Numerical Approach of Coupling Vibration Magneto-convection In Nanofluid

The objective of our work is to visualize numerically the effect of coupling vibratory excitation and magnetic field on cooling an electronic component or a solar cell (originality of our study) in arid and semi-arid area. A square cavity of side H filled with Al2O3-water nanofluid where an electronic component is placed on the bottom horizontal wall is maintained at isothermal hot temperature Th. The top horizontal wall is maintained at a cold temperature Tc. The vertical walls are adiabatic. The equations describing the natural convection flow in the square cavity consist of mass conservation, momentum and energy. For the physical parameters of Al2O3water nanofluid, we use the Brinkman and Wasp model. Transport equations are solved numerically by finite element method. The results are obtained for Rayleigh number Ra= 105, Hartmann numbers between 0 and 100 and vibratory excitation inclination angle between 0° and 90°. The external magnetic field inclination angle varies between 0° and 90° and the Rayleigh number ratio between 0 and 50. Results are presented in the form of heat transfer flux ratio and maximum absolute value of stream function.


INTRODUCTION
The coupling vibration magneto-convection in nanofluid has importance in several areas such as the cooling of electronic systems, power generation, air conditioning, microelectronics.The topic of natural convection in a cavity is of importance because many engineering applications, heating and/or cooling takes place inside the enclosure [1].In the literature, a rich and variety of numerical results have been published on the phenomenon of natural convection in differentially heated shallow enclosures with various wall conditions [2][3][4][5][6][7][8][9][10].
The study of magnetic field effects has attracted attentions of engineers and science due to its wide industrial applications.The problem of heat transfer in the presence of a magnetic field attracts the attention of researchers for a long time.It is little work for nanofluids compared to fluids and liquid metals.Ece and Buyuk [11] illustrated the natural convection flow under a magnetic field in an inclined rectangular cavity for heated and cooled on the adjacent walls.Mahmud and Fraser [12] have investigated magneto-hydrodynamic natural convection flow and entropy generation in a square cavity.On the contrary Grosan et al. [13] has studied effects of magnetic field and internal heat generation on natural convection flow in rectangular cavity filled with porous medium.Pirmohammadi et al. [14] studied the effect of a magnetic field on buoyancy-driven convection in differentially heated square enclosure.They showed that the heat transfer mechanisms and the flow characteristics inside the enclosure depend strongly upon both the strength of the magnetic field as well as the Rayleigh number.Recently, H.R. Ashorynejad et al. [15] were investigated numerically the effect of static radial magnetic field on natural convection heat transfer in a horizontal cylindrical annulus enclosure filled with nanofluid using the Lattice Boltzmann method (LBM).They found that the overage Nusselt number is an increasing function of nanoparticles volume fraction and Rayleigh number, while it is a decreasing function of Hartmann number.Mahmoudi et al. [16] studied the MHD natural convection and entropy generation in a trapezoidal enclosure using Cu-water nanofluid.They found that at Ra =10 4 and 10 5 the enhancement of the Nusselt number due to presence of nanoparticles increases with the Hartman number, but at higher Rayleigh number, a reduction has been observed.In addition, it was observed that the entropy generation is decreased when the nanoparticles are present, while the magnetic field generally increases the magnitude of the entropy generation.
Vibrations are known to be among the most effective ways of affecting the behavior of fluid systems in the sense of increasing or reducing the convective heat transfer.The study of vibrations to the fluid has been the subject of several works, which is not the case for nanofluids [17][18][19][20][21][22].
The objective of our work is to increase the rate of transfer by coupling vibration magnetoconvection to facilitate proper cooling of the electronic component or solar cell and increase their efficiency.

MATHEMATICAL MODEL
The configuration studied is shown in Figure 1.It is mainly based on a square cavity of side H filled with Al2O3-water nanofluid in natural convection.This medium is heated from below and subjected to inclined vibratory excitation and an inclined external magnetic field.The vertical walls are adiabatic and impermeable.The horizontal walls are maintained at constant temperatures and uniform Tc and Th at y = 0 and y = H respectively.In the present analysis, Cartesian coordinate system will be applied.The temperatures of the vertical walls have been considered to be insulated.An electronic compounds or a solar cell is placed in the bottom horizontal wall.The nanofluid is Newtonian, incompressible, and the flow is laminar.The induced magnetic field is assumed to be negligible with respect to the applied magnetic field parallel to gravity.Moreover, it is assumed that both fluid phase and nanoparticles are in thermal equilibrium state and they flow at the same velocity.The Boussinesq approximation is assumed to be valid.
The nondimensional version of the governing system of transport equations are as follow: The thermal diffusivity of nanofluid is given by: The effective thermal conductivity of nanofluids is calculated using the Wasp model [23].
The effective density of a fluid containing suspended particles is given by: ( ) The effective viscosity of a fluid containing a dilute suspension of small rigid spherical particles is given by Brinkman model [24] as: The heat capacitance of the nanofluid can be calculated as: Based upon the previous assumptions and introducing the following dimensionless variables, The governing equations for the problem in dimensionless form are as follows:

Electronic compounds -bw 2 sin(wt)
The expressions for dimensionless parameters are given as: The heat transfer is characterized by the flux ratio between nanofluid and fluid.The average flux ratio at the bottom wall is computed as follows: By replacing Eqn (7) in Eqn (15), we obtain:

NUMERICAL METHOD AND VALIDATION
The numerical results are obtained by solving the system of steady differential Eqns ( 11)-( 14) with appropriate boundary and initial conditions using the Galerkin finite element method.
The two dimensional spatial domain is divided into quadrangle elements and a Lagrange quadratic interpolation has been chosen.Accuracy tests were performed for the steady state results using five sets of uniform grids as shown in Table 1.The mesh is refined near the boundaries and we have adopted 1681 elements.The computational domain consists of bi-quadratic elements which correspond to 41×41 grid.In order to validate our work, we will compare the results obtained with other numerical results available in the literature.The test case considered is the Natural convection flow and heat transfer in a differentially-heated square cavity filled with air.The left and the right side walls of the cavity are maintained at constant temperatures Th and Tc, respectively, with Th > Tc, while its top and bottom walls are insulated.Table 2 shows the comparison of the Nusselt number for different values of the Rayleigh number of other investigators for the same problem.As it can be observed from this table, very good agreements exist between the Nusselt numbers obtained by the present simulation and the results of Khanafer et al [25], G.A.Sheikhzadeh et al [8] and De Vahl Davis [26].
First, we study the effect of the Hartmann number on the average flux ratio and the maximum absolute value of stream function for different values of the inclination angle of vibratory excitation.For this, the Hartmann number varies from 0 to 100, the magnetic field is assumed to be horizontal (θ=0°C) and the Rayleigh number ratio R=10.
The figures (3-a) and (3-b) represent the variation of the average flux ratio as a function of the Hartmann number for different values of the inclination angle of the vibratory excitation for the first half period and the second half-period, respectively.For both curves, the average flux ratio decreases with increasing Hartmann number which leads to a decrease in heat transfer by convection.Also the increase of the inclination angle of the vibratory excitation promotes the conductive regime.Beyond the angle Φ = 75°, the average flux ratio is constant which means obtaining pure conduction.By comparing the two figures, there is a discrepancy in the values of the average flux ratio indicating that the convective mode is favored in the second half period relative to the first.Second, we study the effect of the Hartmann number on the average flux ratio and the maximum absolute value of stream function for different values of the inclination angle of magnetic field.For this, the Hartmann number varies from 0 to 100, the vibratory excitation is assumed horizontal (Φ=0°C) and the Rayleigh number ratio R=10.Figures (5-a) and (5-b) show the variation of the average flux ratio as a function of the Hartmann number for different values of the inclination angle of the magnetic field.For both curves, the average flux ratio decreases with increasing Hartmann number for all inclination angles of the magnetic field.Also the increase of the inclination angle of the magnetic field decreases heat transfer by convection.Beyond Φ=60°, the effect is almost negligible.We note that the heat exchange increases with the Hartmann number to Ha = 50.Beyond this value, the exchange is almost constant.Table 3 summarizes some results of the heat exchange rate.We conclude that the heat exchange rate is low.-a) and (6-b) represent the evolution of the maximum absolute value of the stream function depending on the Hartmann number for different values of the inclination angle of the magnetic field for the first half period and the second half period, respectively.The two curves confirm the previous results.
In this part we study the effect of the Rayleigh number ratio R on the average flux ratio and the maximum absolute value of the stream function.For this, the ratio R varies from 0 to 50, Hartmann number Ha = 30 and the vibratory excitation is assumed horizontal (Φ=0°C).
Figures (7-a) and (7-b) show the average flux ratio as a function of the Hartmann number for different values of the inclination angle of the magnetic field for the first half period and the second half-period, respectively.We note the increase of the ratio R increases the flow ratio in both periods which promotes convective transfer mode.By increasing the ratio R, the average flux ratio increases which confirms the convective mode.As is as one increases the inclination angle of the magnetic field, the average flux ratio decreases.Beyond the temperature θ = 60° C, the inclination angle of the magnetic field has no effect on the flow and therefore report on the convective heat transfer coefficient.
Figures (8-a) and (8-b) represent the evolution of the maximum absolute value of the power function depending on the ratio of the Rayleigh numbers to different values of the angle of inclination of the magnetic field for the first half period and the second half period, respectively.The two curves confirm the previous results.
Figures (9-a) and (9-b) show the variation of the average flux ratio as a function of the ratio of the numbers of Rayleigh to different values of the inclination angle of the vibratory excitation.For both curves, and below of Φ = 45°, the average flux ratio rises with the increase of the ratio R which shows dominance convective regime.Beyond the angle Φ = 45 °, the average flux ratio decreases to a value of R = 10 which means a decrease in heat transfer by convection then becomes constant, confirming the domination of conductive mode.We can conclude a critical inclination angle near Φ=45°.For the angle Φ= 90°, we get the purely conductive regime.Also the second half period promotes convection better than the first half period.
Figures ( 10-a) and (10-b) illustrate the evolution of the maximum absolute value of the stream function depending on the Rayleigh numbers ratio to different values of the inclination angle of the vibratory excitation for the first half period and the second half-period, respectively.The two curves confirm the previous results.

Figure 3 :Figures ( 4
Figures(4-a)  and (4-b) illustrate the variation of the maximum absolute value of the stream function depending on different values of the Hartmann number for different values of the inclination angle of the vibratory excitation in the first half period and the second half period, respectively.The two curves confirm the previous results.Second, we study the effect of the Hartmann number on the average flux ratio and the maximum absolute value of stream function for different values of the inclination angle of magnetic field.For this, the Hartmann number varies from 0 to 100, the vibratory excitation is assumed horizontal (Φ=0°C) and the Rayleigh number ratio R=10.

Figure 4 :
Variation of the maximum absolute value of the stream function depending on the Hartmann number for different values of the inclination angle of the vibratory excitation for: (a) First half period and (b) Second half period.

Figure 5 :
Variation of the average flux ratio as a function of the Hartmann number for different values of the inclination angle of the magnetic field for: (a) First half period and (b) second half period.

Figure 6 :Figure 7 :
Variation of the maximum absolute value of the stream function depending on the Hartmann number for different values of the inclination angle of the magnetic field to (a) First half period and (b) Second half period Variation of the average flux ratio as a function of the Hartmann number for different values of the inclination angle of the magnetic field for: (a) First half period and (b) second half period.

Figure 8 :Figure 9 :
Variation of the maximum absolute value of the stream function depending on the Hartmann number for different values of the inclination angle of the magnetic field to (a) First half period and (b) Second half period Variation of the average flux ratio as a function of the Hartmann number for different values of the inclination angle of the magnetic field for: (a) First half period and (b) second half period.

Figure 10 :
Variation of the maximum absolute value of the stream function depending on the Hartmann number for different values of the inclination angle of the magnetic field to (a) First half period and (b) Second half period

Table 1 :
Dimensionless Boundary Conditions

Table 2 .
Validation of the numerical code in the case of Natural convection

Table 3 .
Some results of the heat exchange rate in the first half period