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The conjugate effects of radiation and joule heating on magnetohydrodynamic (MHD) free convection flow along a sphere with heat generation have been investigated in this paper. The governing equations are transformed into dimensionless non-similar equations by using set of suitable transformations and solved numerically by the finite difference method along with Newton’s linearization approximation. Attention has been focused on the evaluation of shear stress in terms of local skin friction and rate of heat transfer in terms of local Nusselt number, velocity as well as temperature profiles. Numerical results have been shown graphically for some selected values of parameters set consisting of heat generation parameter Q, radiation parameter Rd, magnetic parameter M, joule heating parameter J and the Prandtl number Pr.

The conjugate effects of radiation and joule heating on magnetohydrodynamic (MHD) free convection flow along a sphere with heat generation have been investigated in this paper. The governing equations are transformed into dimensionless non-similar equations by using set of suitable transformations and solved numerically by the finite difference method along with Newton’s linearization approximation. Attention has been focused on the evaluation of shear stress in terms of local skin friction and rate of heat transfer in terms of local Nusselt number, velocity as well as temperature profiles. Numerical results have been shown graphically for some selected values of parameters set consisting of heat generation parameter Q, radiation parameter Rd, magnetic parameter M, joule heating parameter J and the Prandtl number Pr.

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. [

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 [

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

Under the above assumptions, the governing equations for steady two-dimensional laminar boundary layer flow problem under consideration can be written as

With the boundary conditions

The above equations are further non-dimensionalised using the new variables:

The radiation heat flux is in the following form

Substituting (5), (6) and (7) into Equations (1), (2) and (3) leads to the following non-dimensional equations

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

To solve Equations (10) and (11) with the help of following variables

where y is the stream function defined by

Equation (10) can be written as

where is the MHD parameter.

Equation (11) becomes

Along with boundary conditions

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:

Subject to the boundary conditions

In practical applications, skin-friction coefficient C_{f} 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

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 _{f } 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

_{f} increases significantly as the heat generation parameter Q increases and _{f} 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 _{f} increases and