NUMERICAL SIMULATION OF NATURAL CONVECTION IN A POROUS CAVITY FILLED WITH FERROFLUID IN PRESENCE OF MAGNETIC SOURCE

In this study, numerical simulation of natural convection in a porous square cavity filled with Fe3O4-water is investigated. A magnetic source through the left wall of the cavity is also taken into account. Radial basis function based pseudo spectral (RBF-PS) method is applied. The effects of dimensionless parameters Darcy (Da), Hartmann (Ha), Rayleigh (Ra) numbers and solid volume fraction φ are presented both in terms of streamlines, isotherms and vorticity contours and average Nusselt number through the heated wall. Convective heat transfer is inhibited with the rise of Ha, and with the decrease in Da while it is enhanced with the increase in φ and Ra.

(FVM) with SIMPLE algorithm. Their results showed that secondary vortices are formed in case of polarization. Aminfar; Mohammadpourfard and Ahangar Zonouzi [12] utilized the control volume technique with SIMPLEC for simulation of 3D laminar ferrofluid in presence of an electric current through a wire at the bottom of the duct resulting with transverse nonuniform magnetic field. Heat transfer enhancement is obtained in simulation of a water based ferrofluid in a mini channel using FVM in the study of Ghasemian; Ashrafi; Goharkhah and Ashjee [13]. Lattice Boltzmann method (LBM) is performed by Kefayati [14] to analyze the influence of heat dissipation and an external magnetic source on natural convection flow in a cavity filled with kerosene based cobalt. The increase in solid volume fraction of nanoparticles causes heat transfer to decrease at a large Ra = 10 5 . Kefayati [15] used the same method to simulate the same problem in an inclined cavity. The effect of inclination angle on heat transfer is reduced with the increase in solid volume fraction. In studies [16]- [28], control volume based finite element method (CVFEM) is carried out for simulation. In the paper of Sheikholeslami and Ganji [16], governing equations based on ferrohydrodynamic (FHD) and magnetohydrodynamic (MHD) are solved in a semi annulus enclosure with sinusoidal hot wall with a magnetic source. At a low Rayleigh number, the effect of Kelvin force is pronounced. Natural convective heat transfer in a cavity with a circular step in which uniform heat flux is maintained is simulated under the effect of external variable magnetic field by Sheikholeslami [17]. In this study, MHD and FHD terms are added to governing equations. Sheikholeslami; Rashidi and Ganji [18] solve two phase model considering Brownian and thermopheresis in a lid-driven semi annulus under the effect of a magnetic source. The increase in Hartmann and Lewis number causes the Nusselt number to decrease while it increases with Reynolds number. The influence of magnetic field dependent viscosity is examined by Sheikholeslami and Rashidi [19] in a lid-driven semi annulus. Thermal radiation and viscosity of Fe3O4 as a function of magnetic field are taken into account by Sheikholeslami and Shehzad [20] in which an elliptic inner cylinder is settled into the enclosure. The results show that Nusselt number increases as inclination angle of the cylinder increases. Sheikholeslami; Ellahi and Vafai [21] showed that impact of adding Fe3O4 is more pronounced at a lower Rayleigh number. The radiation parameter is taken into account by Sheikholeslami [22] in which impact of a magnetic source in a square cavity including a circular hot cylinder is investigated. Results indicate that temperature gradient augments with the rise of radiation parameter Rd while temperature reduced with the augmentation of Rd. Sheikholeslami also studied [23] free convection in a lid-driven cavity filled with Fe3O4-water. In this study, Columb forces resulting from the electric field on the cavity are also considered. Results report that average Nusselt number enhances with the rise of supplied voltage and the temperature gradient through the hot wall increases as Columb force augments. Ferrofluid flow in a curved cavity with constant heat flux condition for inner wall considering Rd parameter is presented by Sheikholeslami and Shamlooei [24]. Nanofluid velocity increases as Rd increases, and Rd is much more effective on convective heat transfer at large Rayleigh numbers. Sheikholeslami [25] added the radiation source term in energy equation for simulating the ferrofluid free convection concerning viscosity of ferrofluid in a cavity with centered elliptic cylinder. Inner wall temperature increases as Rd increases. Sheikholeslami and Shehzad [26] studied MHD convection in ferrofluid flow and heat transfer under the influence of a radiation source term. Viscosity of ferrofluid as a function of magnetic field is also considered in this study. In a half circular shaped cavity, Sheikholeslami; Hayat and Alsaedi [27] investigated ferrofluid with Kelvin forces in presence of external magnetic field. Results demonstrate that temperature gradient increases if Kelvin forces increase. Sheikholeslami and Rokni [28] showed the improvement of heat transfer with the increase in Lorentz forces considering a circular hot cylinder inside a cavity filled with ferrofluid.
Nanofluids in porous medium are encountered in [29]- [43]. Using FVM with SIMPLE algorithm, Malik and Nayak [29] solved Cu-water nanofluid with Darcy-Brinkmann-Forchheimer model considering the magnetic field. They reported that the rise in Darcy number augments the heat transfer rate. Sheikholeslami [30] conducted the fourth order Runge Kutta integration scheme featuring a shooting technique to analyze the nanofluid flow and heat transfer in a stretching porous cylinder. In this study, KKL model is adopted. Results shows that average Nusselt number increases as the suction parameter increases, and skin friction coefficient has a reverse relation with solid volume fraction. Impact of external magnetic source on free convection of ferrofluid in a porous curved cavity with hot left wall having constant heat flux condition is examined by Sheikholeslami [31] utilizing CVFEM. Sheikholeslami [32] also investigated free convection of ferrofluid with an external magnetic source in a porous curved cavity. In low Darcy number, adding nanoparticles enhances the heat transfer. Non-Darcy model in a nanofluid filled with CuO nanoparticles under the effect of a uniform magnetic field is studied by Sheikholeslami and Ganji [33] considering radiation effect. The results showed that the heat transfer rate increase with the increase in radiation paramater. Sheikholeslami and Bhatti [34] studied Brownian motion impact on viscosity of nanofluid and different shapes of CuO nanoparticles in a porous semi annulus with a moving bottom wall in presence of an uniform magnetic field. They also declared that average Nusselt number increases with the rise in Darcy number. Sheikholeslami [35] simulated the natural convection in a porous media filled with CuO-water and a sinusoidal wall under the effect of a constant magnetic field using Darcy's law and KKL model. Sheikholeslami [36] carried out CVFEM to investigate free convection in a ferrofluid-filled porous curved cavity in existence of magnetic source. In lower value of Darcy number, adding nanoparticles is more meaningful. Sheikholeslami [37] also applied LBM to examine the effect of magnetic field in an open porous cavity filled with CuO-water nanofluid. LBM is also utilized by Sheikholeslami [38] in order to simulate 3D forced convection in a porous, lid-driven cubic cavity. Outputs demonstrate that temperature gradient over hot surface decreases as Hartmann number increases while it augments with the rise of Reynolds and Darcy numbers. In presence of constant magnetic field, natural convection flow in a porous cavity (in Darcy's law) involving centered square cylinder and CuO-water nanofluid (with KKL model) is investigated by Sheikholeslami [39]. Average Nusselt number increases as the size of the inner square cylinder increases. CuO-water nanofluid (in different shapes of nanoparticles) forced convection in a porous semi-annulus under the effect of uniform magnetic field is reported by Sheikholeslami and Bhatti [40]. Results show that platelet shape has the highest rate of heat transfer. The same nanofluid in a porous semi annulus with sinusoidal hot wall and with constant heat flux is examined by Sheikholeslami and Shehzad [41] using KKL and Darcy's models. Adding nanoparticles are more efficient in case of high Hartmann numbers. In an elliptic porous enclosure including a rectangular hot cylinder, CuO-water nanofluid natural convection is presented by Sheikholeslami and Zeeshan [42]. Heat transfer enhancement increases with the increase in rotation angle of the cylinder. 3D nanofluid non-Darcy natural convection with Lorentz forces is reported utilizing LBM by Sheikholeslami [43]. In results, convection dominates if Darcy number, Rayleigh number and solid volume fraction increase, and is reduced with the rise of Hartmann number.
In a technical note prepared by Oztop; Selimefendigil; Abu-Nada and Al-Salem [44], computational methods used in case of curvilinear boundaries are reviewed considering the natural convection in nanofluids. In view of an experimental study by Roy; Asirvatham; Kunhappan; Cephas and Wongsises [45], the efficiency of low-particle concentrated silver-water nanofluid is investigated in a solar flat-plate collector.
In the present study, natural convection flow in a porous Fe3O4-water-filled cavity which has a magnetic source is investigated. The Brinkman-extended Darcy model is simulated utilizing the radial basis function based pseudo spectral (RBF-PS) method with multi quadric (MQ) RBFs. The effects of physical non-dimensional parameters are analyzed. To the author's knowledge, this method is firstly applied to this problem.

PHYSICAL AND MATHEMATICAL SETUP
Two-dimensional, unsteady, laminar, incompressible flow in a porous square cavity filled with ferrofluid is taken into account. The constant thermophysical properties of the fluid are considered except the density variation treated by Boussinessq approximation. The porous medium is homogeneous, isotropic and the fluid and solid of porous medium are in thermal equilibrium. The radiation, induced electric current and displacement currents are neglected.
The problem configuration is depicted in Figure 1.  Thermophysical properties of water and magnetite Fe3O4 are given in Table 1.  The magnetic field intensity and its components are defined as where ( , ) is the location of the magnetic source and is the strength of the magnetic field at the source.
In order to obtain dimensionless governing equations, the non-dimensional parameters are defined as in which the dimensionless parameters Prandtl, Hartmann, Darcy, Rayleigh and Eckert numbers, respectively, are The velocity components in terms of stream function ( = / , = − / ), provide automatically satisfaction of continuity equation, and pressure terms are eliminated applying the definition of vorticity = ∇ × to the momentum equations. Then, in addition to the dimensionless energy equation Eq. (19) stream function and vorticity equations are deduced as

NUMERICAL APPROACH
Pseudo spectral (PS) methods are global and high accurate methods. Radial basis functions based pseudo spectral (RBF-PS) method is based on the approximation of all derivatives in the problem by RBFs. Radial basis functions enable one to use any type of grids, and are attracted by most of the researchers in mesh-free methods. The novel books by Fasshauer [48] and Fasshauer and McCourt [49] involve both theory and applications on RBFs.
An iterative system with the application of the method to the system is built as follows where is the number of iteration, 1 , 2 , 3 , 4 , 5 , 6 are constants seen in Eqs. (19), (21), (22), Ι is the identity matrix of size ( + ) × ( + ) and matrices Initially, stream function, temperature and vorticity are taken as zero everywhere in the computational domain except on the boundary. After solving Eq. (27), the velocity components = +1 and = +1 are computed by = +1 , = − +1 . Then, the boundary conditions for and are inserted. +1 and +1 are used in calculation of matrix in Eqs. (28) and (29). Once Eq.(29) is solved, the unknown vorticity boundary conditions are derived by using the definition of vorticity as = +1 − +1 , and a relaxation parameter The iterations continue until the criterion is satisfied in which 1 = , 2 = , 3 = .
The average Nusselt number through the left hot wall is computed by

NUMERICAL RESULTS
The multiquadric radial basis function = √ 2 + 2 is used. As a difference from the study of Geridonmez Pekmen [10], in computation of ( 2 2 + 2 2 ) in diffusion terms, a perturbed 2 as √ 2 + 2 , in which 0 < = < 1, is tried. Because of this trial, the determination of shape parameter is also done differently.
In this study, the shape parameter is determined by taking the average value of minimum of absolute value of coordinate distances (different than zero) from interior nodes to all nodes. This determination is done with regard to the node distribution given in Figure 2. In this distribution, boundary nodes are Gauss-Chebyshev-Lobatto nodes (from 1 to in which = 4 − 4 for unit square), and interior nodes are the Chebyshev nodes between 0.5( 2 + 3 ) and 0.5( −2 + −1 ).
The base fluid is taken as pure water with Prandtl number 6.8, and Eckert number and the time increment are fixed at = 10 −5 , ∆ = 0.1, respectively. The other dimensionless parameters are in the following ranges: 10 −4 ≤ ≤ 10, 10 ≤ ≤ 50, 10 3 ≤ ≤ 10 5 , 0.04 ≤ ≤ 0.2. Since the inertial effects are not taken into account in this study, these ranges are consistent with the physical aspect as used in the study of Ramakrishna; Basak; Roy and Pop [50]. The validation of the method is shown in Table 2 comparing the average Nusselt number through the heated left wall. The results are in good agreement with the benchmark problem. One of the reasoning in the discrepancies in the results is to use small number of grid points, and the other one may be using the perturbation idea only in diffusion terms.  Table 3. = 80, = 529 for = 10 4 and = 120, = 1089 for = 10 5 are enough to visualize the problem. As noted, the values of are diminished by the rise in the number of nodes.  Figure 3 depicts the heat transfer and flow behavior in different values. Primary vortex in streamlines is diminished pointing to the decrease in the fluid velocity. Also, the intense magnetic field pushes the flow from the left to the right. Isotherms almost become perpendicular to the top and bottom wall in which Lorentz force inhibits the heat transfer. In vorticity contours, the effect from the left wall via magnetic source is noticed as well as the combination of two vorticity vortexes in the center at a large . The influence of Darcy number on heat transfer and fluid flow is described in Figure 4. As is seen, the decrease in porous permeability, a resistance to fluid flow is formed, and the effect of magnetic source on streamlines almost disappears. Isotherms show that the buoyancy-driven convection is pronounced at a large . Vorticity becomes stagnant at the center at a small due to the slow movement of fluid. As expected in Figure 5, the increase in Rayleigh number emphasizes the natural convection, or convective heat transfer in other words. Strong temperature gradient and two elongated vortices in vorticity contours through the left and right walls are noted. The effect of magnetic source on streamlines is realized at a small while it is reduced at = 10 5 . The location of magnetic source is investigated in Figure 6. If the magnetic source is moved closer to the left wall when = 0.5 is fixed, streamlines and vorticity contours form a small peak on the left wall pointing to a small decrease in fluid flow. = 1.05 is a reverse case of = −0.05 on the right wall. If the magnetic source is moved from bottom to top when = −0.05 is fixed, the decrease in the convective heat transfer is exhibited on isotherms while there is no source effect in = 0.05. The primary vortex in streamlines is also decreased and moves a little bit to the right, and vorticity contours are unified and approached to the right when = 0.75.  Table 4. In each cases, convective heat transfer increases with the increase in concentration of nanoparticles. The augmentation in conductive heat transfer is investigated in small and large . Also, heat transfer is decelerated if the magnetic source is moved from left to right or from bottom to top.

CONCLUSION
In this paper, RBF-PS method is carried out for simulation of natural convection in a porous cavity filled with ferrofluid and in presence of a magnetic source. The main goal of the current study is to examine the effect of Darcy number and the magnetic source. The convective heat transfer is declined with the decrease in Darcy number, and with the location of magnetic source close to the left wall or the top wall. The higher values of Hartmann number suppresses the fluid flow and heat transfer due to the retarding effect of Lorentz force while the large Rayleigh number and solid volume fraction value augment the convective heat transfer. Average Nusselt number augments with the rise of the concentration of the nanoparticles. RBF-PS method enable one to use small number of grid points. Also, the idea of the perturbed 2 can also be extended to all matrices formed by in the iterative systems.