Nomenclature.
Abstract
We numerically study the threedimensional magnetohydrodynamics (MHD) stability of oscillatory natural convection flow in a rectangular cavity, with free top surface, filled with a liquid metal, having an aspect ratio equal to A=L/H=5, and subjected to a transversal temperature gradient and a uniform magnetic field oriented in x and z directions. The finite volume method was used in order to solve the equations of continuity, momentum, energy, and potential. The stability diagram obtained in this study highlights the dependence of the critical value of the Grashof number Grcrit, with the increase of the Hartmann number Ha for two orientations of the magnetic field. This study confirms the possibility of stabilization of a liquid metal flow in natural convection by application of a magnetic field and shows that the flow stability is more important when the direction of magnetic field is longitudinal than when the direction is transversal.
Keywords
 Natural convection
 Magnetic field
 Oscillatory
 Cavity
 Liquid metal
1. Introduction
Natural convection of a conducting fluid of electricity contained in a cavity represents an adequate research subject, because of its presence in many industrial processes, especially during the process of crystal growth (Tagawa and Ozoe, 1997). The widespread use of this process in electronic and optical applications had, for consequence, an extended research towards the comprehension and the control of the natural convection in these systems.
With the application of an external magnetic field, it is possible to act on the flows without any physical contact, and thus to remove the fluctuations to control heat and mass transfers, in order to improve the quality of the crystal. For this purpose, the damping magnetic to control the flow induced by a temperature variation was used in several industrials applications [1][8]. Tagawa and Ozoe [1] numerically studied threedimensional natural convection of a liquid metal in a cubic enclosure, under the action of a magnetic field applied, according to the three main directions. Benhadid and Henry [2] studied the effect of a magnetic field on the flow of liquid metal in a parallelepiped cavity, using a spectral numerical method. Bessaih et al. [3] numerically examined the effect of the electric conductivity of the walls and the direction of the magnetic field on the flow of Gallium. Their results show a considerable reduction in the intensity of the convection when the magnetic field increases. Juel and al. [4] had the results of a numerical and experimental study of the effect of the application of a magnetic field in the direction perpendicular to the convective flow of Gallium. Aleksandrova and Molokov [5] considered threedimensional convection in a rectangular cavity subjected to a horizontal temperature gradient and a magnetic field, by an asymptotic model. The effectiveness of the application of the magnetic field depends considerably on the aspect ratio and the value of the Hartmann number. Hof and al. [6] presented an experimental study of the effect of the magnetic field on the natural convection stability in a rectangular cavity of square section, filled with a liquid metal. These authors founded that the vertical direction of the magnetic field is most effective for the suppression of oscillations. This is in good agreement with the work of Gelfgat and BarYoseph [7].
In the present work, we present a threedimensional numerical study on the critical value of the Grashof number
2. Geometry and mathematical model
The geometry of the flow field analysed in this study is illustrated in Figure 1. A liquid metal with a density
The interaction between the magnetic field and convective flow involves an induced electric current
The divergence of Ohm’s law .
By Neglecting the induced magnetic field, the dissipation and Joule heating, and the Bousinesq approximation is valid; and using
Where Gr
The initial conditions impose that the fluid is at rest and that the temperature distribution is zero, and that the electric potential is zero everywhere in the rectangular cavity. Thus, at
At t>0 the boundary conditions of the dimensionless quantities (
The Biot number in equation (6d) is given by
3. Numerical Method
The equations (2) − (5) with the boundary conditions (6a6f) were solved by using the finite volume method [9]. Scalar quantities (
In order to examine the effect of the grid on the numerical solution, a number of grid sizes have been investigated for grid independence: 32
4. Results and Discussion
The sentence of validation consisted in establishing some comparisons with experimental investigations presented in the literature [8]. We compared the temperature distribution
Figure4 (a) and Figure 4(b) show the velocity field of flows in the rectangular cavity for both orientations of the magnetic field. We can see that the fluid moves from the hot wall (at
In this section, we determine the physical instabilities within the flow from natural convection of a low Prandtl number fluid (
From point of view of the dynamic systems, when a system reenters in instability it presents to the beginning an oscillatory or periodic character, then because of the bifurcation phenomenon, this system will become quasiperiodic, and finally it reenters in chaos (or turbulence). The Grashof numbers characterizing the periodic flows are the critical numbers: transition from steady (Figure 5) to timedependent flow (Figure 6(ad)).
The oscillatory aspect (periodic) of the temporal evolutions of three dimensionless components velocity
In absence of the magnetic field
5. Conclusions
A threedimensional numerical study of a low Prandtl number fluid flow inside a rectangular cavity, which is subjected to a uniform magnetic field, has been carried out. The geometry considered here is related to crystal growth by a horizontal Bridgman configuration. The finitevolume method was used to discretize the mathematical model.
In the absence of a magnetic field, the results obtained show that the flow is steady for
Symbol  Quantity  Unity 
A  apect ratio = L/H  

dimensioless magnetic flux density vector  

uiform magnetic flux density  T 
Bi  Biot number = hL/k  

acceleration due to gravity  m/s ^{2} 
Gr  Grashof number 


unitary vector of the direction of 

H  height of the cavity  M 
Ha  Hartmann number 

H  heat transfer coefficient  W /m ^{2} .K 
_{ j → }  electric current density  A/m ^{2} 
K  thermal conductivity of the fluid  W/m .K 
L _{r}  length of the cavity  M 
P  dimensionless pressure  
Pr  Prandtl number 

T  dimensionless temperature  
T *  temperature  K 

ambient temperature  K 
t  dimensionless time  

dimensionless velocity vector  
u, v, w  dimensionless velocities in x, y, and z directions, respectively  
W  width of the cavity  m 
x, y, z  dimensionless transversal, vertical and longitudinal coordinates, respectively  
α  thermal diffusivity of the fluid  m ^{2} /s 
β  thermal expansion coefficient of the fluid  K ^{1} 

dynamic viscosity  Pa.s 

kinematic viscosity of the fluid  m ^{2} /s 
ρ  density of the fluid  kg/m ^{3} 
σ  electric conductivity  Ω ^{1} .m ^{1} 

electric potential  V 
Acknowledgments
The authors gratefully acknowledge the financial support of this work (PhD Thesis) through the project N J2501/03/57/06 provided by the Algerian Ministry of High Education and Scientific Research.
References
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