# Levitated Liquid Dynamics in Reduced Gravity and Gravity-Compensating Magnetic Fields

## Abstract

Dynamic models are used to investigate the behavior of liquid droplets suspended in alternating current and direct current magnetic fields in microgravity and in various configurations providing conditions similar to microgravity. The realistic magnetic fields of solenoidal coils are used for the modeling experiments with electrically conducting (liquid silicon or metal) droplets. At high values of magnetic field, some oscillation modes are damped quickly, while others are modified with a considerable shift of the oscillating droplet frequencies and the damping constants from the nonmagnetic case. On a larger scale, the models are used to investigate the melting and heating process of reactive materials. It is demonstrated how 1 kg of liquid titanium in a traditional “cold” crucible-type furnace can be fully levitated without contact to wall to achieve high superheat of the melt.

## Introduction

From the electromagnetic (EM) levitation experiments1 in an alternating current (AC) magnetic field, it became apparent that the levitated liquid metal is prone to oscillation and instability. The EM and electrostatic levitation experiments with liquid metal droplets show difficulties related to confinement stability and a need for complex correction functions to establish a correlation between the measurements and the droplet material properties.2,3 The observed intense internal fluid flow is in the turbulent regime for earthbound conditions. A combination of AC and direct current (DC) magnetic fields was recently recognized as efficient tool for the magnetically stabilized treatment of molten substances without a contact to contaminating walls.4,5 The intense AC magnetic field required to produce levitation results in turbulent large-scale toroidal recirculation, while the use of a homogenous DC magnetic field, allows the toroidal flow to be damped. However, the magnetically modified turbulence could make the effective viscosity behave in a nonlinear fashion depending on the DC and AC magnetic field intensity.6 The flow in typical melting conditions is approaching the conditions with laminar viscosity and heat transfer when a uniform DC magnetic field exceeds about 4–5 T. In addition to the modified damping, the electrically conducting liquid droplet in a high DC magnetic field behaves quite differently to the nonconducting one: some oscillation modes are damped completely, and other modes are persisting for long times or damped moderately.7

On a larger scale, the magnetic suspension of liquid reactive materials, for example titanium alloys, can be used for high-quality castings. However, it is often difficult to achieve the required superheat in the melt with traditional “cold” crucible-type furnaces due to a partial contact with the water-cooled copper walls.8 If the contact is avoided, then thermal losses would be limited only by radiation and possible evaporation. This would produce a higher superheat and permit investigation of materials at extreme temperatures. At the other extreme, a highly undercooled liquid can be obtained before solidification to a glassy structure in the levitated conditions in absence of nucleation centres. Existing experimental evidence suggests that it is possible to melt and levitate several kilograms of liquid metal.9 In numerical experiments,10 it is demonstrated that full levitation of the liquid metal is achievable but requires careful optimization of the EM force to generate intense tangential flow along the surface away from the bottom stagnation point.

## Magnetic Force Analysis

**j**is the electric current density,

**B**is the magnetic field, χ is the volumetric magnetic susceptibility (e.g., χ = −4.2 × 10

^{−6}for silicon, which is diamagnetic), and

*μ*

_{0}is the magnetic permeability of vacuum in SI units. When a combination of high-frequency AC and DC magnetic fields is used, a time average of the force over the field oscillation period τ (~10

^{−4}s to 10

^{−6}s) is effective at the time scale of typical levitated droplet oscillation period 10

^{0}–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 at double the frequency of the applied AC, in addition to the time-averaged force.

**A**, the magnetic field \( {\mathbf{B}} = \nabla \times {\mathbf{A}} \), the electric potential φ, and the fluid flow-induced part σ

**v**×

**B**.

**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 the liquid related to the total magnetic field and the vector potential

**A**by the Biot-Savart law.12,13

**f**, time-averaged over the AC period, similarly to Eq. 2, can be decomposed in two parts:

**f**=

**f**

_{AC}+

**f**

_{v}. The second, fluid velocity dependent part of the force

**f**

_{v}can include DC and AC time-averaged contributions. Even in a pure AC field, in principle, there is a time-averaged interaction with the more slowly varying velocity field.

**j**= (∇ ×

**B**/μ

_{0}) can be decomposed into the following two parts:

## The Need for Reduced Gravity

In electromagnetic levitation (EML) with AC, the same forces that support the sample against gravity also stir the molten sample. As discussed in the next section, the flows in 1-g are invariably turbulent. The larger, kilogram-scale industrial samples have even higher turbulence intensity. However, in low gravity, the weaker positioning forces allow laminar flow for some conditions.

In two-frequency EMLs such as TEMPUS,14 or material science laboratory electromagnetic levitator (MSL-EML),15 the levitation field is the superposition of two components, each optimized for a different purpose. The positioning field has a large gradient but a low magnetic field. This combination results in a sufficient force with minimal heating of the sample. In contrast, the heating field has a very small gradient, so it produces almost no net force in the axial direction, although it does compress the equator of the sample. The heating field has a larger average magnetic field strength, resulting in larger induced currents and more Joule heating than the positioning field. The heating field also provides much more stirring than the positioning field for comparable currents.

Some of the advantages of reduced-gravity EML are available in the hybrid AC + DC pure DC levitation systems described in the following sections. The hybrid system superimposes a static magnetic field up to about 5 T, greatly reducing fluid flow and forcing the flow to a laminar state. Such systems have been demonstrated to enable oscillation calorimetry to measure not only the specific heat but also thermal conductivity of levitated samples,4 otherwise feasible only in microgravity EML. On the other hand, as described below, the surface oscillations of drops in a hybrid levitator are very different than the free oscillations used to measure surface tension and viscosity. With a large applied DC field, the damping is primarily determined not by viscosity but by the interactions between the conducting sample and the applied field. The pure DC levitation system has no flow driven by the positioning field, but it requires a very large magnetic field, often more than 15 T.

For experiments on solidification, nucleation, and phase selection, other complications from hybrid and DC levitation systems also come into play. While both systems can provide laminar flow, the utility of that situation depends on the particular materials system under study. A large applied magnetic field has been shown20 to shift phase equilibria and transformation kinetics in at least some systems. Such shifts can be problematic for extrapolating the results back to zero-field conditions.

## Transition to Turbulence in EML

Some classes of experiments, e.g., nucleation statistics in pure metals,21 may be unaffected by the internal flow in the samples. However, for other experiments described in the previous section, flow is an essential parameter. There even exist experiments in between, where the flow does not affect the experiment. For example, so long as the flow does not introduce additional damping of surface oscillations, the oscillation experiments are insensitive to the magnetohydrodynamic flow. In this case, it is important to keep the conditions far from the critical Reynolds number *Re*_{c}.

Calculations show that flow in 1-g is always turbulent because of the strong forces needed to support a sample against gravity. In microgravity, both laminar and turbulent flows are possible. Experiments on an 8-mm diameter sample of Pd_{82}Si_{18} used tracer particles to show the state of the flow.22 An example video is available online.23 The sample was heated under constant magnetic conditions. As the temperature increased, the viscosity decreased, increasing the velocity of the internal flow quasi-statically. The internal flow was calculated by a method similar to that described above to estimate the velocity as a function of temperature and time. As the sample melts, the tracers collect in the stagnation line at the equator. At a calculated Reynolds number of about 525, the flow becomes unsteady, with oscillations growing in magnitude and becoming nonlinear as the flow accelerates. Finally, the flow becomes chaotic at a calculated Reynolds number of about 600. These flows are just above the transition from laminar flow and are neither isotropic nor fully developed. However, these flows are chaotic, show enhanced mixing, and definitely not laminar. For still faster flows, the turbulence continues developing; cf. Reynolds number of 10^{4} for the silicon droplet later in this article. Further observations of flow transitions in microgravity EML will enable refinement of the transitional Reynolds number.

## Results for Liquid Silicon Droplet in Solenoid Magnets

The following section gives some examples demonstrating (I) the dynamics of molten silicon droplet (diamagnetic and electrically conducting) when levitated in the high-frequency AC magnetic field and (II) the effect of adding the DC magnetic field in attempts to damp the surface oscillation and the internal turbulent velocity field. The pseudo-spectral code SPHINX6,8,10 is used to obtain the coupled EM and computational fluid dynamics solutions accounting for the magnetic force excited turbulent flow. The solution includes the turbulence damping by the high magnetic field both for the large-scale flow and for the small-scale turbulent properties.8

### AC + DC Field Levitation

*R*

_{0}= 5 mm). The EM force from the interaction with the AC coil supports the droplet against gravity and deforms the shape due to the magnetic force distribution. The coil geometry (Fig. 2) is similar to the previous experimental and numerical investigation,4 where the finite-difference numerical model was applied for a fixed-shape (nonoscillating) spherical droplet and compared to the temperature field change when additional DC magnetic field was introduced. The solution in Ref. 4 was obtained only for the static magnetic field exceeding 4 T, when the flow is sufficiently damped by the action of the magnetic field, and it did not permit to compute the flow at zero or moderate DC magnetic field due to the high-intensity internal flow in the droplet.

^{4}. This flow is mildly turbulent with the effective turbulent viscosity and the thermal diffusion enhanced by the action of the turbulence. Figure 4 demonstrates the effective viscosity distribution as computed from the

*k*–

*ω*model turbulence model.8 Figure 4 shows the violent oscillation pattern observable on the droplet surface

*R*

_{t}(top position) and the dominating oscillation up and down of the center of mass

*Z*

_{t}for the droplet as a whole. The solution accounts for the continuous adjustment of the EM field in the liquid during the development of these oscillations. Clearly, this type of liquid flow and oscillation act detrimentally to any attempts to measure material properties of the levitated liquid sample.

*B*

_{z}= 4 T, the flow intensity decreases and the turbulent viscosity is significantly reduced. The droplet oscillation is quickly damped at this magnitude of the DC magnetic field. The flow is not completely laminar.

### DC Field Levitation

*g*= 0) and a uniform vertical magnetic field, the asymptotic solution7 shows that odd axisymmetric oscillation modes are very moderately damped and the frequencies are reduced significantly. It is not immediately clear if this behavior will stay in the presence of terrestrial gravity and the gradient magnetic field required for the magnetic levitation. The numerical experiments with liquid silicon appear to support the general conclusions of the asymptotic solution as demonstrated by Fig. 7. The

*L*= 3 mode (see the velocity field in Fig. 7) is not damped immediately in the 17 T magnetic field required to levitate the 5 mm radius liquid silicon droplet. The frequency of the

*L*= 3 mode is reduced about √3 times relative to the nonmagnetic case and closely matches the theoretical value.7 The flow is purely laminar as can be seen from the lack of mixing that gives the radiation-cooling-dominated temperature distribution in Fig. 7.

## Liquid Titanium Melt in Cold Crucible

The mechanism of the levitation, and particularly the local magnetic support at the bottom, appears to be dynamic in nature.10 The explanation for the fact that the liquid at the bottom is prevented from leaking and flowing down, is related to the particular velocity field in this region. The bottom vortex in Fig. 8 is maintained by the rotational nature of the EM force \( \left( {curl\;{\mathbf{f}}_{{\mathbf{e}}} \ne 0} \right) \), which drives the fluid tangentially upwards at the side surface of the liquid, away from the bottom stagnation point. Due to the continuity of the velocity field, the outflow at the bottom is redirected to the intense flow upwards along the side surface.

In the levitated conditions, the temperature of liquid metal rises relative to the case when there is a contact to the water-cooled copper crucible. The high heat exchange rates with the intense turbulence at the crucible wall are responsible for the temperature drop in a conventional crucible. The superheat relative to the titanium melting temperature (1667°C) can easily exceed 100° if the metal is fully levitated as shown in Fig. 8.

## Conclusions

Numerical models give useful insight to the levitated liquid droplet dynamics at various combinations of magnetic fields in 1-g, reduced gravity, and a gravity-compensating force field. The more familiar EML in a high-frequency AC field can be improved by adding uniform DC magnetic field of an optimum magnitude to achieve nearly laminar flow conditions required for material property measurements. A high-intensity gradient DC field offers even better conditions to levitate diamagnetic liquid substances (conducting or nonconducting). A large volume of liquid metal can be fully levitated without contact to the wall to achieve high superheat of the melt in a specially designed, traditional “cold” crucible-type furnace.

### References

- 1.E. Okress, D. Wroughton, G. Comenetz, P. Brace, and J. Kelly,
*J. Appl. Phys.*23, 545 (1952).CrossRefGoogle Scholar - 2.I. Egry, G. Lohofer, I. Seyhan, S. Schneider, and B. Feuerbacher,
*Int. J. Thermophys.*20, 1005 (1999).CrossRefGoogle Scholar - 3.D.L. Cummings and D.A. Blackburn,
*J. Fluid Mech.*224, 395 (1991).MATHCrossRefGoogle Scholar - 4.T. Tsukada, H. Fukuyama, and H. Kobatake,
*Int. J. Heat Mass Transf.*50, 3054 (2007).MATHCrossRefGoogle Scholar - 5.H. Yasuda,
*Solidification of Containerless Undercooled Melts*, ed. D.M. Herlach and D.M. Matson (Berlin, Germany: Wiley-VCH Verlag, 2012), pp. 305–320.CrossRefGoogle Scholar - 6.V. Bojarevics and K. Pericleous,
*CFD Modeling and Simulation in Materials*, ed. L. Nastac, L. Zhang, B.G. Thomas, A. Sabau, N. El-Kaddah, A.C. Powell, and H. Combeau (Warrendale, PA: TMS, 2012), pp. 245–252.Google Scholar - 7.J. Priede,
*J. Fluid Mech.*671, 399 (2010).MathSciNetCrossRefGoogle Scholar - 8.V. Bojarevics, R.A. Harding, K. Pericleous, and M. Wickins,
*Metall. Mater. Trans. B*35B, 785 (2004).CrossRefGoogle Scholar - 9.H. Tadano, M. Fujita, T. Take, K. Nagamatsu, and A. Fukuzawa,
*IEEE Trans. Magn.*30, 4740 (1994).CrossRefGoogle Scholar - 10.V. Bojarevics, A. Roy, and K. Pericleous,
*Magnetohydrodynamics*46, 317 (2010).Google Scholar - 11.V. Bojarevics and K. Pericleous,
*Fluid Mechanics and Its Applications*, Vol. 80, ed. S. Molokov, R. Moreau, and H.K. Moffatt (New York: Springer, 2007), pp. 357–374.Google Scholar - 12.L.D. Landau and E.M. Lifshitz,
*Electrodynamics of Continuous Media*, 2nd ed. (Oxford, UK: Pergamon Press, 1984).Google Scholar - 13.W.R. Smythe,
*Static and Dynamic Electricity*(London, UK: McGraw-Hill, 1950).Google Scholar - 14.R. Knauf, J. Piller, A. Seider, M. Stauber, and U. Zell,
*Proceedings of the 6th International Symposium Experimental Methods for Microgravity Materials Science*, ed. R.A. Schiffman and J.B. Andrews (Warrendale, PA: TMS, 1994), pp. 43–51.Google Scholar - 15.G. Lohoefer and J. Piller (Paper presented at the 40th AIAA Aerospace Sciences Meeting & Exhibit, AIAA 2002-0764, Monterey, CA, 2002).Google Scholar
- 16.R.W. Hyers and G. Trapaga,
*Solidification 1999*, ed. S.P. Marsh, N.B. Singh, P.W. Voorhees, and W.H. Hofmeister (Warrendale, PA: TMS, 1999), pp. 23–32.Google Scholar - 17.D.M. Matson, R.W. Hyers, T. Volkmann, and H.-J. Fecht,
*J. Phys. Conf. Ser.*327, 012009 (2011).CrossRefGoogle Scholar - 18.M.C. Flemings, D.M. Matson, W. Löser, R. Hyers, and J. Rogers,
*Science**Requirements Document (SRD) for Levitation Observation of Dendrite Evolution in Steel Ternary Alloy Rapid Solidification (LODESTARS)*, NASA Document LODESTARS-RQMT-0001 (2003).Google Scholar - 19.A.K. Gangopadhyay, R.W. Hyers, and K.F. Kelton,
*JOM*(2012). doi:10.1007/s11837-012-0422-1. - 20.G.M. Ludtka,
*Exploring Ultrahigh Magnetic Field Processing of Materials for Developing Customized Microstructures and Enhanced Performance*, Final Technical Report (Oak Ridge, TN: Oak Ridge National Laboratory, 2005).Google Scholar - 21.W.H. Hofmeister, C.M. Morton, R.J. Bayuzick, and M.B. Robinson,
*Solidification 1999*, ed. S.P. Marsh, N.B. Singh, P.W. Voorhees, and W.H. Hofmeister (Warrendale, PA: TMS, 1999), pp. 75–82.Google Scholar - 22.R.W. Hyers, G. Trapaga, and B. Abedian,
*Metall. Mater. Trans. B*34, 29 (2003).CrossRefGoogle Scholar - 23.R.W. Hyers,
*Meas. Sci. Technol.*16, 394 (2005). doi:10.1088/0957-0233/16/2/010.CrossRefGoogle Scholar