1.1. Problem Definition
The considered physical geometry with related parameters and coordinates are shown in Figure 1(a). A rectangular body with height t and width 
is placed in the center of the enclosure, is supposed to be isothermal at higher temperature than two vertical isothermal walls while the top and bottom walls are insulated. Also, it is also assumed that the uniform magnetic field (
) of constant magnitude 
is applied, where 
and 
are unit vectors in the Cartesian coordinate system. The orientation of the magnetic field form an angle 
with horizontal axis such that 
. The electric current J and the electromagnetic force F are defined by 
and 
, respectively. In this section, 
is equal to zero (Sheikholeslami & Ganji, 2015).
Figure 1. (a) Geometry of the problem; (b) Discrete velocity set of two-dimensional nine-velocity (D2Q9) model
One of the novel computational fluid dynamics (CFD) methods which is solved Boltzmann equation to simulate the flow instead of solving the Navier–Stokes equations is called Lattice Boltzmann methods (LBM) (or Thermal Lattice Boltzmann methods (TLBM)). LBM has several advantages such as simple calculation procedure and efficient implementation for parallel computation, over other conventional CFD methods, because of its particulate nature and local dynamics. The thermal LB model utilizes two distribution functions, f and g, for the flow and temperature fields, respectively. It uses modeling of movement of fluid particles to capture macroscopic fluid quantities such as velocity, pressure and temperature. In this approach, the fluid domain discretized to uniform Cartesian cells. Each cell holds a fixed number of distribution functions, which represent the number of fluid particles moving in these discrete directions. The D2Q9 model was used and values of 
for 
(for the static particle), 
for 
and 
for 
are assigned in this model (Figure 1(b)). The density and distribution functions i.e. the f and g, are calculated by solving the lattice Boltzmann equation, which is a special discretization of the kinetic Boltzmann equation. After introducing BGK approximation, the general form of lattice Boltzmann equation with external force is as follow:
For the flow field:
(1)For the temperature field:
(2) where
denotes lattice time step,
is the discrete lattice velocity in direction
,
is the external force in direction of lattice velocity,
and
denotes the lattice relaxation time for the flow and temperature fields. The kinetic viscosity
and the thermal diffusivity
, are defined in terms of their respective relaxation times, i.e.
and
, respectively. Note that the limitation
should be satisfied for both relaxation times to ensure that viscosity and thermal diffusivity are positive. Furthermore, the local equilibrium distribution function determines the type of problem that needs to be solved. It also models the equilibrium distribution functions, which are calculated with Eqs. (3) and (4) for flow and temperature fields, respectively.
(3)(4) where
is a weighting factor and
is the lattice fluid density.
In order to incorporate buoyancy forces and magnetic forces in the model, the force term in the Equation (2) need to calculate as below (Sheikholeslami & Ganji, 2015):
(5) where
A is
,
is Hartmann number and
is the direction of the magnetic field.
For natural convection, the Boussinesq approximation is applied and radiation heat transfer is negligible. To ensure that the code works in near incompressible regime, the characteristic velocity of the flow for natural 
regime must be small compared with the fluid speed of sound. In the present study, the characteristic velocity selected as 0.1 of sound speed.
Finally, macroscopic variables calculate with the following formula:
(6)According to Bejan (1982), one can find the volumetric entropy generation rate as
(7) where
HTI is the irreversibility due to heat transfer in the direction of finite temperature gradients and
FFI is the contribution of fluid friction irreversibility to the total generated entropy.
In terms of the primitive variables, HTI and FFI become
(8)One can also define the Bejan number,
, as
(9)Note that a 
value more/less than 0.5 shows that the contribution of HTI to the total entropy generation is higher/lower than that of FFI. The limiting value of 
shows that the only active entropy generation mechanism is HTI while 
represents no HTI contribution.
The dimensionless form of entropy generation rate, Ns, is defined as
(10) one finds that
(11) where the dimensionless temperature difference is defined as
(12)The dimensionless viscous dissipation function, addressed in Equation (11), takes the following form
(13)Here, Ge is the Gebhart number which is defined as (Sheikholeslami, Ashorynejad, & Rana, 2016).
(14)Average 
is denoted by
, where the angle brackets show an average taken over the area, as:
(15)Selecting the fluid, trapped between the heated plate and the cavity, as the thermodynamic system, one observes that the amount of heat entered through the heated plate is equal to the one transferred to the surroundings via the isothermal walls. Moreover, one notes that the total volumetric entropy generation rate is obtainable as
(16) where, in terms of
, it reads
(17)Applying perturbation techniques for small values of 
, say 
, one has
(18)The dimensionless entropy generation number can be obtained as
(19)In order to simulate the nanofluid by the lattice Boltzmann method, because of the interparticle potentials and other forces on the nanoparticles, the nanofluid behaves differently from the pure liquid from the mesoscopic point of view and is of higher efficiency in energy transport as well as better stabilization than the common solid-liquid mixture. For modeling the nanofluid because of changing in the fluid thermal conductivity, density, heat capacitance and thermal expansion, some of the governed equations should change. The effective density 
, the effective heat capacity 
, thermal expansion 
and electrical conductivity 
of the nanofluid are defined as:
(20)(21)(22)(23) where
is the solid volume fraction of the nanoparticles and subscripts
and
s stand for base fluid, nanofluid and solid, respectively.
and 
are obtained according to Koo–Kleinstreuer–Li (KKL) model (Sheikholeslami & Kandelousi, 2014):
(24)(25)In order to compare total heat transfer rate, Nusselt number is used. The average Nusselt numbers are defined as follows:
(26)