Conjugate Effects of Radiation and Joule Heating on Magnetohydrodynamic Free Convection Flow along a Sphere with Heat Generation ()
1. Introduction
The conjugate effects of radiation and joule heating on magnetohydrodynamic (MHD) free convection boundary layer on various geometrical shapes such as vertical flat plate, cylinder, sphere etc, have been studied by many investigators and it has been a very popular research topic for many years. Nazar et al. [1], Huang and Chen [2] considered the free convection boundary layer on an isothermal sphere and on an isothermal horizontal circular cylinder both in a micropolar fluid. Molla et al. [3] have studied the problem of natural convection flow along a vertical wavy surface with uniform surface temperature in presence of heat generation or absorption. Miraj et al. [4] studied the effect of radiation on natural convection flow on a sphere in presence of heat generation. Amin [5] also analyzed the influences of both first and second order resistance, due to the solid matrix of non-darcy porous medium, Joule heating and viscous dissipation on forced convection flow from a horizontal circular cylinder under the action of transverse magnetic field. Hossain [6] studied viscous and joule heating effects on magnetohydrodynamic (MHD) free convection flow with variable plate temperature. Alam et al. [7] studied the viscous dissipation effects with MHD natural convection flow on a sphere in presence of heat generation.
In the present work, the effects of joule heating with radiation heat loss on natural convection flow around a sphere have been investigated. The governing partial differential equations are reduced to locally non-similar partial differential forms by adopting appropriate transformations. The transformed boundary layer equations are solved numerically using implicit finite difference method with Keller box scheme described by Keller [8] and later by Cebeci and Bradshaw [9]. The results have been shown in terms of the velocity, temperature profiles, the skin friction and surface heat transfer
2. Formulation of the Problem
A steady two-dimensional magnetohydrodynamic (MHD) natural convection boundary layer flow from an isothermal sphere of radius a, which is immersed in a viscous and incompressible optically dense fluid with heat generation and radiation heat loss is considered. The physical configuration considered is as shown in Figure 1.
Under the above assumptions, the governing equations for steady two-dimensional laminar boundary layer flow problem under consideration can be written as
(1)
(2)
(3)
With the boundary conditions
(4)
The above equations are further non-dimensionalised using the new variables:
(5)
(6)
(7)
The radiation heat flux is in the following form
(8)
Figure 1. Physical model and coordinate system.
Substituting (5), (6) and (7) into Equations (1), (2) and (3) leads to the following non-dimensional equations
(9)
(10)
(11)
where 
is the Prandtl number, 
is the heat generation parameter, 
is the radiation parameter and 
is the joule heating parameter.
With the boundary conditions (4) become
(12)
To solve Equations (10) and (11) with the help of following variables
(13)
where y is the stream function defined by
(14)
Equation (10) can be written as
(15)
where 
is the MHD parameter.
Equation (11) becomes
(16)
Along with boundary conditions
(17)
It can be seen that near the lower stagnation point of the sphere, i.e., ξ » 0, Equations (15) and (16) reduce to the following ordinary differential equations:
(18)
(19)
Subject to the boundary conditions
(20)
In practical applications, skin-friction coefficient Cf and Nusselt number Nu can be written in non-dimensional form as
and 
(21)
where, 
and 
Putting the above values in Equation (21), we have
(22)
3. Results and Discussion
Solutions are obtained for some test values of Prandtl number Pr = 2.00, 5.00, 7.00, 9.00; radiation parameter Rd =1.00, 2.00, 3.00, 4.00, 5.00; heat generation parameter Q = 0.00, 0.05, 0.10, 0.15, 0.20; magnetic parameter M = 0.50, 1.00, 1.50, 2.00, 3.00 and joule heating parameter J = 0.10, 0.50, 1.00, 1.50, 2.00 in terms of velocity and temperature profiles, skin friction coefficient and heat transfer coefficient. The effects for different values of radiation parameter Rd the velocity and temperature profiles in case of Prandtl number Pr = 0.72, heat generation parameter Q = 0.10, magnetic parameter M = 2.00 and joule heating parameter J = 0.50 are shown in Figures 2(a) and 2(b). In Figures 3(a) and 3(b) are shown that when the Prandtl number Pr increases with radiation parameter Rd = 1.00, heat generation parameter Q = 0.10, magnetic parameter M = 2.00 and joule heating parameter J = 0.50, both the velocity and temperature profiles are decrease. For different values of heat generation parameter Q with radiation parameter Rd = 1.00, Prandtl number Pr = 0.72, magnetic parameter M = 2.00 and joule heating parameter J = 0.50 and display results in Figures 4(a,b) that as the heat generation parameter Q increases, the velocity and the temperature profiles increase.
It has been seen from Figure 5(a) that as the magnetic parameter M increases, the velocity profiles decrease up to the position of h = 4.10555 after that position velocity profiles increase with the increase of magnetic parameter. We see that in Figure 5(b) temperature profiles increase for increasing values of magnetic parameter M with radiation parameter Rd = 1.00, Prandtl number Pr = 0.72, heat generation parameter Q = 0.10 and joule heating parameter J = 0.50. It has been seen from Figures 6(a,b) that as the joule heating parameter J increases, the velocity and the temperature profiles increase. In Figures 7(a,b) shown that the radiation parameter Rd increases, both the skin friction coefficient and heat transfer coefficient increase. The variation of the local skin friction coefficient Cf and local rate of heat transfer Nu for different values of Prandtl number Pr while radiation parameter Rd = 1.00, heat generation parameter Q = 0.10, magnetic parameter M = 2.00 and joule heating parameter J = 0.50 are shown in Figures 8(a) and 8(b). From
(a)
(b)
Figure 2. (a) Velocity and (b) Temperature profiles for different values of Rd when Pr = 0.72, Q = 0.1, M = 2.0 and J = 0.5.
(a)
(b)
Figure 3. (a) Velocity and (b) Temperature profiles for different values of Pr when Rd = 1.0, Q = 0.1, M = 2.0 and J = 0.5.
(a)
(b)
Figure 4. (a) Velocity and (b) Temperature profiles for different values of Q when Rd = 1.0, Pr = 0.72, M = 2.0 and J = 0.5.
(a)
(b)
Figure 5. (a) Velocity and (b) Temperature profiles for different values of M when Rd = 1.0, Pr = 0.72, Q = 0.1 and J = 0.5.
(a)
(b)
Figure 6. (a) Velocity and (b) Temperature profiles for different values of J when Rd = 1.0, Pr = 0.72, Q = 0.1 and M = 2.0.
(a)
(b)
Figure 7. (a) Skin friction coefficient and (b) Rate of heat transfer for different values of Rd when Pr = 0.72, Q = 0.1, M = 2.0 and J = 0.5.
(a)
(b)
Figure 8. (a) Skin friction coefficient and (b) Rate of heat transfer for different values of Pr when Rd = 1.0, Q = 0.1, M = 2.0 and J = 0.5.
Figure 9(a) we observed that the skin friction coefficient Cf increases significantly as the heat generation parameter Q increases and Figure 9(b) show that heat transfer coefficient Nu decreases for increasing values of heat generation parameter Q with relevant parameters. It reveals that the rate of heat transfer decreases along the ξ direction from lower stagnation point to the downstream. Figures 10(a,b) shown that skin friction coefficient Cf and heat transfer coefficient Nu decrease for increasing values of magnetic parameter M while radiation parameter Rd = 1.00, Prandtl number Pr = 0.72,heat generation parameter Q = 0.10, and joule heating parameter J = 0.50. From Figure 11(a) we observed that the skin friction coefficient Cf increases and Figure 11(b) shown that heat transfer coefficient Nu decreases for increasing values of joule heating parameter J.