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

  • Vahid AhmadpourEmail author
  • Iraj Mirzaei
  • Sajad Rezazadeh
  • Nima Ahmadi
Review Paper


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 xz plane. The proposed predictions are very helpful in developing methods to improve solidification and casting systems under magnetic field effects.


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/m2)


Heat capacity (J/kg K)

D = (ro − ri)

Annulus width (reference length) (m)

\(\overrightarrow {E}\)

Electric field (V/m)

\(\overrightarrow {F}\)

Body force (N/m3)


Acceleration coefficient of gravity (m/s2)


Annulus height (m)


Liquid fraction


Latent heat of fusion

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

Hartmann number

\(\overrightarrow {J}\)

Electric current density (A/m2)

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

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


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 = Cp(TH − Tm)/Lf

Stefan number


Temperature (K)


Cold wall temperature (K)


Hot wall temperature (K)


Initial temperature (K)


Phase change temperature (K)


Reference temperature (K)


Radial velocity vector (m/s)


Angular velocity vector (rad/s)


Axial velocity vector (m/s)


Coordinate (m)

Greek letters


Thermal diffusivity (m2/s)


Thermal expansion coefficient (K−1)


Coordinate (rad)

\(\overrightarrow {\theta }\)

Angular unit vector

λ = ro/ri

Annulus radial aspect ratio (m)


Dynamic viscosity of fluid (kg/m s)


Magnetic permeability (m/s Ω)


Kinematic viscosity of fluid (m2/s)


Density (kg/m3)


Density at T = To (kg/m3)


Electric conductivity of melt (Ω m−1)



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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  • Vahid Ahmadpour
    • 1
    • 4
    Email author
  • Iraj Mirzaei
    • 2
  • Sajad Rezazadeh
    • 1
  • Nima Ahmadi
    • 3
  1. 1.Faculty of Mechanical EngineeringUrmia University of TechnologyUrmiaIran
  2. 2.Mechanical Engineering DepartmentUrmia UniversityUrmiaIran
  3. 3.Department of Mechanical Engineering, Faculty of Shaheed BeheshtiTechnical and Vocational University (TVU)UrmiaIran
  4. 4.TabrizIran

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