INTRODUCTION
In recent years, numerical simulations have shown the ability of macroscopic solidification models to predict the effects of convection in the mushy zone and bulk liquid on the development of an irregular liquidus front, flow channels in the mushy zone, local remelting of solid and complicated macrosegregation for the solidification of a variety of binary alloys. The cause of macrosegregation is the longrange advection of alloy species due to the relative movement or flow of segregated liquid and solid during solidification. There are numerous causes of fluid flow and solid movement in casting processes: flow that feeds the solidification shrinkage and the contractions of the liquid and solid during solidification; buoyancy forces induced thermosolutal convection flows due to thermal and solutal gradients in the liquid; these forces can either aid or oppose each other, depending on the direction of the thermal gradient and whether the rejected solutes cause an increase or a decrease in the density of the liquid; forced flows due to: applied magnetic fields, stirring, rotation, vibration, etc., (Beckermann, 2002). Therefore, the need to prevent macrosegregation during solidification processing is extremely important.
To prevent macrosegregation all efforts are aimed at controlling fluid flow and solid movement for example: include adjustments to the alloy composition or thermal gradients to induce a stable density stratification in the liquid; include centrifugal forces, or electromagnetic fields to redistribute the flow (Beckermann, 2002).
Macrosegregation is an important defect in steel casting and there have been few macrosegregation studies, where no external magnetic is applied, on ironcarbon and ironcarbon based steel. Ma et al. (2004) have performed a numerical simulation of macrosegregation in Fe0.42wt%C steel ingot during solidification. Liu (2004) has presented numerical modeling for the macrosegregation in the Fe0.2wt%C steel. Also, Singh and Basu (2001a, 2001b) have presented simulations to study the role of double diffusive convection on macrosegregation during solidification of binary Fe1wt%C alloy, where, The effect of thermosolutal convection on extent of macrosegregation profile is discussed.
The modeling of the formation of macrosegregation in steel due to multicomponent
thermosolutal convection was explored in detail by Schneider and Beckermann
(1995a), where, by using a fully coupled multicomponent model, they have shown
that macrosegregation profiles of carbon in multicomponent steel show the same
trend as in binary FeC owing to dominant role of carbon in solutal buoyancy
and thermodynamic equilibrium. Another work performed, by Schneider and Beckermann
(1995b), for study of solidification, where no external magnetic is applied,
the simulation of the austenitic solidification steels containing ten elements
in a rectangular cavity cooled from the side shows the formation of macrosegregation,
channel segregates and islands of mush surrounded by the bulk melt. The global
severity of macrosegregation of an element is found to be linearly dependent
on its partition coefficient.
Moreover, several studies have been carried out where a magnetic field is applied during solidification for different binary alloys. We note here the study due to Sampath and Zabaras (2001), where the system SbGe is considered for study of the thermosolutal and capillary convection. Also, the study carried out by Anwar Hossain et al. (2005) and the study performed by Mittal et al. (2005).
In the present study the influence of an external magnetic field, applied during
solidification process of the system Fe0.42wt%C, on the flows control, especially
the control of the advection of solute in the liquid phase near the front of
solidification (solutal convection flow the main cause of macrosegregation).
MATHEMATICAL MODEL
Consider a mold permeated by a uniform magnetic field B of constant magnitude B_{0} in the axial direction, shown on Fig. 1. The mold is of rectangular geometry with length and height L = 60 mm. This mold, of impermeable to all species walls and noslip velocity conditions were imposed at the walls, contained a molten electrically conducting binary alloy Ironcarbon (Fe0.42wt%C), of a eutectic temperature (T_{e} = 1147°C). The thermophysical properties of this alloy (Ma et al., 2004) are listed in Table 1. All thermophysical properties are assumed constant. The molten alloy was initially isothermal, at a uniform concentration C_{ini}= 0.42% and a uniform temperature T_{ini} = 1528°C.
At time t = 0, the left wall is suddenly cooled to a temperature T_{cold} = 1100°C and maintained at that temperature for times t > 0. The other right, top and bottom walls are insulated and adiabatic. The temperature at the right wall is imposed to be T_{hot} = 1528°C. Under these conditions the directional solidification begins. The motion of the melt is initially driven by the action of the combined of thermalsolutal buoyancy forces and a Lorentz force.
Electromagnetic lorentz force: The concentration of the solute in the liquid phase near the front of solidification must be controlled to ensure the control of macrosegregation which improve the homogeneity and the quality of the solid formed. The control is possible by the application of an external uniform magnetic field in order to suppress fluctuations due to density irregularities provoked by concentration gradients.
Lorentz force F of component (Fx, Fy) drives electromagnetic convection flow is given, in such flow, as follows:
Table 1: 
Thermophysical
properties for Fe0.42wt%C system, (Ma et al., 2004) 

where: q_{e} is the electric charge density of the fluid.
And the electric field intensity E, given in term of the electric field potential φ,
And, J the electric current density that is governed by Ohm’s law:
where: V is the velocity vector of flow field.
In addition to Ohm’s law, the electric current density J is governed by the conservation of electric current that is given, for a moving medium, as:
In addition to the applied magnetic field B of constant intensity B_{0}, there is an induced magnetic field produced by the electric currents in the liquid metal. Since, it is assumed that the walls of the cavity are electric insulators and the magnetic Reynolds number is sufficiently small such that the induced magnetic field is negligible with respect to the imposed constant magnetic field B_{0}, therefore neglecting excess charge i.e.,:
The mold is with electrically insulated boundaries in the present of twodimensional flow the electric potential φ is constant, (Anwar Hossain et al., 2005), therefore:
Assumptions:
• 
We assume that the solidus line, in the FeC binary equilibrium
diagram, as linear. 
• 
As we know the solid phases which can be occur in Fe0.42wt%C steel solidification
are the following: the primary solid phase: γ(Austenitic), the secondary
solid phase: Fe_{3}C(Comentite) and the eutectic component: γ(Austenitic)
+ Fe_{3}C(Comentite) resulted from the eutectic reaction. Therefore,
we assume that only the following eutectic reaction, of solidification,
occurs in the molten alloy. Therefore, no peritectic reaction is assumed: 
• 
Assuming that, these solid phases can be presented by a single solid phase.
Because the present study does not interest to the different solid phases
analyze. 
Also considering the following assumptions:
• 
Twophase flows are described as a mixture of solid and liquid phases, 
• 
Local thermodynamic equilibrium exists between solid and liquid phases
(therefore, the phase diagram is used and the formulation due to Bennon
and Incropera (1987) will be used). 
• 
All of the properties of the mixture can be obtained from each phase component
properties. All transport properties are constant. 
• 
The mushy region, (austenite+ liquid) interface is modeled by means of
isotropic permeability approach, using equation of BlakeKozeny. 
• 
It has no Joule heating effects. 
• 
The intrinsic velocity of the solid phase is neglected. 
• 
The flow of the liquid phase is assumed to be laminar. 
• 
Densities in the solid and liquid are constant except for the variations
in the buoyancy term (we consider the Boussinesq approximation). 
Governing equations: Based on continuum model, (Bennon and Incropera,
1987), governing equations for binary alloy solidification are represented by:
the mass, momentum, energy, species and electrical potential conservation equations
(7 to 12), as reported by Ma et al. (2004) except here we add the Lorentz
force term in the ydirection momentum equation and we have the conservation
of electrical potential equation.
These equations can be written in dimension less form, respectively, (Mechighel and Kadja, 2006). In this dimension less form: we defining characteristic scales for: length, velocity, time, temperature, enthalpy, species concentration and electric potential, respectively as following:
where: m^{ad} is the dimension less slope of the liquidus line, T_{liq}:
is the liquidus temperature and T_{ref}: is the reference temperature.
Therefore, the dimension less coordinates, the dimension less mixture velocity components, the dimension less continuum mixture enthalpy the dimension less continuum mixture composition and the dimension less electrical potential are given, respectively, as following:
where: (x, y): the Cartesian coordinates, (u, v): the velocity components,
h: the enthalpy and C: the concentration.
Also, the dimension less temperature is given such as:
In Table 2 are listed the resulting dimension less numbers:
where: α_{l} the melt thermal diffusivity, v_{l} the melt
kinematical viscosity and g: acceleration of gravity.
The other dimension less parameters include, respectively, the specific heat ratio, the heat conductivity ratio and buoyancy ratio:
In the case where the magnetic field is applied horizontally, therefore:
Table 2: 
Calculated
dimension less numbers using data given by Ma et al. (2004) and
other parameters for Fe0.42wt%C 

The governing equations, in dimension less form are:
where:
ρ 
: 
Is the mixture dimensionless density 
f_{s} 
: 
Is the solid mass fraction 
(ε_{l}, ε_{s}) 
: 
Are the melt and solid volume fractions, respectively. 
While, the subscripts:
s 
: 
Used to identify the solid phase 
l 
: 
Used to identify the liquid phase 
Isotropic model for mushy: The BlakeKozeny equation for calculated the permeability in the mushy zone is given by:
K_{0} is a parameter depending on the morphology and size of the primary dendrite arm spacing: λ_{1}.
The parameter K_{0} is given by Singh and Basu (2001a), such as:
Thermodynamic equilibrium: Since it is assumed that the local thermodynamic
equilibrium exists between the solid and liquid phases, thermophysical properties
of each phase are constant (but different from each other) and the liquidus
and solidus lines on the equilibrium phase diagram are linear. Hence, the supplementary
relations of Bennon and Incropera (1987) can be used.
With the full coupling, in the mushy zone, of the temperature field and species concentration through thermodynamic equilibrium requirements Temperature and concentration is represented through the following equation:
Also, the temperature and solid mass fraction will be calculated from the continuum enthalpy and concentration. Therefore, phase diagram will be used. In addition, local solid and liquid densities are determined and boundary enthalpies are appropriately adjusted.
NUMERICAL SOLUTION
The system of the governing equations, with the coupling of the temperature
and concentrations and the coupling of temperature and mass fraction solid,
has been discretized by means of volume based finite difference method (Patankar,
1980). For the resolution of discretized equations obtained the SIMPLER Algorithm
is used, (Patankar, 1980). And for the resolution of the algebraically system
equations obtained the Thomas Algorithm is used.
The selected mesh size should only be viewed as a compromise between accuracy and computational cost associated with such simulations. A mesh of 50*50 nodes in the domain and a constant dimensionless time step of Δt = 0.005 were used.
RESULTS AND DISCUSSION
Results presented by Ma et al. (2004) for the solidification of the binary system Fe0.42wt%C, results presented by Liu (2004) for the solidification of the binary system Fe0.2wt%C and results carried out by Singh and Basu (2001a and b) for the solidification of the binary system Fe1wt%C, where no magnetic field is applied, shown the large effect of the convective flow driven by thermalsolutal buoyancy forces on the solidifying product: large values of velocities, macrosegreagation and other defects.
For our case we applied a magnetic field for controlling convection flow.
Early stage of solidification: For the convective flow field, under
an external magnetic field, streamlines, liquid isocompositions, isotherms,
solid mass fraction and solutes distribution field corresponding to the early
stage of the solidification process, at t = 10, are presented in Fig.
2ae, respectively.
Convection flow field: Figure 2a shows the coupled effects of electromagnetic and buoyancy driven convection reversal flow.
One can see that horizontal gradients of temperature and solute concentration in the liquid are built up at this early stage of solidification. These two gradients lead to a horizontal gradient of the liquid density. Hence, thermal and solutal forces driven natural convection coupled with Lorentz force driven forced convection (electromagnetically convection) occur in the mold.
For the Fe0.42wt%C alloy, the interdendritic liquid, in the mushy zone, is enriched in solute (Carbon), Fig. 2b, becomes lighter (since the density of Carbon is lower than that of Iron), the effects of solute concentration and temperature on the liquid density are opposite, but soluteinduced convection dominates in the mushy zone, leading a clockwise fluid flow but this flow is extremely weak. This is due to fact that the buoyancy ratio is too small N = 0.59.
The thermal buoyancy force is dominant the melt flow this is due to the fact that the liquid is superheated at the beginning of the solidification and the low value of solutal buoyancy force in the liquid. Therefore, the solutal opposes the thermal buoyancy forces, in the mushy. While, the Lorentz force acts in opposition to the thermal buoyancy force on the liquid flow, Eq. 9. Therefore, the application of the externally magnetic field reduces the velocity vectors and leads to damping out turbulence and intensity of melt flow, thereby improving the homogeneity of the solid formed.
The reversal flow observed in the melt region, Fig. 2a, is a typical feature of natural convection coupled with forced convection in a stratified fluid, (Sampath and Zabaras, 2001; Sampath, 2001).
Solute concentration and temperature fields: Isotherms, in Fig. 2c, are nearly vertical, indicating onedimensional horizontal conduction in solid formed.
In Fig. 2b and e, one can see 2 levels
where the solutes are rejected in the mushy zone, the first level is at approximately
Y = 0.95 and the second, extremely important, is near the bottom of the mold.
The rejected solutes lead to a complicated interface form, Fig. 2b. This is due essentially to the system chosen in this study, characterized by a large Lewis number (Le = Sc/Pr = 1122), for this system the solutal mushy boundary layer is extremely thin and concentration gradients are very high close to the interface, which lead to an increase of the rejected solutes in the mushy. The fluid in the mushy, have an increasing composition, from the nominal value (X_{0} = 0.108) towards to the value (X = 0.36) but not be eutectic (Xe = 1.108), Fig. 2b. While, the temperature is decreasing from the value (1.06) towards the eutectic value (0.0), Fig. 2c.
In Fig. 2e, the dominant thermally buoyancy forces drive
downward the solutes rejected from the mushy zone for to be remelting in the
bulk melt. Likewise, one can see that the Carbon begins to accumulate at the
bottom while the Iron begins to accumulate at the top of the mold.
Solid mass fraction: Figure 2d shows the solid fraction formed for this stage of solidification. The bulk mass fraction of the solid formed; Fig. 2d; need not be eutectic because of the presence of the thin thickness mushy layer and is instead controlled by mass transfer within the fluid.
Intermediate stage of solidification: For the convective flow field,
under an external magnetic field, streamlines, liquid isocompositions, isotherms,
solid mass fraction and solutes distribution field corresponding to the intermediate
stage of the solidification process, at t = 150, are presented in Fig.
3ae, respectively.

Fig. 2: 
Solidification at time = 10, (a) Streamlines, (b) Liquid isocomposition
(c) Isotherms, (d) Solid mass fraction and
(e) Solutes concentration 

Fig. 3: 
Solidification at time = 150, (a) Streamlines, (b) Liquid isocomposition
(c) Isotherms, (d) Solid mass fraction
and (e) Solutes concentration 

Fig. 4: 
Solidification at time = 500, (a) Streamlines, (b) Liquid isocomposition
(c) Isotherms, (d) Solid mass fraction
and (e) Solutes concentration 
Convection flow field: Figure 3a shows the reversal flow, provoked by coupled effects of electromagnetic and buoyancy driven convection, observed in the melt, after the thermally dominated initial transient has passed, at which time the fluid has essentially lost some superheat. For this stage of solidification the mushy is expanding which increases the solutes rejected i.e., solute concentration gradient. While, the temperature gradient is weakening, due to shrinking of melted region. But the thermal buoyancy force driven convection remain the dominating force in the melt (since N = 0.59).
The rejected solutes will transported by the thermally convection flow in the
melt for to be remelting, as shown in Fig. 3e. The Lorentz
force driven forced convection acts in opposition to thermal buoyancy force.
Therefore, the externally magnetic field damped out turbulence and intensity
of melt flow.
From this, it is result that the flow reversal observed in the melt region is weakening in strength and present a typical feature of natural convection coupled with forced convection in a stratified fluid.
Solute concentration and temperature fields: Isotherms, in Fig. 3c, show that the temperature is weakening in the melt region, which indicating that the fluid has lost some superheat.
In Fig. 3b, one can see that the mushy zone is expanding and becoming not uniform.
Figure 3b and e show the compositionally
heavy fluid i.e., Carbonrich fluid settles at the bottom and slowly forms a
stratified layer. This layer continues to grow until the entire fluid composition
evolves toward the eutectic at all levels. This process termed as laminar box
filling, (Sampath, 2001b). While the compositionally lighter fluid, i.e., Ironrich
fluid settles at the top. In addition, in Fig. 3e, one can
see a multiple levels where the solutes are rejected in the mushy, especially
near the bottom, which provoked instabilities in interface growth and influenced
convective melt flow. The interface morphology is instable and of a form extremely
complex, Fig. 3b. This is due to the interdentritic liquid
flow in the mushy zone.
Solid mass fraction: Figure 3d shows the solid fraction
formed for this stage of solidification. The composition of the solid product
shown reflects the evolution of the fluid flow and solute concentration fields.
In Fig. 3e, one can note the complex vertical variation in
the solute concentration due to the melt flow.
Later stage of solidification: For the convective flow field, under
an external magnetic field, streamlines, liquid isocompositions, isotherms,
solid mass fractions and solutes distribution field corresponding to the stage
of the solidification process, at t = 500, are presented in Fig.
4ae, respectively.
Convection flow field: As the mushy zone continues to grow, the influence of solutal buoyancy gradually increases but it is remaining weak, since N = 0.59.
Here also the thermal buoyancy force driven convection is weakening, due to shrinking of melted region and decreasing of temperature gradient, see isotherms Fig. 4c.
For this stage of solidification the solutes rejected are increased. These
solutes will be driven by thermal convection flow in the melt for to be remelting
and will be reintroduced in the mushy, as shown in Fig. 4e.
Here also the Lorentz force driven forced convection flow acts in opposition
to thermal convection flow, which induces that the externally magnetic field
damped out turbulence and intensity of melt flow.
Also, here the flow reversal observed in the melt is a typical feature of natural convection coupled with forced convection flow in a stratified fluid (Sampath, 2001).
Solute concentration and temperature fields: Isotherms, in Fig. 4c, show that the temperature is weakening in the melt region.
Figure 3b and e show the compositionally
heavy fluid settles at the bottom and slowly forms a stratified layer. This
layer continues to grow until the entire fluid composition evolves toward the
eutectic at all levels. While, the compositionally lighter fluid settles at
the top.
In Fig. 4e, one can see that the solutes rejected are more
important for this stage of solidification, which provoked instabilities in
interface growth and influenced convective melt flow. Also, the interface morphology
is more instable and of a form extremely complex, as shown in Fig.
4b.
Solid mass fraction: Also, the composition of the solid product shown
in Fig. 4d reflects the evolution of the fluid flow and solute
concentration fields. Here also we can note the complex vertical variation in
the solute composition due to the melt flow, Fig. 4e.
CONCLUSION
In the present study problem on coupled effects of electromagnetic forced convection flow and buoyancy driven convection flow, in the molten Fe0.42wt%C filled in a rectangular mold has been investigated numerically by employing the finite volume method. Results, obtained, have been shown the positive effect of the external magnetic field in flow control which therefore reduced defects such as macrosegregation, microsegregation etc.
It is being realized that there is tremendous potential in using magnetic fields to control growth conditions in binary alloy systems. Magnetic field enhances microstructural properties through melt flow control, thereby improving the homogeneity of the solid formed.
Finally, it requires to be mentioned that solutions of the present problem could be investigated for other IronCarbon systems, especially Fe1wt%C and for case where the magnetic field is variable and the cases where it is applied in other directions.