Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013 (56–63)

Theorization and Modeling Ⅲ(Review)

Modeling of EML in Combined AC/DC Magnetic Fields as the Basis for
Microgravity Experiments
Koulis PERICLEOUS, Valdis BOJAREVICS and Alan ROY
Abstract
This article contains a review of research activity in the University of Greenwich to model the effects of static or DC magnetic fields on thecharacteristics of EM-levitated melts. The general idea is that the presence of a DC field will damp out the velocities inside a levitated droplet
and so lead to more accurate thermophysical property measurements, especially those of viscosity and thermal conductivity that will otherwise
be affected by turbulence. The technique is in fact used successfully in terrestrial experiments, and this study sets out to examine its
applicability in the case of microgravity. An accurate spectral-collocation numerical scheme is used to couple dynamically the velocity,temperature and magnetic fields, so that internal velocity and liquid envelop changes of a suspended “spherical” droplet can be observed as afunction of applied AC and DC fields, with or without gravity.

1.

Introduction

Since the earliest Electro Magnetic (EM) levitation
experiments in AC magnetic field1) it became apparent that thelevitated liquid metal is prone to oscillation and instability. Both
EM and electrostatic (ES) levitation experiments with liquid
metal droplets show difficulties related to confinement stability
and a need for complex correction functions to establish acorrelation between the measurements and the droplet material
properties2,3). Intense internal fluid flow is visually observed,
apparently being in the turbulent regime for earthboundconditions. This behavior was also observed in numericalsimulations4,5).
In addition to metrology, and usually on a larger scale, there
is demand for melting reactive materials without contamination,for example titanium alloys for high quality castings. However,
it is often difficult to achieve the required superheat in the meltwith traditional ‘cold’ crucible-type furnaces due to a partial
contact with the water cooled copper walls6). In this respect, it
was found that a DC field acting together with the AC induction
field would damp out turbulence and reduce free surface
oscillations thereby increasing superheat. Applications include
the evaporation of levitated melts for coating purposes or the
supply of pure superheated melt for metal powder production 7).
At the other temperature extreme, a highly under-cooled liquid
can be obtained before solidification to a glassy structure in thelevitated conditions in absence of nucleation centers8).
Finally, a combination of AC and DC magnetic fields isrecognized as an efficient technique in stabilizing hightemperature levitated melts, preventing contact to contaminating

walls 9,10). The intense AC magnetic field required to produce
levitation (in full gravity), acts in a thin external region of thelevitated mass (the skin layer) resulting in high shear flow andturbulent large-scale toroidal recirculation. In contrast a static
magnetic field acts uniformly throughout the liquid volume
resisting toroidal flow. Turbulence leads to an apparent viscosity
which behaves non-linearly, and can be 2-3 orders of magnitude
higher than molecular viscosity. This alters the damping rate ofoscillating droplets making the measurement of viscosity very
uncertain11) and it diffuses heat much faster within the melt,
making conductivity measurements impossible. The effect is
non-linear and depends on the relative intensity of the DC andAC fields applied 10). It was found, that in typical terrestrial
melting conditions one can approach the laminar state forviscosity and heat transfer when the applied DC magnetic fieldexceeds 4-5T9). An electrically conducting droplet in a high DC
magnetic field behaves quite differently to a non-conducting one.
For example, the asymptotic solution of Priede12) for highmagnetic field, shows damping of the even axisymmetricoscillation modes, but the odd modes are not damped or are
only moderately damped.
The behavior of liquid melts in a combination AC and DC
field, with or without gravity is the subject of this paper.
Numerical models of the flow within a levitated liquid sample,
coupled with the behavior of the moving free surface give an
insight to the dynamics of levitated droplets of various sizes andmagnetic field intensities. Realistic gradient fields, typical of thesolenoidal coils encountered in superconducting magnets, areused in modeling, to validate the mathematical model against
terrestrial experiments9,13).

Centre for Numerical Modelling and Process Analysis, University of Greenwich, London SE10 9LS(E-mail: k.pericleous@gre.ac.uk)

56− 56 −

Koulis PERICLEOUS, et al.

2.

only in the regions where it is needed. The electric current
distribution in a moving medium of conductivity  is thenNumerical Model

The numerical model (code SPHINX) uses an adaptive meshformulation of the spectral collocation method with a
Chebyshev grid used for the radial direction and Legendre
nodes in the vertical direction. The model uses a coordinatetransformation for the free surface, which allows the problem tobe solved on a unit sphere.

2.1

Electromagnetic Force Calculation

The starting assumption for theoretical models attempting topredict the free surface behavior of EM-levitated droplets isbased on the idea of an infinitely thin skin layer where the highfrequency AC magnetic field penetrates. This then replaces the
EM force distribution by the so-called ‘magnetic pressure’,
acting on the free surface. Such an approximation effectively
removes any dependence on the electrical conductivity of thelevitated material. The skin-layer approximation was tested
against direct numerical solutions14) for typical experimental
frequencies and material properties (Al, Ni, Si). The results
indicate that for medium frequencies (<100 kHz) the
deformation and velocity field are significantly different in a
10mm droplet compared to the asymptotic skin-layer case. For
higher frequencies (>100 kHz) a relatively good correspondence
for the droplet shape can be obtained. The velocity field within
a levitated liquid material is controlled by the volumetric EM
force distribution; it is important to understand how various
approximations to this force field can be obtained from amathematical point of view. The general expression for the EMforce acting on a levitated fluid volume becomes rather
complex15) if one accounts for variable electrical conductivity
and magnetic properties. For uniform properties within theliquid, the commonly used expression for the force is:f  j  B  ( B2 / 20 ) (V  B) :j   (t A    V  B)  jAC  jv

−4.2×10−6 for silicon which is diamagnetic) and 0 is themagnetic permeability of vacuum in SI units. When acombination of high frequency AC and DC magnetic fields isused, a time-average of the force over the field oscillation
period  (~ 10-4 - 10-6 s) is effective with a time scale for alevitated droplet oscillation period being 100 - 10-2 Hz. However,
the oscillating part of the high frequency AC field is still present
within the fluid, adding a time-dependent vibrating force atdouble the frequency of the applied AC, in addition to the time
averaged force.
In the work presented, the electromagnetic force iscomputed from an integral equation representation. This has theadvantage that the far field boundary conditions are notexplicitly required, and the electromagnetic field can be solved(2)

The j AC part of the current is induced in the conducting
medium even in the absence of velocity. The governing integral
equations can be obtained from the electric current distribution
in the source coils and the unknown induced currents in theliquid, related to the total magnetic field and the vector potential
A by the Biot-Savart law15,16):

A (r ) 04

 | r  r ' | d r 'j(r ')(3)

V( j0)Equations (2) and (3) can be solved in the axisymmetric case forharmonic fields as in5).
The induced current in the liquid drop depends on its
instantaneous free surface shape and needs to be recomputed as
the shape changes during oscillation. The electromagnetic forcef, time-averaged over the AC period, can be decomposed in two
parts: f  f AC  fv as in (2). The second, fluid-velocitydependent part fv, can include both DC and AC time-averaged
contributions. The electromagnetic force associated to theelectric current j  (  B / 0 ) can itself be decomposed into
two parts:f  j B
 (  B / 0 )  B  ( B 2 / 20 )  (B)B / 0

(4)

The first term on the right hand side, being a gradient, is the‘magnetic pressure’, which can be incorporated into a modified
pressure function:(1)

Where, j is the electric current density, B the magnetic field,
 is the volumetric magnetic susceptibility (e.g.,  =Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013

given by the magnetic vector potential A, the magnetic fieldB   A , the electric potential φ and the fluid flow induced partpm  p  B2 / 20   B2 / 20(5)That also includes the force on the magnetic material. For adiamagnetic,   0 , the magnetic contribution gives anadditional ‘compression’ to the liquid droplet. However unlike
the induced current component, this magnetic force is notconcentrated in the skin-layer. The magnetic pressure function isused to modify the free surface boundary condition for the
diamagnetic part of the force. The AC field induced part isalways solved explicitly within the fluid volume to account for
both the rotational and the gradient parts in (4).

2.2Computing the Flowfield

To determine the flowfield within a levitated liquid mass, thenumerical model solves the momentum and mass conservation
equations (6), with the modified pressure term defined in (5),which will also include the effect of gravity:− 57 −

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Modeling of EML in Combined AC/DC Magnetic Fields as the Basis for Microgravity Experiments

t V   V   V   1Pmod   e   V  VT    1 J  B .

(6)

V  0Note, that e in (6), represents the effective viscosity, which may
contain the turbulent contribution. The boundary conditions for(6) at the free surface relate normal stress to the surface tension,en  Π  en  K ,(7)

continuity, and the tangential stress conditions:V  0 ,en  Π  e 1  0 , en  Π  e 2  0

In addition to the stirring and levitation forces, the induced
current in the droplet generates heat, through the Joule term qJ =J2/ (W/m3). Heat transfer within the droplet is thencharacterized by the transport equation

4
n  (kT )   T (Tamb
T 4)

3.

Example Simulations

curvature. The free surface shape R(θ,t) is updated at each time
step using the kinematic condition:

3.1

Silicon Droplet in AC/DC Field(9)

The magnetic field in a solenoid is modeled as a superpositionof the magnetic field generated by axisymmetric coil filamentsof finite cross-section (see Fig. 1 as an example). The
electromagnetic body force requires in general an additional
equation to be solved for the electric potential (only in the 3Dcase). The equation for the electric potential is obtained fromthe charge conservation  J  0 . Taking the divergence of theelectric current density, gives:

2E   V  B (10)This equation is solved subject to the condition at the
instantaneous drop surface: J  en  0 .
In most situations, the flowfield in the droplet becomesturbulent. For this reason, a turbulence model needs to be used.
The ‘k-ω’ turbulence model, where k is the kinetic energy ofturbulence and  the turbulent frequency, modified to include
the effects of DC magnetic field damping17) is used,t k  v k   [(   k T )k ]  G   * k  2 m k B 2 / (11)t  v    [(    T ) ]   G / k    2 - m B 2 / with ‘no flux’ boundary conditions prescribed at the surface:

 n k  0,

 n  0.

(12)

Starting with simulations performed under terrestrial
conditions, the following section contains the results ofcomputations for an axisymmetric molten silicon droplet(diamagnetic, electrically conducting). It reproduces the method
reported in Tsukada et al. 13), which uses periodic laser heating
to determine the thermal conductivity of an electromagneticallylevitated droplet. The static DC field is used to suppress
convection. Figure 1 shows the model setup and Fig. 2, theinstantaneous spectral collocation mesh used within the droplet.
Dynamic spectral computations with the code SPHINX werecompared against experimental data from9,13) and in static
situations against the commercial code COMSOL18). Since
under the conditions of the experiment the droplet tends tooscillate along the vertical axis, we examine the sensitivity ofthe measured temperatures to displacement within the levitatingcoil. In this situation, the levitating coil also provides the heatrequired to maintain the droplet in a liquid state. The difference
in stability as the gravity is removed is also examined.
Tsukada et al.13) report results on temperature/fluid flow inthe droplet for the case when the laser is absent.
Assuming the same geometry, material properties and operating
conditions, these results have been closely replicated with
COMSOL.

Fig.1 A typical terrestrial arrangement used for themeasurement of thermal conductivity in a liquid
droplet, with an AC levitating coil positioned in thecore of a solenoid DC magnet

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Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013(14)

Where, T is the Stefan-Boltzmann constant.

to the free surface, Π is the stress tensor and K is the surface

t R  en  V  en

(13)

The last term in (14), represents the latent heat. Heat is lost byradiation at the surface, as given by the boundary condition:(8)

Where, en , eτ 1 , eτ 2 are the unit vectors normal and tangential C p  tT  V .T   . C peT  qJ  qL

Fig. 2 The mesh used in spectral simulations− 58 −

Koulis PERICLEOUS, et al.

The solution in Fig. 3 shows a scalar plot of |B| andstreamlines of magnetic flux. The current is confined to a thinskin and concentrates on the inner surfaces of the coils where
the contours are tightly bunched. The highest values of |B| arefound in the space enclosed by the bottom coil turn which has
the smallest radius and leads to higher flux confinement. The
top two turns of the coil have the opposite sign to the lower set
since they enclose flux which is oppositely directed.

Figure 4 shows scalar plots of vz and temperature in thedroplet when a uniform DC magnetic field of 4T is applied.
The results closely match, qualitatively and quantitatively, those
produced in13). Two circulating vortices are present and thehighest temperatures form a circular plateau in the lower half ofthe droplet.
Tsukada et al.13) report the variation in phase shift according
to the choice of thermal conductivity when the laser frequency
is 0.1 Hz.

These results have been reproduced here, and Fig. 5(left) shows the temperature modulation state at the bottom of
the droplet over one cycle of laser heating with frequency
flaser=0.1Hz

forvarious thermal

conductivities

k=Cp.

Figure 5 (right) shows how the phase shift between thesewaveforms and that of the laser (taken as the reference) depends
on the choice of k. The phase shift becomes less as the valueof k is increased.
Figure 6 shows how the (time-averaged) axial force on thedropletRe( Br J * ) / 2 dV  mg varies according to its z-

Fig. 3 COMSOL model of electromagnetic levitator(f=200 kHz, Ipeak = 375A)

position on the symmetry axis of the coils, with and without the
influence of gravity. The zero-crossings of the curves indicate
positions where the force is balanced.
In earthboundconditions there is one stable and one unstable equilibrium,
whereas in zero-gravity there is one stable and two unstableequilibria. The equilibria are indicated by the hollow (stable)
and solid (unstable) circles. The results in Fig. 6 can be used
to generate phase-space portrait diagrams in (z,vz) spacedescribing the motion when the droplet is displaced off its
equilibrium, by fitting a spline through the graphs and applying
a standard ode solver to evolve the following pair of differential
equationsforwardandbackwards
intime:

dVz Fz ( z )
dz

 g,
 Vz
dt
m
dt
Fig. 4 Computed scalar plots of temperature T andvelocity component Vz, in a spherical droplet.

Fig. 5 Temperature modulation state (left) and phase shift as a function of k (right) when flaser=0.1Hz

Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013

− 59 −

59

Modeling of EML in Combined AC/DC Magnetic Fields as the Basis for Microgravity Experimentsbound. The unstable equilibria are associated with hyperbolic
fixed points whose stable and unstable manifolds are indicatedby the arrows.
The unstable equilibria in zero-gravity are of relevancebecause, irrespective of the presence or absence of gravity, it isat these positions that Joule heating in the droplet approaches
local maxima and where strong coupling occurs. The gradientof the heating curve at the earthbound stable equilibrium isquite high – found to be approximately 8 W/mm - which would
suggest that even small amplitude oscillations could causesignificant fluctuation in the temperature, highlighting theimportance of eliminating axial motion when measuring thephase shift in the temperature oscillation. Since any droplet
displacement also affects the impedance of the levitating coil,variations in power would also affect the measurements4).
The COMSOL simulations presented in the stability analysis
are for a rigid, spherical droplet and the flowfield was assumed
laminar due to the presence of a DC field. However, true
dynamic simulations are needed to account for changes in the
shape of the droplet due to the time-varying forces acting on it,
and turbulence must be included as it will modify the internal
flow and temperature distribution. The spectral code SPHINX
was used for this purpose.
The plots of velocity and temperature (without laser heating

Fig. 6 The axial force on the droplet as a function of z.

or DC field) in Fig. 8 were obtained using a peak current of
530A in the excitation coil. Without the DC magnetic field,
intense turbulent flow develops in the interior of the droplet.
Compared to the plot of temperature in Fig. 4,where the flow
was laminar and assumed damped by the strong DC magnetic
field, turbulence enhances the effective thermal diffusion andreduces the temperature variation throughout the droplet from(a) g=9.81m/s2

12 to just over 3.

The average temperature is about 85C

lower, due to the adjustment in operating conditions that makes
the deformed droplet rest higher in the field and so experiences
lower levels of Joule heating. Fluid velocities of 0.1 m/s are
realized, making a Reynolds number of the order 104. In this
scenario, the droplet assumes the more realistic diamond shapewith dimensions as indicated in Fig. 8.
Now consider what happens when the static magnetic field isgradually increased. At moderate field strengths when Bdc=1T,
the flow intensity actually increases compared to the situation(b) g=0m/s2
Fig. 7 Phase space portraits in earth-bound (top) and zerogravity (bottom) conditions

when Bdc=0 T. This is apparent from Fig. 9 which shows
large-scale circulation velocities up to 0.2 m/s. In this scenario
the mean large-scale flow is readjusted in a pattern along thePhase space portraits for earthbound and zero gravity conditions

vertical DC magnetic field (compare Figs. 8 and 9), theturbulence is damped by the magnetic field and the effective

are given in Fig. 7. The closed curves indicate oscillatory
motion around the stable equilibria, while the dotted curves
represent “fly-by” trajectories where the droplet is temporarilycaught in the field of the coils but does not remain permanently

60

Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013viscosity reduced to a near- laminar value. Compared to Fig. 8,
the temperature variation throughout the droplet has increased
from 3 to 6 - presumably because the shift in the position of
the vortices reduces convective heat transfer to the top of the droplet.

− 60 −

Koulis PERICLEOUS, et al.

7.6mm

Increasing further the strength of the DC field suppresses
more effectively the radial component of velocity and generally
decreases convection. Figure 10 shows the velocity andtemperature field at Bdc=5T where conditions approachinglaminar flow and heat transfer are achieved. Thermo-capillaryeffects become noticeable on the surface of the droplet which
are not easily damped by the action of the vertical DC field lines.
The temperature variation throughout the droplet is now 11

8.8mm

and velocities of around 0.006 m/s occur on the symmetry axis.
This compares well with the earlier results in Fig. 5 andsuggests that, while the average temperature in the droplet islower due to the adjustment in shape and position, the spatial
distribution of T is similar to that obtained for a spherical
droplet.

3.2
Fig. 8 Temperature and velocity, no DC field

Fig. 9 Velocity/temperature field when Bdc= 1 T.

In this case the proposed design for the MSL-EML device isevaluated for microgravity operation. It features two ‘pancake’coils using a dual frequency supply, one to provide heating andthe other to stabilize the suspended charge in flight (Fig. 11) .
Each turn of the top coil carries AC current at a frequency of
146 kHz, Ieff = 139 A (‘+I’ current at the top and ‘-I‘ at thebottom ) – the positioning field. The same coil is also carrying
AC current at a frequency of 370 kHz, Ieff = 0 - 150A all in thesame direction – the heating field. Initially the sphere is solid atroom temperature, heats due to the action of the two fields
melting after 10-11s and the heating current is then reduced toIeff = 50A.
To calculate the stability of this system, the solid copper
sphere was displaced first vertically along the coil axis, and thenradially, with the positioning field on, but no heating field. The
restoring force was calculated and the results given in Figs. 12
and 13. There is good correspondence between laboratory
measurements19) and the COMSOL simulations for thisconfiguration. In both directions the system is stable withdisplacements up to 5mm in either direction, but beyond theselimits it becomes unstable.

Fig. 11 Schematic diagram of the MSL-EML levitator
consisting of one pair of 3-turn pancake coils,
containing a 4mm suspended copper sphere(dimensions in mm)

Fig. 10 Velocity/temperature field when Bdc= 5 T

Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013

A Copper Sphere in the MSL-EML Levitator− 61 −

61

Modeling of EML in Combined AC/DC Magnetic Fields as the Basis for Microgravity Experiments

4.00E-03
COMSOL (Filaments Ipeak=197A, f=146kHz)

3.00E-03

DLR MSL1-HPS test

2.00E-03
1.00E-03
0.00E+00
-10

-5

0

5

10

-1.00E-03-2.00E-03-3.00E-03-4.00E-03

Fig. 12 Computed vertical force on a copper sphere as afunction of vertical displacement from mid-position
when a lifting-coil current of Ipeak=197 A, f=146 kHz
is applied – experimental results by Lohofer19)
8.00E-03

Fig. 14

Steady-state conditions, B=0T, fheat =360kHz, Iheat=
50A, fpos =146kHz, Ipos= 139A

Fig.15

Flow field and contours of turbulent viscosity (left),
contours of temperature (right) – heating field off,
B=0T

Fx (x, Z=0) (COMSOL 3D, filaments, 197A peak)Fx DLR MSL1-HPS test

6.00E-03
4.00E-03
2.00E-03
0.00E+00
0

5

10

15

20

25

30

-2.00E-03-4.00E-03-6.00E-03-8.00E-03

Fig. 13

Computed radial restoring force on copper sphere as afunction of radial displacement from coil axis when alifting-coil current of Ipeak=197 A, f=146 kHz isapplied – experimental results Lohöfer 19)

However, in practice the heating and positioning currents act
together, at least until the sphere has reached the required
temperature and once the sphere has melted, it no longer
remains perfectly spherical. Since the heating current flows unidirectionally in both coils, what is the effect on stability?
To answer these questions, the problem was simulated
dynamically using the spectral code SPHINX. Figure 14 first,
shows the resulting flow vectors and temperature contours in theliquid copper sphere with both positioning and heating fields on.
The strong convection, coupled with the excellent thermal
conductivity of copper, keep the temperature almost uniform,
although the influence of the heating field can still be seenaround the droplet equator. The flow exhibits toroidal vortices
that ‘pump’ the fluid towards the poles, then returning as a faststream within the thin skin layers placed directly facing the coils.
The shape of the droplet becomes elliptic, stretched slightly
along the axis.
In contrast, when the heating field is switched off, thespherical shape returns, but the flowfield (Fig.15) looks

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Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013

completely different. The number of toroidal vortices has now
doubled, with strong jets flowing towards the center from midlatitude positions and returning to the surface along the poles
and the equatorial plane. The temperature is still fairly uniform,
although the effects of radiation cooling down slightly thesurface of the sphere are now apparent. There is someturbulence also generated along the surface due to shear in theskin layers; however the value of the effective viscosity remains
quite low – encouraging for viscosity measurements.
When a DC field of 1T is superimposed onto the positioning
field (Fig.16), then the flow is damped (note the scale of vectors
is reduced from 0.03m/s to 0.015m/s), with only the equatorial
vortices still maintaining some strength. At the same time, theviscosity drops to the laminar value, and the temperature in thedroplet remains uniform to within 0.1oC. Increasing the DC
field even further to 5T (Fig. 17) -the value recommended
above for terrestrial measurements, leads to a total suppressionof convection within the droplet, although fast jets appear along
the surface in regions of high temperature gradient, due tothermo-capillary action.

− 62 −

Koulis PERICLEOUS, et al.

Fig. 18
Fig. 16

Flow field and contours of turbulent viscosity (left),
contours of temperature (right), with DC field addedat B=1T and heating field off.

Flow field and contours of turbulent viscosity (left),
contours of temperature (right), at g=0, with DC field
added at B=1T and heating field off.

the droplet remains nearly spherical, whilst under full gravity it
becomes egg-shaped due to the higher levitation force needed.

References
1)
2)
3)
4)
5)
6)
7)

Fig. 17

Time history of the top polar point of the droplet as
the DC field increases from 0T to 5T

It is not just the internal flow that is affected by the DC
magnetic field, but also the motion of the droplet center of mass
within the coils and its surface oscillations. Plotting the top
polar point position against time, we can observe in Fig. 18 thatas the magnetic field strength increases from 0T to 1T, themotion of the droplet surface is rapidly damped in microgravity.

4.

9)
10)

11)
12)
13)

Conclusions

This brief review demonstrates the use of modeling to
identify the role of static magnetic fields in the EM levitation of
metallic liquid droplet, both in microgravity and underterrestrial conditions. It clear that an imposed DC field has an
immediate effect on internal flow structure. At low level (0-1T)
it is turbulence that is suppressed first, together with large
vortical structures. As the field strength increases, the flow isdamped altogether and the droplet oscillates as a solid body. A
field of 5T was needed to suppress surface oscillations in fullgravity, but 1T was sufficient in microgravity. In microgravity

Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 20138)

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