We describe a novel approach to the mathematical modeling and computational simulation of fully three-dimensional, electromagnetically and thermally driven, steady liquid-metal flow. The phenomenon is governed by the Navier-Stokes equations, Maxwell’s equations, Ohm’s law, and the heat equation, all nonlinearly coupled via Lorentz and electromotive forces, buoyancy forces, and convective and dissipative heat transfer. Employing the electric current density rather than the magnetic field as the primary electromagnetic variable, it is possible to avoid artificial or highly idealized boundary conditions for electric and magnetic fields and to account exactly for the electromagnetic interaction of the fluid with the surrounding media. A finite element method based on this approach was used to simulate the flow of a metallic melt in a cylindrical container, rotating steadily in a uniform magnetic field perpendicular to the cylinder axis. Velocity, pressure, current, and potential distributions were computed and compared to theoretical predictions.

1.
Prescott
,
P. J.
, and
Incropera
,
F. P.
,
1993
, “
Magnetically Damped Convection During Solidification of a Binary Metal Alloy
,”
ASME J. Heat Transfer
,
115
, pp.
302
310
.
2.
Walker
,
J. S.
,
Henry
,
D.
, and
,
H.
,
2002
, “
Magnetic Stabilization of the Buoyant Convection in the Liquid-Encapsulated Czochralski Process
,”
J. Cryst. Growth
,
243
, pp.
108
116
.
3.
Hughes
,
M.
,
Pericleous
,
K. A.
, and
Cross
,
M.
,
1995
, “
The Numerical Modeling of DC Electromagnetic Pump and Brake Flow
,”
Appl. Math. Model.
,
19
, pp.
713
723
.
4.
Verardi
,
S. L. L.
,
Cardoso
,
J. R.
, and
Costa
,
M. C.
,
2001
, “
Three-Dimensional Finite Element Analysis of MHD Duct Flow by the Penalty Function Formulation
,”
IEEE Trans. Magn.
,
37
, pp.
3384
3387
.
5.
Vives
,
Ch.
, and
Ricou
,
R.
,
1985
, “
Fluid Flow Phenomena in a Single Phase Coreless Induction Furnace
,”
Metall. Trans. B
,
16
, pp.
227
235
.
6.
Spitzer
,
K.-H.
,
Dubke
,
M.
, and
Schwerdtfeger
,
K.
,
1986
, “
Rotational Electromagnetic Stirring in Continuous Casting of Round Strands
,”
Metall. Trans. B
,
17
, pp.
119
131
.
7.
Barz
,
R. U.
,
Gerbeth
,
G.
,
Wunderwald
,
U.
,
Buhrig
,
E.
, and
Gelfgat
,
Yu. M.
,
1997
, “
Modelling of the Isothermal Melt Flow Due to Rotating Magnetic Fields in Crystal Growth
,”
J. Cryst. Growth
,
180
, pp.
410
421
.
8.
,
H.
,
Henry
,
D.
, and
,
S.
,
1997
, “
Numerical Study of Convection in the Horizontal Bridgman Configuration Under the Action of a Constant Magnetic Field. Part 1. Two-Dimensional Flow
,”
J. Fluid Mech.
,
333
, pp.
23
56
.
9.
,
H.
, and
Henry
,
D.
,
1997
, “
Numerical Study of Convection in the Horizontal Bridgman Configuration Under the Action of a Constant Magnetic Field. Part 2. Three-Dimensional Flow
,”
J. Fluid Mech.
,
333
, pp.
57
83
.
10.
Witkowski
,
L. M.
,
Walker
,
J. S.
, and
Marty
,
P.
,
1999
, “
Nonaxisymmetric Flow in a Finite-Length Cylinder With a Rotating Magnetic Field
,”
Phys. Fluids
,
11
, pp.
1821
1826
.
11.
Witkowski
,
L. M.
, and
Walker
,
J. S.
,
2002
, “
Numerical Solutions for the Liquid-Metal Flow in a Rotating Cylinder With a Weak Transverse Magnetic Field
,”
Fluid Dyn. Res.
,
30
, pp.
127
137
.
12.
Moffatt
,
H. K.
,
1965
, “
On Fluid Flow Induced by a Rotating Magnetic Field
,”
J. Fluid Mech.
,
22
, pp.
521
528
.
13.
Davidson
,
P. A.
, and
Hunt
,
J. C. R.
,
1987
, “
Swirling Recirculating Flow in a Liquid-Metal Column Generated by a Rotating Magnetic Field
,”
J. Fluid Mech.
,
185
, pp.
67
106
.
14.
Fujisaki
,
K.
,
2001
, “
In-Mold Electromagnetic Stirring in Continuous Casting
,”
IEEE Trans. Ind. Appl.
,
37
, pp.
1098
1104
.
15.
Natarajan
,
T. T.
, and
,
N.
,
2004
, “
Finite Element Analysis of Electromagnetic and Fluid Flow Phenomena in Rotary Electromagnetic Stirring of Steel
,”
Appl. Math. Model.
,
28
, pp.
47
61
.
16.
Natarajan
,
T. T.
, and
,
N.
,
1999
, “
A Methodology for Two-Dimensional Finite Element Analysis of Electromagnetically Driven Flow in Induction Stirring Systems
,”
IEEE Trans. Magn.
,
35
, pp.
1773
1776
.
17.
Natarajan
,
T. T.
, and
,
N.
,
2002
, “
A New Method for Three-Dimensional Numerical Simulation of Electromagnetic and Fluid-Flow Phenomena in Electromagnetic Separation of Inclusions From Liquid Metal
,”
Metall. Mater. Trans. B
,
33
, pp.
775
785
.
18.
Meir
,
A. J.
, and
Schmidt
,
P. G.
,
1999
, “
Analysis and Numerical Approximation of a Stationary MHD Flow Problem With Nonideal Boundary
,”
SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
,
36
, pp.
1304
1332
.
19.
Meir
,
A. J.
, and
Schmidt
,
P. G.
,
2001
, “
On Electrically and Thermally Driven Liquid-Metal Flows
,”
Nonlinear Anal. Theory, Methods Appl.
,
47
, pp.
3281
3294
.
20.
Meir, A. J., and Schmidt, P. G., 1998, “Analysis and Finite Element Simulation of MHD Flows, With an Application to Seawater Drag Reduction,” Proc. International Symposium on Seawater Drag Reduction, J. C. S. Meng, ed., Naval Undersea Warfare Center, Newport, RI, pp. 401–406.
21.
Meir, A. J., and Schmidt, P. G., 1999, “Analysis and Finite Element Simulation of MHD Flows, With an Application to Liquid Metal Processing,” Proc. Fluid Flow Phenomena in Metals Processing, 1999 TMS Annual Meeting, N. El-Kaddah et al., eds., Minerals, Metals, and Materials Society, Warrendale, PA, pp. 561–569.
22.
Schmidt
,
P. G.
,
1999
, “
A Galerkin Method for Time-Dependent MHD Flow With Nonideal Boundaries
,”
Commun. Appl. Anal.
,
3
, pp.
383
398
.
23.
Bakhtiyarov
,
S. I.
,
Overfelt
,
R. A.
,
Meir
,
A. J.
, and
Schmidt
,
P. G.
,
2003
, “
Experimental Measurements of Velocity, Potential, and Temperature Distributions in Molten Metals During Electromagnetic Stirring
,”
ASME J. Appl. Mech.
,
70
, pp.
351
358
.
24.
Hughes, W. F., and Young, F. J., 1966, The Electromagnetodynamics of Fluids, Wiley, New York.
25.
Meyer
,
J.-L.
,
Durand
,
F.
,
Ricou
,
R.
, and
Vives
,
C.
,
1984
, “
Steady Flow of Liquid Aluminum in a Rectangular-Vertical Ingot Mold, Thermally or Electromagnetically Activated
,”
Metall. Trans. B
,
15
, pp.
471
478
.
26.
Brezzi, F., and Fortin, M., 1991, Mixed and Hybrid Finite Element Methods, Springer, New York.
27.
Girault, V., and Raviart, P.-A., 1986, Finite Element Approximation of the Navier-Stokes Equations, Theory and Algorithms, Springer, New York.