramamoorthy [2] has obtained the steady flow of an incompressible fluid between two concentric rotating cylinders and compared the classical hydrodynamic velocity with the magnetohydrodynamic velocity in the presence of a radial magnetic field by neglecting the induced magnetic field. Further, an exact solution for the magnetohydrodynamic flow between two rotating coaxial-cylinders under a radial magnetic field has been obtained by Arora and Gupta [3]. Shaarawi and Sarhan [4] have studied laminar free convective flow in an open ended vertical concentric annuli with rotating inner walls. An investigation for the fully developed free convective flow in vertical annuli with two isothermal boundaries has been obtained by Joshi [5]. Singh et al. [6] have obtained the exact solutions for the fully developed natural convection in open ended vertical concentric annuli with mixed kind of thermal boundary conditions of an electrically conducting fluid under a radial magnetic field. Lee and Kuo [7] have discussed the fully developed, laminar flow in annuli ducts imposed by constant wall temperature at the boundaries by assuming the flow to be Newtonian.
In 2002, Mahmud and Fraser [8] have discussed irreversibility analysis of concentrically rotating annuli in terms of entropy generation in the fluid flow and heat transfer inside cylindrical annuli. An analytical solution through perturbation method and Fourier transform for the natural convection in concentric cylinders with a porous sleeve has been obtained by Leong and Lai [9]. Barletta et al. [10] have discussed the fully developed parallel flow in an annular region filled with a porous medium surrounding a straight cylindrical electric cable. Nada, et al. [11] have investigated heat transfer enhancement in horizontal annuli using nanofluids. Further, Shaija and Narasimhan [12] have investigated the coupled action of conduction, natural convection and surface radiation inside a horizontal annulus numerically.
Most of the above studies on hydromagnetic free convective phenomena involve the cases in which the induced magnetic field is neglected. However, in several physical situations the induced magnetic field has considerable impact and it will be necessary to include its effect in the fluid-dynamical equations. Singh and Singh [13] have investigated effect of induced magnetic field on natural convection in vertical concentric annuli. A study of the time dependent magnetohydrodynamic couette flow in a porous annulus has been made by Jha and Apere [14]. Kumar and Singh [15] have discussed effect of the induced magnetic field on free convection when the concentric cylinders heated/cooled asymmetrically.
In this paper the fully developed laminar free convective flow in open ended vertical concentric annuli has been considered in the presence of a radial magnetic field. The inner cylinder is taken to be magnetic conducting with constant heat flux on it while the outer cylinder is non-conducting and is at a constant temperature. The induced magnetic field is taken into account arising due to the motion of an electrically conducting fluid. The transport equations for the present situation are obtained in the form of simultaneous ordinary differential equations. These equations are first obtained in the non dimensional form and solved analytically for the velocity, induced magnetic field, temperature field, skin-friction and induced current density. The expressions for fluid flux and induced current flux in non-dimensional form have been obtained. The governing equations are also solved for the singular case   2 Ha  . The effects of governing physical parameters in the model are shown on the graphs and tables.

MATHEMATICAL ANALYSIS
We have considered here steady laminar fully developed free convective flow of an electrically conducting fluid in the vertical concentric annulus of infinite length. The z -axis is taken as the axis of the co-axial cylinders and r denotes the radial direction measured outward from the axis of the cylinder. The radius of the inner and outer cylinders is taken as a and b respectively. The applied magnetic field of the form 0 aB r   is taken directed radially outward.The physical model of the considered problem is shown in the figure 1. In the present physical situation, the inner cylinder is taken to be electrical conducting while the outer cylinder as non-conducting. The inner cylinder is at constant heat flux while the outer cylinder is at constant temperature. As the flow is fully developed and cylinder is of infinite length, the variables describing the flow formation depends only on the co-ordinate r and as a result the velocity and magnetic fields are given by The boundary conditions for the velocity, induced magnetic field and temperature field are In the above equations, u is the velocity of fluid, g is the acceleration due to gravity,  is the coefficient of viscosity, e  is the magnetic permeability, T  is the temperature of fluid, f T  is the ambient temperature,  is the kinematic viscosity of the fluid and  is the coefficient of volume expansion.
then the governing equations in non-dimensional form are obtained as follows: The corresponding boundary conditions for the velocity, induced magnetic field and temperature field in dimensionless form are obtained as Additional non-dimensional physical parameters appearing in the above equations are the Hartmann number The skin-friction in dimensionless form at outer surface of inner cylinder and the inner surface of outer cylinder are obtained from Eq. (11), and are given by The mass flux and the induced current flux of the fluid through the annuli are given by -

Ha 
The expressions (11) and (12) for the velocity and induced magnetic field contain the term due to which there is a singularity at

RESULTS AND DISCUSSION
The outcomes from the numerical computations of analytical solutions are illustrated via figures and tables in order to analyze the behaviour of the physical parameters on the transport processes resulting from natural convection between coaxial cylinders. Figures 2-4 show the velocity profiles for the different values of the parameter  . Fig. 2  Ha , the velocity of the fluid decreases.    also shows that the induced magnetic field decreases with increase in the value of Hartmann number. Comparing the graphs of the induced magnetic field profiles given in the Figs. 5, 6 and 7, we find that as the ratio of outer radius to the inner radius increases the induced magnetic field increases.
Figs. 8-10 show the behaviour of induced current density for different values of the parameters  and Ha .

 
The nature of induced current density is found to decrease with increasing the value of Ha , and it increases with increasing the ratio of outer and inner radii,  .
The numerical values of skin-friction at the outer surface of the inner cylinder and at the inner surface of the outer cylinder are given in table 1.