A.D. Sneyd and H.K. Moffatt- Fluid dynamical aspects of the levitation-melting process

8/3/2019 A.D. Sneyd and H.K. Moffatt- Fluid dynamical aspects of the levitation-melting process

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,J . Fluid Meck. (1982), vol.Printed i n Great Brita in

117 , pp. 45-70

Fluid dynamical aspects of the levitation-melting processBy A. D. S N E Y DUniversity of Waikato, Hamilton, New Zealand

A N D H. K. MOFFATTDepartment of Applied Mathematics and Theoretical Physics,Silver Street, Cambridge

(Received 28 May 1981)When a piece of metal is placed above a coil carrying a high frequency current, theinduced surface currents in the metal can provide a Lorentz force which can sup-port it against gravity; a t the same time the heat produced by Joule dissipation canmelt the metal. This is the process of levitation melting, which is a well-establishedtechnique in fundamental work-in physical and chemical metallurgy. Most theoreticalstudies of magnetic levitation have dealt only with solid conductors and sohaveavoidedthe interesting questions of interaction between the free surface, the magnetic fieldand the internal flow. These fluid dynamical aspects of the process are studied in thispaper.A particular configuration that is studied in detail is a cylinder levitated by twoequal parallel currents in phase; this is conceived as part of a toroidal configurationwhich avoids a difficulty of conventional configurations, viz the leakage of fluidthrough the magnetic hole a t a point on the metal surface where the surfacetangential magnetic field vanishes. The equilibrium and stability of the solid circularcylinder is first considered; then the dynamics of the surface film when meltingbegins; then the equilibrium shape of the fully melted body (analysed by means of ageneral variational principle proved in $5); and finally the dynamics of the interiorflow, which, as argued in 5 2, is likely to be turbulent when the levitated mass is of theorder of a few grams or greater.

1. IntroductionIf a piece of metal is placed in an alternating magnetic field, electric currents will be

induced, interacting with the magnetic field to produce a Lorentz force which cansup-port the metal against gravi ty. A t the same time, the ohmic heating due to the inducedcurrents can cause melting and result in a blob of liquid metal levitated by the appliedfield. This process, first suggested by Muck (1923) and reviewedt by Peifer (1965),hasseveral advantages over the usual method of crucible melting: most obvious amongthese is that the liquid metal does no t come into contact with a crucible wall so there isno danger of contamination, particularly by carbon and sulphur (in many metallurgicalexperiments see, for example, El-Kaddah & Robertson 1978 it is important to pre-pare very pure specimens as even small traces of impurities can affect the physical

t An excellent, although somewhat inaccessible, review of the literature up to 1975 isprovided by Stephan (1975).

www.moffatt.tc


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46 A . D . Sneyd and H . K . MoffattAxis o fsymmetry

Levitated solidbefore melting

Levitated molten

- 2 c d

( 0 ) ( bFIQURE. (a) ypical axisymmetric levitation device (after Okress et al. 1952);( b ) dealised two-dimensional levitation device with parallel wires at P and Q carrying equal currents I cos w t , andimage currents - I cos wt at the image pointsP, .properties); also the stirring of the liquid metal by the Lorentz force is an efficientmethod of mixing different metals in the production of alloys.

Levitation melting has been achieved by many experimenters, for example, Okresset al. (1952).A typical levitation device is sketched in figure 1a ; he lower coil providesthe main levitation force and the upper coil helps stabilise the metal against horizontaldisplacements. The frequency of the alternating current passed through the coils istypically of order 104 o 105Hz and a t such high frequencies the metal behaves as aperfect conductor, confining the field penetration to a thin surface layer; the metal is ineffect supported by the magnetic pressure distribution over its surface.

Such levitation devices suffer from a number of problems. Firstly, in the configur-ation of figure 1a , the (approximately tangential) magnetic field on the metal surfacevanishes a t the highest and lowest points. At the lowest point the metal will tend toleak through the magnetic hole, and in some experiments the lower end becomeselongated and drips.? This is generally the crucial factor in limiting the mam of liquidmetal that can be levitated. The levitated metal is also subject to a number of in-stabilities, and may rotate or vibrate rapidly. I n spite of these difficulties, levitationmelting has been successful for masses of order 100 g or more of various metals, andseveral commercial devices are available.

Some metallurgical processes also involve what might be called partial magneticlevitation -parts of the surface of a body of liquid metal being supported by magneticfield and others resting on solid supports. For example in the continuous casting ofaluminium, the vertical walls of a column of the liquid metal are supported magneti-cally, while the base of the column rests on a solid ingotwhichisgraduallymoveddown-wards as more metal solidifies a t the interface (Moreau 1980).

t This leakage is of course inhibited to some extent by surface tension. Moreover, varioustechniques have been tried, involving the use of two sets of coils carrying currents in timequadrature, for plugging the leak in a time-averagedmanner (Zhezherin 1959).


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Fluid dynamical aspects of the levitation-melting process 47Theoretical studies of levitation have so far dealt mainly with solid conductors.?

Piggot & Nix ( 1966) have analyzed levitation of an infinitely long solid circular cylinderby a pair of equal and opposite alternating line currents, calculating stability bound-aries and the rate of heating, and comparing theoretical and experimental results. Itwas found that (as is in fact generally true) the levitation force increases with th efrequency, eventually tending to the perfectly conducting limit, and tha t the rate ofheating increases indefinitely with frequency. Brisley & Thornton ( 1963) have carriedout similar calculations for a solid sphere levitated by a number of coaxial circular coils.Harris & Stephan (1975 ) (see also Stephan 1975) have carried out a wide range ofexperiments on complete and partial levitation using sodium surrounded by a siliconeoil, the metal having a low melting point and the buoyancy of the oil reducing thenecessary levitation force. Some of these experiments illustrate a dramatic folding instability of the metal surface, which has also been discussed by Zhezherin (1959).Volkov (1962)has analyzed the stability of a plane layer of liquid metal supported by arapidly travelling magnetic field and has found that the fastest growing instabilitiesare those which extend along the magnetic field lines (i.e. the crests of the surfaceripples running parallel to the magnetic field) but that these can be stabilised bysufficiently strong surface tension or magnetic field.

Magnetic levitation of liquid metals presents three interacting problems -thedetermination of the magnetic field, the unknown free surface shape, and the internalfluid motion. Most previous theoretical work -in particular the studies of solid levi-tation -has avoided the difficultyof the unknown freesurface, and the closestapproachto solving all three coupled problems seems to have been the paper by Volkov (1962))but even here only linearised departures from a simple equilibrium are involved. I nthe present paper we make some at tempts at solving the coupled problems but only insituations where one of the three effects can be neglected. For example, we may s tudythe coupling between the free surface shape and the magnetic field under the assump-tion that internal fluid motion can be neglected, or we may calculate the internal fluidmotion under the assumption that the free surface shape is predetermined by strongsurface tension. We consider in detail only two-dimensional problems, and in par-ticular, levitation of a n infinitely long cylinder by two parallel line currents in phase(figure 1b ) .This geometry1 (which can be thought of as approximating a torus of largemajor radius) eliminates the difficulty due to the neutral magnetic field point whichmust occur a t the bottom of a simply connected liquid-metal body in an axisymmetricconfiguration, and also enables us to use conformal transformation. We assumethroughout that the alternating frequency is large, so that the magnetic field in theconductor is confined to a thin surface layer. The field lines outside the conductor areas sketched in the figure.

Section 2 uses the approximate formulae of Sneyd ( 1979) to show tha t the effect ofthe Lorentz force can be thought of as a magnetic pressure (which provides the levi-tation force) and a surface source of vorticity (which will generate internal fluid motion).Estimatesof the strength of the internal flow are also given and it is concluded th at thisflow is likely to be turbulent in situations of interest. As a prelude to the study of fluid

t A parallel study to that reported here has been carried out by Mestel (1982) and cross-reference to this study will be made at appropriate points in the text.1 The suggestion that a cylindrical configuration may be advantageous is not new; it has beenexploited in the so-called boat crucible which levitates a cylinder of finite length (Peifer 1965).


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48 A . D . Sneyd a,nd H . K . Moffattlevitation, $ 3 examines the levitation of a solid circular cylinder by parallel linecurrents in phase, calculating stability boundaries, and $ 4examines the surface flowthat develops during the initial stages of melting. Section 5 develops the general theoryof fluid levitation when internal flow can be neglected, and it is shown th a t equilibriumcan be determined by means of a variational principle involving gravitation, surfacetension and magnetic energies. This principle is used to determine the shape for thespecial case of the cylindrical geometry. Finally, in $6, the dynamics of the turbulentflow within the levitatkd sample is considered; a uniform eddy viscosity is assumed,and a simple low Reynolds number analysis is used to give a first indication of thestructure of the mean flow.

2. Order-of-magnitude considerationsA number of metals that have been melted in the levitated sta te (Peifer 1965) are

listed in table la , together with some relevant physical properties (Smithells 1967).The masses levitated in useful contexts are generally in the range l-lOOg, and thetypical span L of the levitated drop is generally in the range 5-30 mm.

We suppose that currents in the external coils produce a magnetic fieldB = Re (B(x) iot). (2.1)

6 = (2wp,u)-4 (2.2)For large w, this penetrates a small distance O(S) into the drop where

where U is the electrical conductivity of the drop, and ,U = 4n x lO-' (SI units). Wesuppose that64L. (2.3)

Table l b includes values of S for w/2n = lOSHz, and it will be clear that (2.3) isnormally satisfied at frequencies of this order of magnitude.

Under condition (2.3), he field B(x) n (2.1)is determined by t he equationsVAB= U J(x), V . B = 0, (2.4)

where Re (J(x) iut) represents the current that is concentrated in the external coils;the boundary conditions (to eading order in SlL) are

B . n = O on S , B = O ( r 3 ) a t CO, (2.5)where S is the surface of the drop, B is clearly uniquely determined. If B = B, on X(where B, is a tangential field) the mean magnetic pressure on S is

P,, = (4,UrPB,.% (2.6)and, as indicated in Q 1 , i t is essentially this pressure field which must support thesample. I n order of magnitude, we must have

P M L 2 mg, (2.7)wherem N L3p is the mass of the sample; equivalently, if B, is a typical magnitude ofIB,l, then, from (2.6)and (2.7)

B,/(POP)t- gL)4 (2.8)


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Fluid dynamical aspects of the levitation-meltingprocess 49T,,, 10-3p 10ea A 1 0 5 ~ I O ~ V io-ec, 10-5~, YMetal "C kg/m3 0-lm-l ma/s ma/s m2/s J/kg/"C J/kg N/"

A1 660 2.37 5.00 0.159 3.58 1.9 10.84 3.88 0.915cu 1083 8.24 4.74 0.168 3.23 0.55 4.94 2.04 0.135Ga 29.8 6.10 3.87 0.206 1.35 0.33 4.08 0.80 0 - 735In 157 5.03 3.02 0.264 2-17 0.24 2.74 0.28 0.559Li 180 0.508 4-17 0.190 2.14 1.9 42.3 4.16 0.398Pb 327 10.6 1.05 0.758 1-01 0.25 1.52 0.23 0.480TABLEa. Physical properties of some liquid metals at (or just above) the melting point Tm(from Smithells 1967, converted to SI units); p = density, U = conductivity, A = (,u,,u)-',K = thermal diffusivity, v = kinematic viscosity, c, = specific heat, L, = latent heat of fusion,y = surfacetension.

Metal mrn T J/m2/s mm/s 5 8 8 mmA1 0.5 0.02 7 . 5 ~04 0.08 230 2.8 125 1 . 9 8 ~0-' 2.6cu 0.5 0.03 2 . 7 ~06 0.16 170 3.1 62.5 4 . 5 ~ 0-' 61Ga 0-6 0.03 2 . 2 ~@ 0.45 3-4 7.4 22 5.2 x 10-2 8 - 3Li 0.6 0.008 1 . 8 ~04 0.08 14 4.7 125 5 -2 x 1OWa 1 - 3Pb 1.1 0.04 7 . 3 ~05 3.0 7 9.9 3.3 9.1 x 10-2 22In 0- 0.03 2.9 x 106 1.5 11 4.6 6.6 6 . 9 ~O-' 13

TABLE b . Deduced orders of magnitude for levitation of the liquid metals of table 1a,with w = 211 x 105 Hz, L = 10 mm.

i.e. the Alfvbn velocity must be of the same order as the 'free-fall' velocity (gL)$flevitation is to occur.Let us now suppose that the flow inside the drop is characterised by a typical

velocity uo.We may then expect dynamic pressure variations of order pu;, and anupper limit on uo s given by the condition

(2.9)since otherwise centrifugal 'forces would lead to fragmentation of the drop. The mag-netic Reynolds number RM = p0uuoL then satisfies

pui 5 Z) M -PO,RN 5 P o 4 W L , (2.10)

i.e. RAI5 0.1 for the fluids of table 1 with L 5 30 mm. This means that the field B andcurrent j in the fluid volume V may be calculated neglecting the fluid motion, i.e. as fthe sample were solid.

The Lorentz force in the surface layer has a mean par t-F = @e(j* A B), (2.11)and an oscillating part of frequency 2w. The latter has a negligible effect when w islarge, the response being limited by fluid inertia (aided by viscosity). We can thereforefocus attention on the mean part (2.11). Boundary-layer methods (Sneyd 1979,particularly equations (2.17), (2.19))show tha t, t o leading order in 8/L,-F = 2&1pMe-W8n, (2.12)

V A F = - 2 6 4 ( n A Qp,) e-2wJ8, (2,13)


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50 A . D . Sneyd and H . K . Moffuttwhere w is the distance from S in the direction of the inward normal n. ntegrating(2.12) and (2 .13) across the magnetic boundary layer shows that the net effect of theLorentz force is to provide (i) a surface pressure distribution pL,f as expected) and(ii) a surface source of vorticity - A VpJr.The surface pressure controls the shape ofthe sample and provides the levitating force, while the surface source of vorticitygenerates a rotational flow within the sample.Let us first obtain some orders of magnitude that would follow from an assumptionthat the velocity field u(x) s laminar and steady; it is purely meridional in the axi-symmetric configuration o f figure 1a , and two-dimensional in the configuration offigure 1b ; in either case the streamlines are closed. The Navier-Stokes equations maybe written in the form

w A U + V ( $ U + p / p + g . X ) = p-l~-VvAO, (2.14)where w = V A U, nd integration round any closed streamline C gives

(2 .15)where A is a surface spanning C. This equation provides an estimate for uo; or supposethat C passes partly through the magnetic boundary layer and part ly through thefluid core, and let wo- uo/Lbe a typical value of IwI ; hen within the surface layer?

Iv Awl W O / & (2 .16)(whereas in the core region, IV A wI - w o / L ) , nd (2.15) provides the estimateP W O l S ) L (PO 9-l (BtIL)6L,or equivalently

u0N w0L - B;G/pvp,- LS/v .The corresponding Reynolds number is

(2.17)

R = uoL/v- gL28/v2, (2.18)and this is generally very large for L N 10 mm or greater. With v - 10-6niz/s (seetable 1) and with L N 10 mm, 8- 1 mm, we findR - 106!Flow a t such large Reynoldsnumbers is of course likely to be unstable; the assumption of steady laminar flow there-fore appears to be untenable.

Suppose then th at the interior flow is turbulent, with bot,h mean an d fluctuatingparts characterised by velocity scale uo(which is of course no longer given by (2 .17) ) .The force will then be balanced primarily by the gradient of Reynolds stresses oforder put; (2 .15) is then replaced by

(2.19)t This estimate should be contrasted with the corresponding situation in a layer on a rigidsurface, within which I V A U - uO/S,V A w 1 - U , /@ - w oLIP. Mestel(l981) ha: arrived atthe same estimate (2.17)through consideration of the rate of work of the force field F.


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Fluid dynam ical aspects of the levitation-melting process 51

---f cos wt

( a ( bFIQURE. (a) lan view, ( b )cross-section, of levitated torus.

where U is the turbulent velocity; this gives an estimate pu: - Bg/,uo,or using ( 2 . 8 ) ,U0 - (gL)*. ( 2 . 2 0 )This estimate is much less than the estimate (2 .17) ;it is moreover (just)consistent withthe limit (2 .9 ) set by the requirement tha t dynamic pressures be contained by themagnetic pressure a t the surface. It seems therefore th at turbulence limits the level ofinternal velocity to the ree-fall scale, which is in any case maximal for containmentpurposes. The associated Reynolds number is R - 3000 (again using the data ofTable 1 ) .

As regards modelling the turbulence in this complicated context, i t seems unlikelythat one can do better (in the first instance) than assume a uniform eddy viscosityv, - uo , the associated Reynolds number R , = uoL f v, being then (by definition) oforder unity. A low Reynolds nuniber analysis of the mean flow should then give aqualitatively correct description, and this is the procedure we shall adopt later (see 5 6).

The possible presence of turbulence within the levitated drop is of great practicalimportance, as it will play a crucial role in mixing melt constituents and yielding ahomogeneous product -one of the frequently quoted merits of the levitation-meltingprocess (see, for example, Peifer 1965) . Experimental evidence for the presence ofturbulence is limited to observations of random ripples on the surface of levitatedsamples (Block & Theissen 1971 ; Stephan 1975); i t seems likely th at such ripples aresimple surface perturbations associated with sub-surface turbulence, which cannot bedirectly detected.

3. Levitation of a solid circular cylinder by equal parallel currents in phaseWe first consider the levitation and stability characteristics of the configuration of

figure 1 b , wherein the levitated body is a cylinder of circular cross-section (for themoment assumed solid). The solution t o the external field problem (2.4), (2.5), s theneasily solved by the method of images. With the notation of figure 1 b , the image systemfor currents I coswt a t P and Q are currents - I coswt at the image points P, , and(possibly) a current l l ( t ) a t the centre of the cylinder. The tota l current flowing alongthe cylinder is then

Itot I , - 1 coswt .


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52For the caseof a cylinder that is curved in the form of a torus Ywith RI, is in fact zero for the following reason.? Consider the flux

A. D . Sneyd and H . K . Moffatta (figure2a),

Q, = S l 3 .as,where 8, spans the curve C in figure 2a, passing along the axis ofT Since B z 0insideZ e have equally

Q, = J ' s , B d S ,where C' s a circle on the surface of9,s shown. Now

d @ / d t = - E.dx = 0,I,sinceE w 0 inZ and so Cg = 0 (all fields being periodic). Now, for largeR, sing a well-known formula for the mutual inductance of two circles,

Q, wp,RIcoswt In-+ln- +p oR j l n( : s, (:)as,whered,, d,, d are defined in figure 2b; hence, letting R -+ 0, with d,, d,, d fixed, we have

21coswt+ I j d S = 21coswt+1,, = I, = 0, (3.1)J Aas stated above.Vertical equilibrium and stability

The lift force on the cylinder can be most simply calculated as he force on the imageline currents, and has an average vertical componentF, given by

where (figure 1b ) k = a / c , q = H / c and the anglesa, are as indicated in figure 4(a);hence

Figure 3 shows graphs of G ( 7 ,k ) against q for various k. The behaviour of G depends onwhether k < 1 or k > 1, i.e. on whether the cylinder diameter is smaller or greater thanthe gap between the wires. When k < 1, G attains a maximum, a t 7= q l ( k )say, andwhen k > 1,G tends to infinityas q + ( k 2- )6 at which point the cylinder is in contactwith the wires.

Vertical equilibrium is given by Fv =mg wherem is the cylinder mass/unit length,or from (3.2),and the equilibrium is vertically stable providedG ( 7 ,k ) = 2nmgc/p0I 2 = W say, (3.4)

a G / q < 0. (3.5)t For a straight cylinder of finite length, I t& = 0 and so Il = 2 1 cos ot. The levitation forceper unit length of cylinder is much less in this case than for the case of a torus.


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Fluid dynamical aspects of the levitation-melting process3 r \ I

G

2

W1

6 3

1 7 ) 2FIGURE . The function G ( 7 ,k),given by equation (3-3),for various values of k. If k c: 1, th eequation G ( 7 , k) = W can have two solutions; one of these (8)epresents a vertically stableequilibrium and one (U) an unstable equilibrium. If k > 1, there is one stable equilibrium (2').The vertical dashes represent the maximum value of 7 for horizontal stability.For any given W, when k > 1, there is evidently a unique solution of (3.4)and this isstable; when k < 1, there are two solutions of (3.4) for 7, but only the larger valuesatisfying

represents a position of stable equilibrium.T > Tl (k ) , (3.6)

Horizontal stabilityIf the cylinder is displaced horizontally through a distance s (figure 4a), then thehorizontal component of force on the cylinder due to the line current a t P which tendsto restore equilibrium) is given by

The condition for horizontal stability is (dFH/ds),=,, > 0, or, from ( 3 . 7 ) , after somecalculation,

r ] < ( 1 +kz))t= y , ( k ) say. (3.8)When k < 1, stable equilibrium is possible only if

T1(k) < 7 < 7 % ( k ) ,w, w > w,,or equivalently if

(3.9)(3.10)

where W, = G(y2, k).Table 2 lists yI,q2, W, nd W, for 0-05 < k < 0.95. The rangeW,- W, of possible cylinder weights is extremely narrow for k 5 0.8,bu t widens rapidlyfor larger k.On the curves of figure 3, the point 7, is marked by a vertical dash.


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54 A . D . Sneyd and H. K. Moffattk w,

0.05 1-000 00.1 1-000 10.15 1400060.2 1.000 190.25 1.000460.3 1.000940.35 1.001 690-4 1.002 820.45 1.004410.5 1.006570.55 1.009430.6 1.013 170.65 1*018010-7 1.024320-75 1.032 740.8 1.044 790-85 1.068 650-9 1.200860-95 1.618 32

w,1~000001-000011.000061*000191*000461.000931.001 671.002 761.004 251.006231.008741.011841.015571.019941.0251.030751.037 191.044 331.05216

711.001 211404891.010931.019011.028 891.040 271.052 681.065 71.078771.091 351.102671.111681.116841.115 521-102171.060430.814 2210484 1920-319 122

7%1.001 251.004991.011191.01981.030 781.044031-059481.077031.096 591.118031.141 271.166 191.192691-220661.251.280631.312441.345361.37931

TABLE. Stability bounds for a evitated cylinder, ~ E Ia unction of the geometrical parameter k.If k B 1, the sample will clearly become unstable when it melts (see $5below and,

particularly, figure 8);only values of k of order unity are therefore of practical interest,and for this reason we have restrictedattention to the range0.5 < k < 1.5insubsequentcomputations.

Magnetic pressure distributionIt is a straightforward matter to calculate B, and hence p&, n the cylinder surface.In the notation of figure4b,we find

whereand

(3.11)(3.12)

1' (3.13)1+'(O) = ( 1 -K2 ) 2 l+Kz-2Kcos(O-a) 1+K2-2Kcos(O+a) 'with K = k(1+ 72)-4, a = cot-' 7.The corresponding expression for - n A V p M (which

appears in (2.3)) s -n A V pM= a-lpaIof(0) , (3.14)where 2 is a unit vector along the cylinder axis. The function f (6 )which plays animportant part in the subsequent theory, is shown in figure5 or various values of thegeometrical parameters k and 7.

The case k = 7 = 1This case will be useful by way of example. With k = 7 = 1, we have K = 2-4 anda = an.From (3.3) and (3.4), this is a possible equilibrium if W = G(1, 1) = 2, andit is stable. The functionf(O)given by (3.12) simplifies in this case to

3- COS8 (3.15)


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Fluid dynamical aspects of the levitation-m elting process 55

Q C C P( a ) ( b )

FIGURE . The levitated cylinder; (a) onstruction for determining horizontal stability ;( b )notation for determination of p M ( 8 ) on the cylinder surface, and the streamfunction$ (86).

1so

50

0 e lrFIGURE. The functionf(8) for various values of k and 7.

4. Flow in a surface film of molten metal4.1. General considerationsIn this section, we study the initial stages of melting, the intermediate phase betweensolid and fluid levitation, when the solid is covered by a layer of molten metal so thinthat viscous forces dominate and the methods of lubrication theory can be employed(see figure 6).


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56 A . D . Sneyd and H . K . Moffattw = h w = o

FIGURE. Notation for the thin film analysisof $4.The mean rate of Joule heating per unit volume is

y/u = B& -awla/,u:&zu,and so the mean rate of Joule heating per unit surface area (orequivalently the flux ofelectromagnetic energy through the surface) is

J OIn order of magnitude, using (2.8))we have

q J hl pgLh/&* (4.2)Now the total heat supplied per unit time is -Laq , and, provided the variation intemperature throughout the sample isnot too great, this is of order ( p L 3 ) ,AT, hereA T is the increase in temperature in unit time and c, is the specific heat of the metal;hence the time t, to raise the temperature from 0 "C to the melting point T, is

tm PLCsTm/qJ* (4.3)This is compared with the diffusion time t, in table 1b ; for the best conductors, t, t,,and for the others, t, and t, are of the same order of magnitude. This means that thermaldiffusion is generally strong enough to spread the heat fairly uniformly through thesample during the melting process. Once melting begins, the rate of advance of theliquid/solid interface is

where Lf is the latent heat of fusion, and the time to complete the melting is(see also table 1b).the layer (in the lubrication approximation) is

8 - Q J l P L f , (4.4)tme1t - LIO (4.5)Using the expression (2.12) orP,he normal componentof the equation of motion in

- p / h + (2pN/b) e-Mla +p g .n = 0. (4.6)


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Fluid dynam ical aspects of the levitation-meltingprocess 57Here, by virtue of ( 2 . 8 ) , he gravitational term is smaller by a factor 0 ( 6 / L ) han themagnetic term and may be neglected. Since p = 0 on w = 0 (neglecting a surfacetension contribution), p = p M ( - -2W ). ( 4 . 7 )If the layer thickness h is small compared with S, then ( 4 . 7 )gives

p 2 ( w / S ) p M , ( 4 . 8 )while, if h $ 6 , pw pM throughout the fluid layer except in the magnetic skin.the same lubrication approximation)

Substitution of ( 4 . 7 ) n the tangential projection of the equation of motion gives (in

( 4 . 9 )v ( P u / a w 2 )= (1- -zwla) VS P M -PgS,where gs = g-n(g.n), V, = V - n(n.V). The solution of ( 4 . 9 ) satisfying U = 0 onw = h and = 0 on w = 0 s

1 1U = - ( h 2 - w 2 ) g s + - [ 2 ( h - w ) ( d - w - h ) - S 2 ( e - z w l a - e - z h / s ) ] s p f i f , ( 4 . 1 0 )2v 4PVand the corresponding flux Q n the layer is

Q = u dw = 3v (g,-p- lF ( V p A f ) , ( 4 . 1 1 )where

38 9P ( x )= 1 - - (2x2- 1 + ( 2 x + l ) e - 9 . ( 4 . 1 2 )The two limiting cases mentioned above correspond here to the limiting behaviour

( 4 . 1 3 )Allowing now for the advance of the liquid/solid interface due to melting, the fluid

conservation equation takes the form( 4 . 1 4 )

It isclear that, when h is sufficiently small, the melting term in ( 4 . 1 4 )dominates, and hgrows in proportion to pM.When h = O(h,,), where

h, = (L2Av/Lfa)), ( 4 . 1 5 )the melting and convection terms in ( 4 . 1 4 )are of the same order of magnitude; andwhen h % h,,, the convection term V .Q dominates; h,, is generally in the range 0.01-0.1 mm (see table l b ) .

There is of course also an upper limit on h for the validity of the lubricationapproximation. From ( 4 . 1 1 ) , the flux within the layer is of order h 3 g / v , andthe Reynolds number is then - h 3g / v2 .The lubrication approximation requires that( h 3 g / v 2 ) ( h / L ) 1 , i.e.

h 5 L v z / g ) j= h, say. ( 4 . 1 6 )


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5 8 A . D . Sneyd and H . K . Moj fa f tFor the typical liquid metals of table 1, this gives the astonishingly low estimateh , N 0.1 mm ! It is hard to believe that inertial effects could become importan t forh 5 1 mm, and yet this does seem to be an inevitable consequence of the very smallkinematic viscosity of liquid metals.It follows from these orders of magnitude tha t, for so long as lubrication theory isvalid, the approximation h < 6 is also likely to be satisfied, and then (4.11) becomes

(4.17)Here, t he magnetic effect is smull compared with the gravitational effect, and the layersimply drains under gravity as soon as i t is formed, the levitating force being locatedpredominantly within the solid metal.This conclusion is rather uninteresting from a fluid dynamical point of view. A moreinteresting behaviour can however arise if the field frequency is increased to the pointa t which 6 < h , when (4.1 ) becomes

(4.18)Now the magnetic pressure term competes with gravity on an equal basis in deter-mining the film evolution.

h3Q N 5 &-P-'VP>,)*

4.2. Surface layer in the levitated cylinderThis behaviour is well illustrated by the case of the levitated cylinder, for which p , , isgiven by (3.14) and (3.15).Hence from (4.11), the flux & (O ) in the layer is given by

Q - - h3g( s i nO+FM f ' (0 ) ) , ( h $- 6),3v 8Wkand (4.14)becomes

(4.19)

(4.20)from which the evolution of the film may be computed.

For the particular case k = 7 = 1, (and W = ), using (3.13), (4.15) becomes (with= case) (8pS-12p+ 13)(@'- 1 2 ~ + 5 ) ~ (4.21)

When h 9 6, o tha t F ( h / S )N 1 , this flux is positive for p > 0.108 (i.e. for 101 < 83.8O),and i t is negative for 101 > 83.8O.The fluid in the layer is then driven by the combi-nation of magnetic pressure and gravity towards the points 0 = -& 83.8O. Obviouslythe layer thickness builds up according to (4.20)in a neighbourhood of these pointsuntil lubrication theory ceases to be valid.

5. Magnetostatic analysis 5.1. General considerationsIn this section, we shall obtain some exact results concerning the shape of a levitateddrop, under the assumption that effects associated with the interior motion may beneglected; the shape is then determined by a static balance between gravitational,surface tension and magnetic forces.


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Fluid dynamical aspects of the levitation-meltingprocess 59In the fluid core, where P = 0, the pressure is hydrostatic, i.e.

P = P'o-Pgx, (5-1)where x is vertically upwards and p, is constant. In the magnetic boundary layer

p = pM(- - zwp ) +YK,where we now include the effect of surface tension y; K is the sum of the principalsurface curvatures. Since (5.2) tends to (5.1) as w/6+ m, we obtain the na.tura1boundary condition on S,f

yK+pJ1+pgx = cst. on S. (5.3)We now aim to convert (5.3) nto a variational principle, which will be useful for the

computation of stable equilibria. This principle involves the energies

associated with the gravity, surface tension and magnetic field.$ where As is the areaof S, and P is the exterior domain. Suppose that the surface S is perturbed by a smallamount 6hn subject to the constraint that the volume of liquid remains constant, i.e.

n n6 d7 = 6hdS = 0 ,J v Jand subject also to the condition that the currents in the external coils remain atconstant amplitude. Then the associated changes in U and U, are

&Ur= ySAs = y KShdS,J8 Jand the change in U, isNow let B = B, +B,, where B, is the field due to the currents in the external coils inthe absenceof any levitated material (so that 6B, = 0),and B, is the field due to theinduced currents in V, so that VAB, = 0, i.e. B, = Vq5, say, in 7.Then

J A B* . 6 Bd + = B * . V # ~ ~ T =-V.(B*g%,)d7V J? V=Ss(n.B*)q5,dS = 0,

t With B = V+ and p~ = ( 4 , ~ & l ( V $ ) ~ , 5.3) is identical with the dynamic boundary con-dition a t he free surface of astea dy incompressible rrotational inviscid flow with velocity potential4, provided (4pJ1 is identified with (2p)- ' . In both cases, + is a harmonic function satisfyinga + / h = 0 on S ; but when q5 represents a velocity potential, its domain is the interior of S,whereas in the magnetic case its domain is the exterior.$ To get a inite value for UM,he d istribution of current in the external coils must be regardedas continuous, so that the sources for B are not singular. Alternatively, the singular par t can besubtracted out, as done later in $5.3.3 F L M 117


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GOand so (5.7)becomes

A . D . S n e y d and H . K . Moffntt

SU& = - PM ShdS.1818

It follows from (5.6)and (5.8) that6 ( ~ + u , - & ~ ) (YK+r)M+pgx) 6hdS.

Hence the boundary condition (5.3), ogether with the constraint (5.5), mplies that6 ( u g + u y - u M ) = 0, (5.10)

and conversely, the variational statement (5.10), subject to ( 5 4 , implies (5.3).Note the appearance of the minus sign in (5.10):the variation in the total energy(U, +Uy+U,) is not zero, but is equal to 2SU,,, the work done in the external circuitsto maintain currents of constant amplitude.

Note also that the principle (5.10)holds also for a solid levitated body, with U, = 0.It has been used in this form by some authors (e.g. Hatch 1965) o analyse positions ofequilibria; but the above proof applied to general fluid equilibria appears to be new.?5.2. The e v i t a t e d j u i d cyl inder; strong surface tension

For the levitated cylinder problem, we may first obtain an analytic indication of theequilibrium shape, by supposing that surface tension is strong so that the cross-sectionis approximately circular. Specifically, let

m g l y = e < 1, (5.11 )and let the equation of the cylinder surface1 be

1r = a, i + ~ ncosnO+O(se) ,[ n =a (5.12)the n = 1 term being omitted since this corresponds simply to a vertical displacementof the centre. The cross-sectional area is

and so, in the linearized analysis that follows, we may take a,, = a.In equation (5.3),using (3.11)and (3.12),

= ~ ( c o ~ 6 + ~ f ( t 9 ) )+ O ( @ ) ,nu

(5.13)

(5.14)t A variational principle analogous to (5.10) has been established by Brancher & SeroGuillaume (1981) in the context of a f e r r o m g n e t i c fluid subjected to a sta tic magnetic field, and

under the influence of gravity and surface tension.$ A fluid cylinder, levitated in the manner envisaged here, might be subject t o longitudinalinstabilities ; hese could presumably be stabilized by the application of a sufficiently stronglongitudinal high frequency field, as in analogous plasma contexts; for present purposes weignore them.An approximation of the form (5.11), (5.12) has been adopted also by Mestel (1982) in aninvestigation of the effect of interior motion on free surface shape.


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Fluid dynamical aspects of the levitation-melting process 61

8 8 8 8

( 0 ) ( 6 )FIGURE . Cross-sectional shapes of levitated molten cylinder. (a) alculated by perturbation7 = 1.2, E = 15; k = 0.8, 7 = 1.2, 8 = 30. ( 6 ) Calculated numerically using variationalW = 1.096.procedure: k = 0.5 , 7 = 1 . 1 , E = 16; ---- k = 0.5 , 7 = 1.1, E = 32; --- = 0 . 8 ,principle: (k = 1 ) - = 1, W = 1.07; ----- r = 0.3, w 1.081; --- = 0.1,and the curvature K is given by

K = (r2+ 2rf2-rr") ( r2+r '2 ) - )1= a-1 1 + S x n2- 1)ancosnO+O(s2).( n = 2

Hence, to order 8, (5 .3)gives(5.15)

(5.16) LIf we now expand f (0) as a Fourier cosine series

m

n = l. f(O) Bfo+ Z fncosnO,then, from (5 .1 ,

an = - f* (n= 2,3 , ...). (5.18)i ) 8 n k W ( n 2 - 1) '(5.17)

6A numerical method was used to calculate the f,, and hence to obtain the surfaceperturbation for various values of k and 7 ( W being given by ( 3 . 3 ) , (3.4)); figure 7 ashows cross-sections calculated in this way. The effect of magnetic pressure in theregions closest to the line currents is quite evident.

5.3.T h e levitatedfluid cylinder; weak surface tensionWhen surface t,ension s weak, the perturbation from circular shape will be large, and anumerical method must be adopted. We shall use the variational principle (5.10) odetermine the shape. It is convenient also to adopt complex variable techniques.

'The plane of cross-section of the fluid cylinder is taken to be the complex z-planeand for convenience the x-axis chosen as the axis of symmetry. Applying the RiemannMapping Theorem to the exterior ofX shows that there exists a unique analytic function

3 . 2


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6 2 A . D . Xneyd and H . K . Moffatt[(z) which transforms the free surface S to the unit circle 151 = 1 in such a way tha t[(CO) = CO, {'(a)s real. The inverse function z([) is also analytic and can be expandedas a Laurent series

mz = C , { + c , + C n C - n .n= 1By symmetry the c , and C, are all real.Since the cross-sectional area insideS is fixed,

41mS zdz = naa.sSubstitution from (5.19) gives

since [E = 1 on S, nd evaluation by Cauchy's Theorem givesmC: = a2+ n c f .n = l

(5.19)

(5.20)Our strategy will be to choose the cn as independent variables, and to expressU ;;r U + Uy-U,, in terms of these variables; Numerical minimization will determinethe c, .Express ion for U, :

Substitution from (5.19)givesm

n = lv, = ~ p g ~ 2 ~ ( c l c o s e +

0and evaluating the various trigonometric integrals

7 T mm-, = 7 r q c 0 - ~ c , n c f - n C , 2 nc ,c ,+ , -q (m + 3n)cmcnc,+, .PS n =l n = l m = l n = l ( 5 . 2 1 )Express ion for U,

where E is the length of S. It willnot be possible to write an algebraic formula for 1 interms of the c , - or example when c , = 0 for n > 2 then S is an ellipse and the expres-sion for the perimeter of an ellipse in terms of its major and minor axes involves ellipticintegrals- o an approximate method must be used.

U,= Yl'

Let r ( [ ) be an analytic function with a Laurent expansionm ( 5 . 2 2 )

( 5 . 2 3 )such that


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Fluid dynamical aspects of the levitation-meltingprocess 6 3Substitution of ( 5 . 2 2 ) nto ( 5 . 2 3 )shows that the coefficients r , can be determined bythe recursive scheme r0=4,1 = 0 ,

I L C l .-L , (n= 2 , 3 , ...) , I! 2m=

Then ,

so

Expressionfor U,:(6.24)

The field near each line current gives an infinite contribution to the magnetic energy,but these singularities can be removed by defining

L& = l i m ( L 1 B2dxdy+ -1nrr - + ~~ P O, 477 ( 5 . 2 5 )where2,s the exterior of the cylinder excluding two circles of radius r centred on theline currents a t P and &. If f ( z ) s the complex potential for B, henand the magnification of the conformal transformation from the z-planeto the [-planeis Id[ /dzl so tha t if [ = U +i~Transformation of the integral in ( 5 . 2 5 )gives

( 6 . 2 6 )where2;.s the region of the [-plane corresponding tocorresponding small circles around the line currents. Now and r the radius of the

lnr = lnr+ln(r/r) + nr+ln Idz/d[I, as r + 0,so ( 5 . 2 6 )can be written in the form

( 5 . 2 7 )where U& is the magnetic energy of the field in the [-plane. This simple relation be-tween,UJfnd Ug s what makes a conformal transformation method so convenient.

the point P in the [-plane, corresponding to P , was located byNewton iteration on the equation z p = z([~,); hen U& could be found by imagemethods.

To calculate

MinimizationThe series (5.19)was truncated after N terms, so that U = U ( c o , 1, . .,c N ) .The series

( 5 . 2 2 )was truncated after 15 terms, since retention of further terms had no effect on


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64 A . D . Sneyd and H . K . NqffattN3 1.04325 1.038 396 1-04289 1.036017 1.042 84 1.036009 1.04284 1.03600

W ,:maximum W for stable levitation W, minimum W or stable levitation

TABLE. Convergence of results for increasing N ; k = 0.8, 7 = 1.0.

the results, The function U was minimised using the NAG subroutine E04DFF, whichis based on a modified-Newton algorithm, and the calculations carried out on theUniversity of Waikato Vax 11/780 computer. The programme was checked by settingN = 0, which constrains the cylinder cross-section to be circular, and the results oftable 2 were reproduced. Convergence was checked by running the pragramme forvarious N , and table 3 shows a typical set of results. For the bulk of the calculations, Nwas set equal to 4, in which case each minimization took approximately 3 seconds ofC.P.U. ime.

Horizontal stability could be tested in a simple way. OnlyUw s affected by a rigidhorizontal displacement of the fluid cylinder, which is represented mathematically bythe addition of a purely imaginary constant to (5.19). Since dz /dg is unaffected, thestability depends only on the first term of (5.27) o the criterion for horizontal stabilityis just (3.8) applied to the [-plane cylinder.

The maximum W (dimensionless cylinder weight), for which levitation is possible,with fixed values of the other parameters, was determined by increasing W graduallyuntil no minimum of U could be found.Resultsthe cross-sectional area is na8,W = 2ncmg/p,-, 12, nd

r = n y c / p , ~= w/, (5.28)which provides a dimensionless measure of surface tension relative t o magnetic forces.Figure 7 b shows three cross-sections (fork = 1.0)which show the effect of decreasing l?.When I' = 1 the perturbation from the circular shape is still small, and as r decreasesthe change in shape is qualitatively similar to that predicted by the linear theory of$5.2.

For given k and I, there is a range of values of W over which stable levitation canoccur, the width of this range increasing to a maximum as I' tends to infinity (a twhichpoint the results are the same as for the rigid cylinder). Figure 8a shows stabilityboundaries in the range 0 < r < 1fork = 0.7,i.Oand 1.5.Fork c -788 table levitationis possible with I' = 0, but for k > 0.788 it is possible only if I' > r,(k) (see Figure 8 b ) .If r < rm(k)he lateral spread becomes so great that the fluid in effect spills over theedges of the magnetic well; k then decreases to the value k, a t which Fm(k1)w l? andsurface tension can then just retain the sample within the stability limit.

The lower stability limit for W is determined by the horizontal stability criterionreferred to above. If W is decreased (by decreasingm,keeping other parameters fixed),the sample rises until criterion ( 3 . 8 )(adapted to the (-plane) is violated.

The dimensionless parameters which can be independently varied are k = a/c,where


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Fluid dynamical aspects of the levitation-meltingprocess 65

. 1.04478

I Stable levitation / i t

I= 1.5I \t table levitation

0.778 1 .o k 1.5( b )FIGURE. (a)Regions of stable levitation for k = 0.7, 1.0, 1.5. Note the different vertical scalein each diagram. ( b )Graph of r m ( k ) ; table levitation is possible if r > rm(k).6. Fluid motion in the levitated cylinder

In order to obtain some qualitative understanding of the motion within the cylinder,we shall ignore the departure of the cylinder cross-section from a circle (aprocedurethat is strictly justifiable only if surface tension is strong, i.e. I& 1). From (3.14) thesurface vorticity generation is, in the cylindrical geometry, a-lpAfo(S),he functionf(S) eing as shown in figure 5, for various k. There are apparently two distinct possi-bilities: (i) f k is small,f(0) ecreases monotonically from 0 = 0 to B = n, o that the


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66 A . D. neyd and H . K . Moffattsign of vorticity generation does not change in this interval; (ii) for larger k, f ( 0 )reaches a maximum a t some intermediate point, where the vorticity generation there-fore changes sign. One would expect the resulting fluid motion to consist of two eddiesin case ( i) , and four in case (ii), as is in fact found (see figure 9 below, which shows onlythe right-hand half of the flow domain).

We have argued in $ 2 th at the flow is likely to be turbulent when L - IOmm andgreater, and tha t a unifarm eddy Viscosity vT - U,, may then provide the dominantmechanism of momentum transfer. The Reynolds number based on vT is of orderunity, and a low Reynolds number analysis is likely to give a reasonable qualitativedescription of the mean flow. This, a t any rate , is the approach we now ad0pt. t

The streamfunction $ ( r , 0) of the mean flow then satisfies the (inhomogeneous) bi-harmonic equation

The circle r = a is a streamline on which the tangential stress vanishes, and so

Now the right-hand side of (6.1)issignificant only within the magnetic boundary layer,within which V4 z a4/ar4, and so a particular integral of (6.1) s

th e general solutich (with appropriate symmetry) being4 = 5 (A , , ( : )n -+ B , ( : )n+2) sinne.n= 1IThe boundary conditions (6.2) then determine the coefficients A , and B,, in terms of

the Fourier coefficientsf, off(0)see 5.17). Retaining only leading order terms in th esmall parameter &/a, he solution for ( a- ) / 6 > 1 is

m+ = ( 1-R s ) fn Rn in no,'PVT n = lwhereR = r /a .conjugate series

The series in (6.5) may actually be summed! To do this, consider h t he complexcoE ( R , B )= g f o + fnRncosnO

n = lfor which, from (5.19) and (3.11))

It can be shown by elementary trigonometry th at on the circle r = a ,( P - a 2 ) r ; 2 = 2Zr;1cosBi-1 (i = 1,2), (6.8)

t This is to be contrasted with t,he approach of Mestel(l982)who uses numerical methods todetermine the corresponding laminar flow in a phere at high Reynolds number.


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Flu id dynamical aspects of the levitation-meltingprocess 67

FIGURE. Streamlines @ = cst. where @ is given by the low Reynolds number solution (6 .13);(a) = 0 . 5 , 7 = 1.1; ( b )k = 0 .95 , 7 = 1.0.where P i, Oi are defined in figure 4b, nd that

where c - ( z ~ - u ~ c o s ~ u ) / C ~ ,l = 1(IP+a~-2a~cos~a)/C4,]C , = (12-a2)2/C4, (6.10)

- C, = 2(14+a4-2a212cos2a).Substitution of (6.8)and (6.9)in (6.7)gives

= Wl, , , ~ , , , )> say, (6.11)where

(6.12)

SinceE ( R ,0) and V are both harmonic in R < 1and equal on R = 1, they must be equalthroughout R < 1 . The complex conjugate of (6.11) is obtained by replacing cosinesby sines; hence (6.5)may be written

D1 = 8l(C,1- 1)+4P/(12- U'),Do = 4 +8Z2C0- Z2/(l2- '). D , = 8Z2C,,

Figure 9 shows streamline patterns for two values of (k, ), which have the structureanticipated from the argument based on the sign of vorticity generation. However,'


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68 A . D . Sneyd and H . K . Moffntt

FIQURE0. Graphs off(@ and the surface velocity V, (O) : ( a )k = 0 . 5 , ~ 1.1; ( b ) k = 0.76,71= 1 . 2 ; (e ) k = 0 .95 , 7 = 1.0. Note that the zero of I$ is displaced towards B = 0, relative tothe zero off(@.the point a t which the surface velocity

v, = a-l(a$pR)R=l, (6.14)changes sign does not quite coincide with the point wheref(8) = 0 figure 10) but isslightly displaced towards 8 = 0.

7. DiscussionThe results of Q $ 3 , 5 nd 6 show the feasibility of levitating a two-dimensional liquid

metal cylinder in the magnetic field due to parallel line currents in phase, which canbe thought of as representing in cross-section a fluid torus levitated by two circularline currents (figure 2 a ) . This geometry avoids the crucial difficulty associated withpractical levitation devices,viz thGt surface tension is necessary t o prevent fluid fromleaking through the lower neutral magnetic-field point; and indeed we have found thatlevitation is possible with zero surface tension, and that there is (in principle) no limitt o the mass of liquid metal t hat can be levitated with a sufficiently high current, An


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Fluid dynamical aspects of the levitation-melting process 69immediate qualification is however appropriate: although this system can be madestable with respect to perturbations in the plane of cross-section, it would be subjectto longitudinal instabilities due to surface tension (which would tend to divide thetorus into drops) and due to t he tendency of the torus to sink lower into the magneticfield at any point where the cross-sectional area was increased by longitudinal inflow.Nevertheless such instabilities could be eliminated by the application of a sufficientlystrong longitudinal magnetic field- the same technique that is used to eliminate theplasma pinch instability - and a toroidal levitation device could probably be made towork.

There are however a number of obvious imperfections in the analysis that we havebeen able to develop. Firstly, the shape calculation was based on the variationalprinciple (5.10) which is valid only if dynamic pressure effects are negligible. A com-plete solution of the problem clearly requires inclusion of dynamic pressure in (5.1)andthis requires solution of the dynamical problem within a free surface that is stronglydistorted by magnetic pressure- formidable problem even if the flow were known tobe laminar. Secondly, accepting that the flow is likely to be turbulent, there is theproblem of improving significantly on the assumption of uniform eddy viscosity,adopted in 9 6 . A similar problem of turbulence modelling in flows with closed stream-lines driven by rotational forces arises in related contexts (see, for example, Hunt &Maxey 1980), but we appear to be still far from an adequate solution. Of course, thenormal component of fluctuating velocity falls to nearly zero (exactly zero if surfaceripples are ignored) a t the free surface, and so a decrease in eddy viscosity near the freesurface is to be expected. This effect should perhaps be incorporated in a more realisticanalysis.

Finally, there are questions of global stability associated with the fact that inpractice it is voltage, rather than current, tha t is generally prescribed in the externalcoils. Perturbations in the shape of the levitated sample lead to an associated per tur-bation in the mutual inductance between coils and sample, and so to the possibility ofdynamic instability involving this coupling.

All of these effects require further analysis in conjunction with experiments aimedat providing some detailed information concerning the velocity field within the sampleand on its surface.

This work was initiated a t the School of Mathematics, University of Bristol, in1978/9 during A.D.S.s enure of an S.R.C. Senior Visiting Fellowship, Grant no. GR/B04969. Professor M. R . Harris of the University of Newcastle-upon-Tyne drew ourattention to the comprehensive study of Stephan (1975),and provided helpful criticismand comments which are gratefully acknowledged.

R E F E R E N C E SBLOCK, . R. & THEISSEN, . 1971 Electromagnetic levitation melting -method of crucible-BRANCHER,. P. & SEROGUILLAUME,0. 1981 Sur 16quilibre des ferrofluides avec interfaceBRISLEY,W. & THORNTON,. S . 1963 Electromagnetic levitation calculations for axiallyEL-KADDAH,. H. & ROBERTSON, D. c. C. 1978 The kinetics of Gas-Liquid metal reactions

free melting. Elektrowiirme Int. 29, 349.libre. Rapport interne, LEMTA, Nancy, France.symmetric systems. Brit. J. AppE. Phys. 14, 682.involving levitated drops. Metall. Truns. 9B, 191.


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70 A . D . Sneyd and H.K. MojfattHARRIS, . R. & STEPHAN,. Y. 1975 Support of liquid metal surface by alternating magneticfield. IEEE Trana. MACt-11, 1508.HATCH, . J. 1965 Potential-well description of electromagnetic levitation. J . A w l . Phys. 36,44.HUNT,J. C. R. & MAXEY, M. R. 1980 Estimating velocities and shear stresses in turbulentflows of liquid metals driven by low frequency electromagnetic fields. In MHD Flows and

Turbulence, (ed. H. Branover & A. Yakhot), Israel Universities Press, Jerusalem.LIUHTHILL, . J. & WHITHAM,. B. 1955 On kinematic waves. I. Flood movement in longrivers. P T OC. oy. Soc. A, 229, 281.MESTEL,J. 1982 Magnetic levitation of liquid metals. J. Fluid Mech. 117, 2 7 .MOREAU, . 1980 Applications m6tallurgiques de la magn6tohydrodynamique. In Proc. XVthInt. Cong. Theor. Appl. Mech. (ed. F. P. J. Rimrott & B. Tabarrok), North Holland.MUCK,0. 1923 German Pat. No. 422004, Oct. 30 1923.OKRESS,E. O.,WROUOHTON,. M., COMENETZ,., BRACE,P. N. & KELLY, . C. K. 1952PEIFER, . A. 1965 Levitation melting, a survey of the state-of-the-art,J. Metals 17, 487.PIUGOT,. S. & NIX,G. F. 1966 Electromagneticlevitation of a conductingcylinder. PTOC..E.E.113,1829.SMITHEUS,C. J. 1967 Metals Reference Book 4th. edn. vols. I and 111, Butterworths, London.SNEYD,A. D. 1979 Fluid flow induced by a rapidly alternating or rotating magnetic field.STEPHAN, . Y. 1975 Support of liquid metal suIfaces by al ternating magnetic fields - a nVOLKOV, . F. 1962 Stability of a heavy conducting fluid contained by a rapidly varyingZHEZHERIN,T. F. 1959 Problems of magnetohydrodynamics and plasma dynamics. Izv.A N

Electromagnetic levitation of solid and molten metals. J. A p p l . Phys. 23, 83.

J . Fluid Mech. 92, 35.experimental and theoretical study. Ph.D. thesis, University College, London.magnetic field. Sowiet Phys. - Tech. Phys., 7, 22.Latv. SSR. iga. 279.

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A.D. Sneyd and H.K. Moffatt- Fluid dynamical aspects of the levitation-melting process