# Investigation of solid/liquid interface evolution in the solidification process of liquid metal in an annulus crucible at the presence of static magnetic field: numerical study

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## Abstract

This paper studies the effect of constant magnetic field on solidification of a conductive liquid that filled a vertical three-dimensional annulus-shaped crucible. The computational domain of the proposed geometry is a vertical annulus cavity; the inner and outer sides of the mentioned cavity have constant temperature, but top and bottom sides have been thermally insulated. The governing equations have been solved numerically using finite volume method, so obtained results illustrated acceptable conformity with available reported experimental and numerical data. The investigation has been carried out for various vital parameters, such as Hartmann and Rayleigh numbers. The magnetic field effect on the melted flow field and interface of solid/liquid form and position has been discussed with more details. The numerical results have shown a strong relation between interface shape, Rayleigh number and magnetic field strength. Then, the applied magnetic field leads to vanishing axisymmetric condition of solid/liquid interface. Finally, it has been conducted that strongest stabilization of the flow field and solid/liquid interface occurs at the *x*–*z* plane. The proposed predictions are very helpful in developing methods to improve solidification and casting systems under magnetic field effects.

## Keywords

Magnetic field Solidification Electric field Natural convection Solid/liquid interface## List of symbols

*A*=*L*/*D*Aspect ratio of annulus

- \(\overrightarrow {{B_{o} }}\)
External magnetic field, Tesla (Wb/m

^{2})*C*_{p}Heat capacity (J/kg K)

*D*= (*r*_{o}−*r*_{i})Annulus width (reference length) (m)

- \(\overrightarrow {E}\)
Electric field (V/m)

- \(\overrightarrow {F}\)
Body force (N/m

^{3})*g*Acceleration coefficient of gravity (m/s

^{2})*L*Annulus height (m)

*f*_{l}Liquid fraction

- h
_{f} Latent heat of fusion

- \(Ha = D B_{0} \sqrt {\frac{\sigma }{\rho \nu }}\)
Hartmann number

- \(\overrightarrow {J}\)
Electric current density (A/m

^{2})- \(Nu = \frac{D}{{T_{H} - T_{C} }}\left. {\frac{\partial T}{\partial r}} \right|_{{r = r_{o} }}\)
Nusselt number

- \(\overline{Nu} = \frac{1}{2\pi }\int\limits_{0}^{2\pi } {\int\limits_{0}^{H} {Nu \partial \theta \partial z} }\)
Average Nusselt number

*P*Pressure (pa)

- \(Pr = \frac{\nu }{\alpha }\)
Prandtl number

- \(\overrightarrow {r}\)
Radial unit vector

- \(Ra = \frac{{g\beta (T_{H} - T_{m} )(r_{o} - r_{i} )^{3} \ }}{\nu \alpha }\)
Rayleigh number

- \(R_{\text{m}} = \mu_{0} \sigma VD\)
Magnetic Reynolds number

*Ste*=*C*_{p}(*T*_{H}−*T*_{m})/*L*_{f}Stefan number

*T*Temperature (K)

*T*_{c}Cold wall temperature (K)

*T*_{H}Hot wall temperature (K)

*T*_{ini}Initial temperature (K)

*T*_{m}Phase change temperature (K)

*T*_{o}Reference temperature (K)

*u*Radial velocity vector (m/s)

*v*Angular velocity vector (rad/s)

*w*Axial velocity vector (m/s)

*z*Coordinate (m)

## Greek letters

*α*Thermal diffusivity (m

^{2}/s)*β*Thermal expansion coefficient (K

^{−1})*θ*Coordinate (rad)

- \(\overrightarrow {\theta }\)
Angular unit vector

*λ*=*r*_{o}/*r*_{i}Annulus radial aspect ratio (m)

*μ*Dynamic viscosity of fluid (kg/m s)

*μ*_{0}Magnetic permeability (m/s Ω)

- \(\nu\)
Kinematic viscosity of fluid (m

^{2}/s)*ρ*Density (kg/m

^{3})*ρ*_{o}Density at

*T*=*T*_{o}(kg/m^{3})*σ*Electric conductivity of melt (Ω m

^{−1})

## Notes

## References

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