1.1. Problem Definition
The physical model along with the important geometrical parameters and the mesh of the enclosure used in the present CVFEM program are shown in Figure 1 (Sheikholeslami, 2014). The inner circular wall is maintained at constant hot temperature 
, the two horizontal walls are thermally isolated and other walls are maintained at constant cold temperature 
(
>
). To reach the shape of inner circular and outer rectangular boundary which consists of the right and top walls, a supper elliptic function can be used as follows;
(1)When 
and 
the geometry becomes a circle. As 
increases from 1 the geometry would approach a rectangle for 
and square for 
. In this study we assumed that 
.
For the expression of the magnetic field strength it can be considered that two magnetic sources at points 
and 
. The components of the magnetic field intensity (
) and the magnetic field strength (
) can be considered as (Sheikholeslami and Rashidi, 2015):
(2)(3)(4) where
the magnetic field strength at the source (of the wire). The contours of the magnetic field strength are shown in Figure 2. In this study magnetic source is located at
and
.
Figure 1. (a) Geometry and the boundary conditions; (b) the mesh of enclosure considered in this work.
The flow is two-dimensional, laminar and incompressible. The magnetic Reynolds number is assumed to be small so that the induced magnetic field can be neglected compared to the applied magnetic field. Using the Boussinesq approximation, the governing equations of heat transfer and fluid flow for nanofluid can be obtained as follows:
(5)(6)(7)(8)The terms 
and 
in (6) and (7), respectively, represent the components of magnetic force, per unit volume, and depend on the existence of the magnetic gradient on the corresponding x and y directions. These two terms are well known from FHD which is the so-called the Kelvin force. The term s
and 
appearing in (6) and (7), respectively, represent the Lorentz force per unit volume towards the x and y directions and arise due to the electrical conductivity of the fluid. These two terms are known in MHD. The principles of MHD and FHD are combined in the mathematical model presented in Sheikholeslami and Ganji (2014) and the above mentioned terms arise together in the governing equations (6) and (7). The term 
in Equation (8) represents the thermal power per unit volume due to the magneto caloric effect. Also the term 
in (8) represents the Joule heating. For the variation of the magnetization 
, with the magnetic field intensity 
and temperature 
, the following relation is used Loukopoulos and Tzirtzilakis (2004):
where
is a constant and
is the Curie temperature.
In the above equations, 
is the magnetic permeability of vacuum 
, 
is the magnetic field strength, 
is the magnetic induction 
and the bar above the quantities denotes that they are dimensional. The effective density (
) and heat capacitance
of the nanofluid are defined as:
(10)(11) where
is the solid volume fraction of nanoparticles. Thermal diffusivity of the nanofluid is
(12) and the thermal expansion coefficient of the nanofluid can be determined as
(13)The dynamic viscosity of the nanofluids is given is
(14)The effective thermal conductivity of the nanofluid can be approximated by the Maxwell–Garnetts (MG) model as (Sheikholeslami, 2014):
(15) and the effective electrical conductivity of nanofluid was presented as below:
(16)By introducing the following non-dimensional variables:
(17) where in
Equation (17)
and
. Using the dimensionless parameters, the equations now become:
(18)(19)(20)(21) where
and
are the Rayleigh number, Prandtl number, Hartmann number arising from MHD, temperature number, curie temperature number, Eckert number and Magnetic number arising from FHD the for the base fluid, respectively. The thermo physical properties of the nanofluid are given in Table 1 (Sheikholeslami and Ganji, 2014).
Table 1. Thermo physical properties of water and nanoparticles
|  |  |  |  |  |
| Pure water | 997.1 | 4179 | 0.613 | 21 | 0.05 |
 | 5200 | 670 | 6 | 1.3 | 25000 |
(Sheikholeslami and Ganji, 2014)
The stream function and vorticity are defined as:
(22)The stream function satisfies the continuity Equation (18). The vorticity equation is obtained by eliminating the pressure between the two momentum equations, i.e. by taking y-derivative of Equation (20) and subtracting from it the x-derivative of Equation (19).
The boundary conditions as shown in Figure 1 are:
on the inner circular boundary
on the outer wall boundary
on all solid boundaries
(23)The values of vorticity on the boundary of the enclosure can be obtained using the stream function formulation and the known velocity conditions during the iterative solution procedure.
The local Nusselt number of the nanofluid along the hot wall can be expressed as:
(24) where
r is the radial direction. The average Nusselt number on the hot circular wall is evaluated as:
(25)To estimate the enhancement of heat transfer between the case of 
and the pure fluid (base fluid) case, the heat transfer enhancement is defined as:
(26)