Compositional and thermal equilibration of particles, drops ...perso.ens-lyon.fr/stephane.labrosse/sites/default/files/...Rising/Falling Particles, Drops, and Diapirs 2.1. Chemical

Article

Volume 12, Number 10

20 October 2011

Q10014, doi:10.1029/2011GC003757

ISSN: 1525‐2027

Compositional and thermal equilibration of particles, drops,and diapirs in geophysical flows

M. UlvrováLaboratoire de Géologie de Lyon, École Normale Supérieure de Lyon, 46, Allée d’Italie, F‐69364 LyonCEDEX 07, France (martina.ulvrova@ens‐lyon.fr)

N. Coltice and Y. RicardLaboratoire de Géologie de Lyon, Université de Lyon 1, Bat Geode, 43 Boulevard du 11 Novembre1918, F‐69100 Villeurbanne, France (nicolas.coltice@univ‐lyon1.fr; ricard@ens‐lyon.fr)

S. LabrosseLaboratoire de Géologie de Lyon, École Normale Supérieure de Lyon, 46, Allée d’Italie, F‐69364 LyonCEDEX 07, France (stephane.labrosse@ens‐lyon.fr)

F. DubuffetLaboratoire de Géologie de Lyon, Université de Lyon 1, Bat Geode, 43 Boulevard du 11 Novembre1918, F‐69100 Villeurbanne, France ([email protected])

J. VelímskýDepartment of Geophysics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2,180 00 Prague 8, Czech Republic ([email protected])

O. ŠrámekDepartment of Physics, University of Colorado at Boulder, 390 UCB, Boulder, Colorado 80309‐0390,USA ([email protected])

[1] Core formation, crystal/melt separation, mingling of immiscible magmas, and diapirism are fundamen-tal geological processes that involve differential motions driven by gravity. Diffusion modifies the compo-sition or/and temperature of the considered phases while they travel. Solid particles, liquid drops andviscous diapirs equilibrate while sinking/rising through their surroundings with a time scale that dependson the physics of the flow and the material properties. In particular, the internal circulation within a liquiddrop or a diapir favors the diffusive exchange at the interface. To evaluate time scales of chemical/thermalequilibration between a material falling/rising through a deformable medium, we propose analytical lawsthat can be used at multiple scales. They depend mostly on the non‐dimensional Péclet and Reynoldsnumbers, and are consistent with numerical simulations. We show that equilibration between a particle,drop or diapir and its host needs to be considered in light of the flow structure complexity. It is of funda-mental importance to identify the dynamic regime of the flow and take into account the role of the innercirculation within drops and diapirs, as well as inertia that reduces the thickness of boundary layers andenhances exchange through the interface. The scaling laws are applied to predict nickel equilibrationbetween metals and silicates that occurs within 130 m of fall in about 4 minutes during the metal rain stageof the Earth’s core formation. For a mafic blob (10 cm diameter) sinking into a felsic melt, trace elementequilibration would occur over 4500 m and in about 3 years.

Components: 6500 words, 5 figures, 1 table.

Keywords: equilibration; fluid mechanics; magma; mixing; modeling; segregation.

Copyright 2011 by the American Geophysical Union 1 of 11

http://dx.doi.org/10.1029/2011GC003757

Index Terms: 1009 Geochemistry: Geochemical modeling (3610, 8410); 1043 Geochemistry: Fluid and melt inclusiongeochemistry; 8145 Tectonophysics: Physics of magma and magma bodies.

Received 15 June 2011; Revised 29 August 2011; Accepted 30 August 2011; Published 20 October 2011.

Ulvrová, M., N. Coltice, Y. Ricard, S. Labrosse, F. Dubuffet, J. Velímský, and O. Šrámek (2011), Compositional and thermalequilibration of particles, drops, and diapirs in geophysical flows, Geochem. Geophys. Geosyst., 12, Q10014,doi:10.1029/2011GC003757.

1. Introduction

[2] Bubbles and crystals travel through differenti-ating magmas; metal drops and diapirs fell throughmolten silicates during the formation of Earth’score [Stevenson, 1990; Rubie et al., 2003; Samueland Tackley, 2008; Monteux et al., 2009]; andsometimes, coexistent immiscible magmas or metalsseparate to reach gravitational equilibrium [Dawsonand Hawthorne, 1973; Dasgupta et al., 2006, 2009;Morard and Katsura, 2010]. Differential motiondriven by gravity is a prerequisite for planetary dif-ferentiation at all scales. While a phase travelsthrough and deforms the other, chemical and thermaldiffusion proceed towards thermodynamic equilib-rium. Depending on material and flow dynamics,non‐equilibrium fractionation could result frominefficient mass/heat transfer from one phase to theother during travel. In order to quantify the time scaleof thermodynamic equilibration, it is necessary tomodel deformation of both phases and transportdynamics.

[3] A generic physical description of these differ-entiation mechanisms can be formulated by therise/fall of chemically (or thermally) distinct parti-cles, drops or diapirs through a viscously deform-ing medium. In this paper, the term particle refersto a small self‐contained body significantly moreviscous than the surroundings and possibly solid,while drop and diapir are defined by self‐containedbodies as viscous or less viscous than the sur-roundings and possibly inviscid. A diapir is alarge‐scale body with approximate sphericity, andwe use the term of drop for small‐scale body, whensurface tension controls the sphericity. The purposeof this paper is to review and propose analyticallaws that describe the chemical/thermal equilibra-tion of a traveling particle, drop or diapir, that canbe used at multiple scales and applied to a varietyof geological problems. This chemical equilibrationis that of minor or trace elements, migrating acrossthe surface of the traveling sphere, assuming thatthe major element mineralogies, inside and outsidethe sphere, do not change. We first draw attention

to results that are often overlooked in the geosci-ence literature though acknowledged in engineer-ing and mass/heat transfer communities. Indeed,the mass and heat transfer between a liquid drop/solid particle and a viscous surrounding mediumhas been described for various industrial purposes[Clift et al., 1978]. We then extend their use andcouple them to concentration models inside andaround the spherical body. We propose scalinglaws for the time of equilibration for 4 differentregimes: a particle with and without inertia, a drop/diapir with and without inertia. Then, we proposetimes of chemical equilibration during core for-mation and silicate melt differentiation.

2. Models for the Equilibration ofRising/Falling Particles, Drops,and Diapirs

2.1. Chemical and Thermal TransferFrom a Sphere[4] We restrict ourselves to the study of an indi-vidual spherical particle, drop or diapir of radius Rin steady‐state motion with terminal velocity Ut. Inthe following, the subscript “o” denotes the prop-erties outside the sphere and “i” inside the sphere.The viscosity ratio between the falling/rising bodyand its host liquid is Rm = mi/mo and we speak of“particles” when Rm = mi/mo � 1 and “drops” or“diapirs” when Rm = mi/mo ] 1. All the notationsand parameters can be found in Table 1.

[5] In order for the dispersed phase to keep itssphericity, the interfacial force has to exceed dis-rupting forces, i.e. the viscous force and inertia,that tend to deform the sphere. The smallest dropsor diapirs coalesce (they are swept by largerspheres traveling faster), the largest deform, stretchand eventually break‐up. For highly viscous flows,the capillary number Ca = moUt/g (g is interfacialtension), being the ratio between the viscousstresses and the interfacial tension, reaches a criti-cal value for break‐up conditions that depends on

GeochemistryGeophysicsGeosystems G3G3 ULVROVÁ ET AL.: COMPOSITIONAL AND THERMAL EQUILIBRATION 10.1029/2011GC003757

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the viscosity ratio Rm across the surface. Thiscritical capillarity number Ca is about 0.1 forhigh viscosity ratios and larger for low viscosityratios, for which the drop/diapir stretches andforms a slender shaped body difficult to fragment(see Stone [1994] for a review). For low viscosityflows, perturbations of the interface generateRayleigh‐Taylor and Kelvin‐Helmoltz instabilitiesthat ultimately break up the drops [Kitscha andKocamustafaogullari, 1989]. This situation hap-pens when the Weber number We = roUt

2R/g (rois external density), which is the ratio betweeninertia and interfacial tension, reaches valuesaround 10 [Wierzba, 1990].

[6] For a body sinking or rising through a viscousmedium, two non‐dimensional numbers control thedynamics of chemical equilibration of the travelingsphere with its surroundings: (1) the Reynoldsnumber Re = RUtro/mo that describes the effect ofinertia to viscous force and (2) the Péclet numberPe = RUt/Do that relates the diffusion time to theadvection time in the host liquid, D standing forchemical diffusivity. As both Re and Pe include theterminal velocity, it may be confusing to use

simultaneously the two numbers and we introducetheir ratio, also called the Schmidt number Sc =Pe/Re = mo/(roDo). When the spherical body andits host liquid have different physical properties,the ratios of internal to external diffusivities RD =Di/Do and viscosities Rm = mi/mo, have to beconsidered.

[7] Starting from non equilibrium initial conditions,the sphere and its surroundings tend to chemicallyequilibrate by microscopic diffusion and macro-scopic stirring. The stirring, i.e., the advection ofconcentration by the flow, occurs outside andpossibly inside the sphere, due to the circulationforced by the shear stress at the surface of the drop.

[8] We assume that the initial concentration c ofsome trace element outside the sphere is uniformand equal to c∞ while the concentration inside thedrop is equal to c0. The dimensionless transportequation governing this process, assuming materi-als are incompressible, is written as

@C

@t¼ r � D

Pe

#

C � vC� �

; ð1Þ

Table 1. Variables and Parameters of the Studied System Together With Expressions for the ProposedEquilibration Times t

Parameter Notation Unit

Viscosity of the host liquid mo Pa sViscosity inside the sphere mi Pa sDiffusivity of the host liquid Do m

2 s−1

Diffusivity inside the sphere Di m2 s−1

Density of the host liquid ro kg m−3

Density of the sphere ri kg m−3

Radius of the sphere R mTerminal velocity Ut m s

−1

Initial concentration within the sphere c0 mol m−3

Concentration at infinity c∞ mol m−3

Dimensionless Number Notation Expression

Viscosity ratio Rm mi/moDiffusivity ratio RD Di/DoReynolds number Re RUtro/moPeclet number Pe RUt/DoSchmidt number Sc Pe/Re = mo/(roDo)Sherwood number Sh −h #Cosurfi/hCosurfiRegime Equilibrium Timescale

Drop: low Re, low Rm � ¼ Pe3 K0:461 1þR�ð Þ�1=2Pe1=2 þ1

10RD

� �Drop: high Re, low Rm � ¼ Pe3 K0:79 Pe1=2 þ 110RD

� �Particle: low Re, high Rm � ¼ Pe3 K0:64 Pe1=3 þ 3�2RD

� �Particle: high Re, high Rm � ¼ Pe3 K0:6 Pe1=3Re1=6 þ 3�2RD

� �


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where C stands for the normalized concentrationof any minor element of interest, i.e., C = (c − c∞)/(c0 − c∞) and D for the dimensionless diffusioncoefficient being 1 outside the drop andRD = Di/Doinside. To scale the quantities back to numbers withdimensions, the dimensionless distance has to bemultiplied by the radius of the sphere R, and thetime by the advection time R/Ut. The initial non‐dimensional concentrations are one inside and zerooutside.

[9] The local chemical equilibrium implies that theconcentration ci

surf on the inner side of the sphere iscontrolled by thermodynamics to be Kcosurf, whereK is the partition coefficient at the surface (K =cisurf/co

surf). The final equilibrium is reached whenthe outside concentration is homogeneous and equalto c∞ and the inside concentration also homoge-neous but equal to Kc∞. The normalized concen-tration inside the sphere, Ci, evolves therefore from1 to (K − 1)c∞/(c0 − c∞).

[10] The chemical and thermal diffusion of a trav-eling sphere is the subject of numerous studies inthe chemical/heat transfer literature [e.g., Clift et al.,1978; Levich, 1962] that we can only briefly intro-duce here. Usually the mass transfer coefficient ofthe sphere is defined as the Sherwood number:

Sh ¼ �R

#

csurfo� �csurfo� �� c∞ ¼ �

#

Csurfo� �Csurfo� � ; ð2Þ

where hcosurfi and hCosurfi, and h #cosurfi and h #Cosurfiare the average concentrations at the surface of thesphere and average gradients of concentration nor-mal to it, with and without dimensions. The minussign in equation (2) insures the positivity of Sh.Notice also that although the gradient of concen-tration h #cosurfi can have any sign, the normalizedgradient h #Cosurfi is always negative. Because of thesimilarity of heat and diffusion equations, all theresults on diffusion relating Schmidt and Sherwoodnumbers have thermal counterparts where the massflux is equivalent to the Nusselt number Nu =−R

#

Tsurf/(Tsurf − T∞), and the Schmidt number tothe Prandtl number Pr = mo/(ro�o), where T is tem-perature and � thermal diffusivity.

[11] A very large number of semi‐empirical equa-tions predicting Sh, can be found in the literature,based on experiments and physical analysis.However, simple boundary layer theories can bedeveloped to quantitatively describe mass fluxes atthe interface in the different flow regimes. Thegeneral method is to express the velocity at thesurface of the sphere, estimate the shape of the dif-fusion layer and perform careful averaging over

the sphere [Levich, 1962; Ribe, 2007]. The gen-eral results for high enough Pe can be written inthe form

Sh ¼ aScmPen ¼ aScmþnRen; ð3Þ

where a, m and n are constants for a givenregime. The exponents can be found by scalingarguments as summarized hereafter and the pre-factor estimated analytically, numerically orexperimentally:

[12] 1. When Rm ] 1, i.e., in the case where theinternal viscosity is of the same order or smaller thanthe external viscosity, the external flow experiencesthe surface of the drop/diapir as a free‐slip boundarycondition. The surface velocity in the referenceframe of the drop/diapir is therefore of order Ut,the transport term of the advection‐diffusionequation v.

#

C is of order UtC/R (transport alongthe surface of the sphere) and is balanced by adiffusion term Dor2C of order DoC/d2 across adiffusion boundary layer of thickness d (diffusionperpendicular to the surface of the sphere).Therefore the diffusion boundary layer is of order(d/R)2 / 1/Pe, and, as Sh = R/d

Sh ¼ aPe1=2 ¼ a Sc1=2Re1=2: ð4Þ

The details of the external velocity field control thediffusion layer and hence the expression of theconstant a. (1) For low Re flows, inertia is negli-gible, the external velocity is given analytically bythe Rybczynski‐Hadamard expression [Acrivos andGoddard, 1965] and following Levich [1962] weobtain

a ¼ 0:461ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

1þR�

s: ð5Þ

(2) At very large Re, the flow becomes irrotationaland the analytical expression of the potential flowyields a = 0.79 [Clift et al., 1978].

[13] 2. When Rm � 1, the sphere behaves rigidly.In its own reference frame, the surface velocity iszero then increases away to Ut. Two cases mustthen be considered. First, at low Re, there is noviscous boundary layer as viscous forces dominateeverywhere in the domain. As a consequence thevelocity increases from 0 to Ut over the distance Rso that the velocity is of order Utd/R at the distanced. The balance between advection and diffusionnow gives (Utd/R)(C/R) / DoC/d2, which leads to(d/R)3 / 1/Pe and thus to

Sh ¼ aPe1=3 ¼ a Sc1=3Re1=3; ð6Þ


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where a = 0.64 [Levich, 1962]. Second, at large Re,the situation is more complex. In this case, there isa viscous boundary layer of thickness d′ where theinertia term of the Navier‐Stokes equation roUt

2/Ris balanced by the viscosity moUt/d′

2. The diffusiveboundary layer is therefore embedded in a viscouslayer of thickness d′ / R Re−1/2. The velocity atthe distance d of the particle surface is of orderUtd/d′ = UtdRe

1/2/R. The balance between diffu-sion and advection is now (UtdRe

1/2/R)(C/R) /DoC/d

2 which leads to (d/R)3 / 1/(Re1/2Pe) andthus to

Sh ¼ aRe1=6Pe1=3 ¼ aSc�1=6Pe1=2 ¼ aSc1=3Re1=2; ð7Þ

where a = 0.6 [Ranz and Marshall, 1952].

[14] 3. For a non moving body, the diffusionequation can be solved exactly and Sh = 1. This is aspecial case, that does not obey the asymptoticequation (3) valid for high Pe. In the intermedi-ate regime where Pe is small, various empiricalexpressions for each specific case can be found inthe literature. For example, for Re � 1, Clift et al.[1978] propose Sh = 1 + (1 + a4/3Pe2/3)3/4 in thecase Rm ] 1 (which generalizes equation (4)) andSh = 1 + (1 + a3Pe)1/3 in the case Rm � 1 (whichgeneralizes equation (6)). These expressions arecumbersome and the cases where diffusion dom-inates advection not very interesting physically. Fornumerical applications, the reader should use themaximum of the asymptotic equation (3) and of thediffusive limit Sh = 1.

2.2. Equilibration Time Scales[15] Once the Sherwood number is known (theaverage concentration gradient), to compute theevolution of the concentration within the sphere,we must now relate hCosurfi to hCii, the averageconcentration of the spherical body. Hence weintegrate the diffusion equation (1) to get

@ Cih i@t

¼ 3Pe

#

Csurfo� � ¼ �3 Sh

PeCsurfo� �

: ð8Þ

The thermodynamic equilibrium at the surfaceimplies ci

surf = Kcosurf, or in term of normalized

concentrations,

Csurfi ¼ KCsurfo þ K � 1ð Þc∞

c0 � c∞ : ð9Þ

[16] The concentration diffuses from the surfacewhere the concentration gradient is h #Cosurfi/RD.

Therefore a reasonable profile for the radial con-centration inside the drop is

Ci rð Þ ¼ Csurfi� �þ #Csurfo

� �RD f rð Þ: ð10Þ

The function f(r) characterizes the concentrationprofile and verifies the conditions at the surfaceand the center of the sphere: f(1) = 0, f ′(1) = 1and f ′(0) = 0. This function should also satisfysome positivity constraint as the real concentrationci(r) should be everywhere positive. We do notimpose such a condition. Our models implies thatthe concentration near the surface of the sphere isalways positive and this controls the average con-centration (related to the integral of Ci(r)r

2) whichis always positive as we see below.

[17] Equation (10), averaged over the volume of thesphere, gives for the radial average concentration

Cih i ¼ Csurfi� �� #Csurfo

� �b RD ¼ C

surfo

� �K þ Sh

b RD

� �

þ K � 1ð Þ c∞c0 � c∞ ; ð11Þ

where the second equality uses averaged equation (9),and where b is positive and given by

1

b¼ �3

Z 10

f rð Þr2dr: ð12Þ

[18] Combining equations (8) and (11) we predict anexponential homogenization of the concentration

@ Cih i@t

¼ � 1�

Cih i � K � 1ð Þ c∞c0 � c∞

� �

with � ¼ Pe3

K

Shþ 1bRD

� �:

ð13Þ

With real dimensions, the solution is simply

ciðtÞh i ¼ c0 exp � UtR� t� �

þ Kc∞ 1� exp � UtR� t� ��

: ð14Þ

These last expressions show that the average con-centration in the sphere reaches exponentially theasymptotic equilibrium value and provides an esti-mate of the characteristic homogenization time t asa function of K, Pe, Re,RD andRm (which controlsthe appropriate expression of Sh). It also shows thatthe average concentration is always positive, inde-pendently of the choice of f(r), hence b, that weestimate for two limiting cases:


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[19] 1.WhenRm� 1, there is no recirculation insidethe sphere. When diffusivity of the outer material islarge, 1/(bRD) � K/Sh, the time of equilibration indimensional quantities is t = R2/(3bDi), which isthe same as the classical value obtained for diffu-sion in a sphere with imposed surface concentra-tion R2/(p2Di) [Carslaw and Jaeger, 1959], whenb = p2/3. This is obtained for f(r) = −sin(pr)/(pr),which indeed verifies f ′(0) = f(1) = 0, and f ′(1) = 1. Inour equilibration experiments, the concentration isnot imposed at the sphere surface but at infinity.The diffusion toward the sphere must also proceedsoutside the sphere and using equation (13) withSh = 1 we predict for the equilibration time of astatic sphere (with real dimension)

�s ¼ K R2

3Doþ R

2

�2Di: ð15Þ

The equilibration of a sphere with concentrationmaintained at infinity is indeed slower than whenthis concentration is imposed at the surface anddepends on internal and external diffusivities andon the partition coefficient.

[20] 2. When Rm ] 1, there is an internal recircu-lation inside the drop/diapir, the concentration atthe center is close to the concentration at the sur-face because of the efficient inner transport. Thus,the radial concentration profile within the fluidsphere must also satisfy f(0) = 0. The simplestpolynomial function that verifies all four conditionsf(0) = f ′(0) = f(1) = 0, and f ′(1) = 1 is f(r) = r2(r − 1)which results in b = 10.

[21] For the convenience of the reader, the expres-sions for the equilibration times in the differentregimes are summarized in Table 1.

3. Numerical Examples

[22] The goal of this section is to compare fullnumerical solutions for the time of equilibrationwith the analytical laws proposed above. Hence werun 2D axisymmetric numerical simulations of theincompressible Navier‐Stokes equation coupledwith the mass transfer equation. The experimentsare performed using the finite element method(FEM) implemented in the Elmer open software(CSC IT–Center for Science, 2010, available athttp://www.csc.fi/english/pages/elmer). The com-puting domain consists of an axisymmetrical cyl-inder with height and diameter of 40 R. In thecenter of the cylinder is a motionless sphere ofradius R. Constant inflow of magnitude Ut parallelto the axis of symmetry together with zero con-centration boundary condition are prescribed at thebottom of the cylinder, neglecting thus the influ-ence of other drops. Free‐slip boundary conditionfor velocity, and zero concentration are imposed atthe sides of the cylinder. At the top, a flow parallelto the symmetry axis is forced, and a Neumannboundary condition of zero concentration gradientis prescribed. Finally, at the surface of the sphere,zero normal velocity and zero tangent traction areprescribed. A jump in concentration is imposedaccording to the choice of the partition coefficient,while the concentration flux remains continuous.

Figure 1. Maps (close‐ups) of the nondimensional velocity relative to the average drop velocity (arrows) and con-centration (color) for falling (a) particle in Re < 1 flow, (b) drop in Re < 1 flow (in to out viscosity ratio Rm = 10−3)and (c) particle in a higher Re flow (being 50 here). The spheres fall under their own weight and start with a com-position of 1 (light grey) and the surrounding material has a zero concentration initially (green). All models havePéclet number Pe = 2800 and equal diffusivity in and out of the sphere (RD = 1). Snapshots are taken after afalling distance of 36 (Figure 1a), 41 (Figure 1b) and 283 (Figure 1c), times the sphere radius.


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[23] Using FEM allows us to refine the mesh in theboundary layer around the sphere in order to have agood resolution for the velocity and transportequations, together with the refinement in the wakewhere more complicated structures of the flowappear at high Re. We use up to 80 000 meshnodes. For each of the simulations, the time ofequilibration t is computed through a least‐squaresfit of the time series of hCi(t)i. We explore itsdependence on the non‐dimensional numbers Pe inthe range 102–105, Re in the range 0–170 anddiffusivity and viscosity ratios RD and Rm, in therange 10−1–103 and 10−3–103, respectively.

[24] Typical flows are shown in Figure 1 (velocityin the sphere reference frame is depicted by arrows,concentration by color scale). For a rigid sphereand low Reynolds number (panel a, Rm = 100,Re = 0.1), the flow is a typical Stokes flow,homogenization proceeds from the surface, and atail is emitted in the wake of the sphere. When theinternal viscosity is reduced keeping the same lowReynolds number (panel b, Rm = 10−3, Re = 0.1), acirculation is induced within the drop with veloci-ties comparable to the terminal velocity, and wenote two minima for the concentration, along thesymmetry axis and at the surface. As Re increases(panel c, Rm = 104, Re = 50), the symmetry of theflow breaks down and a vortex is generated behindthe sphere. The variety of flows, within and outsidethe sphere, is the expression of the diversity ofregimes for chemical and heat transfer. The tran-sition from the drop/diapir case, in which the innercirculation is pronounced, and the particle case,where the sphere acts as a solid, occurs for Rmbetween 1 and 500 in our calculations.

[25] In Figure 2 we depict the average radial con-centrations corresponding to cases with and with-out internal recirculation, for the situations andtimes of the cases in Figures 1b and 1c (the case inFigure 1a, without recirculation is comparable tothe case in Figure 1c for what concerns the averageradial concentration). Although the fits are notperfect, the analytical profiles capture the behaviorof the numerical solutions. The quality of theapproximations are increasing with time, as equili-bration proceeds. As the average concentrations inthe sphere involves Ci(r)r

2, and are thereforemostly controlled by the concentration near r = 1,more accurate fits are not needed.

[26] To benchmark the quality of the predictivelaws proposed above, we compute the evolutions ofthe concentrations in numerical simulations. These

evolutions can be closely matched by exponentials,as predicted. In Figure 3, the Pe dependence pro-posed above reproduces the results of the simula-tions in the various cases, at low and high Re. Theproposed analytical expressions are in very goodagreement with the numerical experiments. For thedrop/diapir case, the analytical model with inviscidflow gives a lower bound for the time of equili-bration since viscosity should slow motion inboundary layers. In Figure 3, the role of the cir-culation within the drop/diapir is expressed by theshorter equilibration time for the fluid sphere caserelative to the solid case. Indeed, the circulationwithin the sphere produces efficient stirring thatgenerates stronger chemical/thermal gradients whichdiffuse away more rapidly. As Pe increases, thenon‐dimensional time of equilibration increases. Tokeep the same reference time scale for the non‐dimensionalization (fixing the values for Ut and R)while increasing Pe implies that diffusivity has tobe decreased. It is then expected that keeping thevelocity constant and decreasing the diffusivitythwarts equilibration.

[27] The observed role of Re in the simulations isalso consistent with our predictions with andwithout internal circulation, as seen in Figure 4.Increasing Re leads to a decrease of the dimen-sionless time of equilibration. At high Re, veloci-ties can be larger and stronger velocity gradientsare allowed which favor a faster mixing. Indeed,the higher the Re the thinner the boundary layeraround the sphere, and thus the more efficient thediffusion across the drop interface. However, thismechanism is somewhat modest since a limitedreduction of the equilibration time by a factor of2 requires more than 3 order of magnitude higherRe. The onset of the wake instability does notgenerate significant changes in the equilibrationstyle mostly because the flux of elements/heat isdominated by the fluxes at the front of the spherewhile it remains close to zero in the wake.

[28] Figure 5 shows t as a function of RD for afixed Pe for the drop/diapir and particle cases atlow and high Re. Again the numerical results showa good agreement with our theoretical predictions:when RD > 1, diffusion inside the sphere is moreefficient than outside. Hence, diffusion in the hostliquid is the limiting parameter for equilibrationand t does not depend on RD, cf. equation (13).For RD < 1, diffusion in the sphere is the limitingparameter and as a consequence t decreases withRD. As explained above, the analytical model for the


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high Re drop regime represents a lower bound for theanalytical model since inviscid fluid is considered.

4. Discussion and Conclusions

[29] We presented approximate analytical modelsto predict equilibration times for spherical particles,drops and diapirs traveling through a viscouslydeforming surroundings due to buoyancy forces.Numerical simulations for a wide range of parametersconfirm our predictive laws that can be used ingeophysical problems at any scale. Small differencesbetween analytical and numerical predictions canhowever be noticed (particularly visible in Figure 4where we use a vertical linear scale). This might bedue to the several assumptions of the analyticalmodels (asymptotic expressions and choices ofsimple radial profiles) or of the numerical simula-tions (finite size of the computation domain).

[30] We showed that it is fundamental to take intoaccount the flow structure and hence evaluate thecorrect regime for a given situation. The existenceof an internal circulation within the spherical bodyis essential since it significantly reduces the timeneeded for equilibration. Compared to the purelydiffusive systems, advective motion gives rise tothinner boundary layers and thus raises concentra-tion gradients. Consequently, diffusion transportinside the spherical body and whole equilibrationare more efficient than for a particle or motionless

drop/diapir. When inertia dominates over the vis-cous forces, the boundary layer is even thinnerspeeding up further diffusion across the rim.Concerning the role of diffusivity and viscosity, ahigh diffusion rate of the surrounding host liquidalways favors a rapid equilibration. The role ofthe external viscosity is more complex. The time ofequilibration decreases both when the externalviscosity is too low (in which case no stirringoccurs within the drop) and when it is too large (inwhich case the terminal velocity and the internalvelocities also decrease). The cases where theinternal and external viscosities are close, i.e., thetransition between drops (internal recirculation)and particles (no internal recirculation), are difficultto predict analytically. In the low Re number limit,the flow can be expressed as a function of Rm bythe Rybczynski‐Hadamard formulae but a choicehas to be made for the value of b (b = 10 for a drop,b = p2/3 for a particle). The situation is even morecomplex at high Re numbers where the real solu-tion lies in between the two analytical cases.

[31] The predictive laws are scale‐independent andcan be applied to various geophysical settings with-out computing the full mass transfer problems. Asmall scale problem is hybridization of mafic blobfalling in a more silicic melt. We choose typicalnumbers as those of Grasset and Albarède [1994]:200 kg m−3 density excess than the felsic surround-ings [Huppert et al., 1982], viscosities of 500 Pa s

Figure 2. Average concentrations in the sphere as a function of normalized radius, corresponding to the cases inFigures 1b and 1c (solid lines). The profile approximations (dashed lines), equation (10) with f(r) = r2(r − 1) andf(r) = −sin(pr)/(pr), are in reasonable agreement with the simulations, particularly near r = 1.


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and 25000 Pa s for the mafic blob and host silicicmelt and viscosity, respectively, and 10 cm for theblob diameter. The Rybczynski‐Hadamard formulagives a terminal velocity of Ut = 5.3 · 10

−5 m s−1

which implies Re = 2 · 10−7. The correspondingdynamic regime is that of a drop traveling at low Re.To compute the time of equilibration we use typical

diffusion coefficients of 10−12 m2 s−1 for traceelements for both liquids. As a consequence, Pe isabout 2.7 · 106. We arbitrarily choose a partitioncoefficient of 2 between the two melts which meansthat a trace element will be twice more abundant inthe mafic enclave than in the silicic melt after fullequilibration. The mafic blob is much less viscousthan the surrounding melt. Stirring inside the bodythus enhances the hybridization rate and the char-acteristic time of equilibration is 2.7 years corre-sponding to a falling distance of 4535 m. This is asignificantly shorter time than if equilibration pro-ceeded only by static diffusion obtained fromequation (15). Without any movement the equili-bration could be attained in about 60 years. Notice,that many people would use the classical expressiont = R2/(p2Di) that gives for this case an equilibra-tion time of 8 years, but is physically inappropriateas it neglects the diffusion in the surroundings ofthe sphere and the partition coefficient.

[32] For small iron droplets falling through a sili-cate magma ocean during early planetary differ-entiation, we use values similar to those given byRubie et al. [2003], with a drop size of R = 0.5 cm.The most uncertain and critical parameter is theviscosity of molten silicates composing the magmaocean ranging in a wide interval 10−4–100 Pa s.Choosing 0.01 Pa s gives us a terminal velocityUt =0.6 m s−1 using the work of Brown and Lawler[2003] for high Re flows. The viscosity of irondroplets is fixed at mi = 0.01 Pa s [Vočadlo et al.,2000] and we choose a partition coefficient ofK = 30, which would be that for nickel at a pres-

Figure 3. Nondimensional time of equilibration as afunction of Péclet number (Pe) for drops and particles,at Re = 0 and Re = 50. In these simulations, the ratioof internal to external diffusivity is RD = 1000 and forthe drop Rm = 0.1. The analytical relationships aredepicted by dashed lines.

Figure 4. Nondimensional time of equilibration as afunction of Re for drops and particles. In these simula-tions the Péclet number is Pe = 2800, ratio of internalto external diffusivity is RD = 1000 and for the dropcase, ratio of in and out viscosities is Rm = 0.1. The ana-lytical relationships for low and high Re are depicted bydashed lines. The transition from low to high Re regimehappens around Re = 1.

Figure 5. Nondimensional equilibration time as a func-tion of diffusivity ratio RD for drops and particles, atlow and higher Reynolds number Re (Re = 0 and 50here). In these simulations the Péclet number is Pe =2800 and for the drop viscosity ratio Rm = 0.1. The ana-lytical relationships are depicted by dashed lines.


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sure of 50 GPa [Li and Agee, 2001]. The diffusivityof nickel in liquid iron is set to Di = 10

−8 m2 s−1

which is estimated from self‐diffusion in liquidFe at high pressure [Dobson, 2002]. The diffu-sion coefficient in molten silicate is chosen tobe Do = 10

−9 m2 s−1, imposing a diffusion ratio of10. Using parameters above results in Pe, Re, andRm of 3 × 106, 1000, and 1, respectively. Hence,the regime is that of a drop in a high Re fluid.In this case equilibration should be attained in4 minutes with a traveled distance around 126 m.This distance is certainly shorter than the depth ofa magma ocean that could be generated froman impact with a Mars‐sized object [Tonks andMelosh, 1992]. Our predictions are of the sameorder of magnitude as the results obtained byRubie et al. [2003] considering the uncertaintieson the parameters. However, our theory takes intoaccount the flow within the drop and effect ofhigh Re, and thus proposes an intrinsically fastertime of equilibration and shorter distance thanRubie et al. [2003].

[33] Further applications can be made (crystal set-tling in granitoids or ignimbrites, immiscible silicateand carbonatitic melts segregation etc…) using thecorrect proposed predictive relationships. How-ever, we have made 3 main assumptions that haveto be considered as limitations:

[34] 1. First, sphericity was assumed, which isknown to be matched for drops when the surfacetension dominates and for low Re, and for solidshaving a spheroidal shape. The deformation ofdiapirs and drops by viscous stresses along theboundaries can lead to peculiar shapes possiblyskirted and with instabilities leading to break‐up.More complex calculations have to be performed tofollow the shape evolution in such case.

[35] 2. A second assumption involves non‐interactingbodies. In the case of the previously advocatedmetallic rain in magma oceans the coalescence andinfluence of neighboring droplets have to be takeninto account [Ichikawa et al., 2010].

[36] 3. Third, we assumed pure buoyancy drivenflow. The settling of particles and drops can beinfluenced by the effect of rotation or other forcesthat we have not considered in the present study.

Acknowledgments

[37] This work has been supported by the Agence Nationale dela Recherche under grant ANR‐08‐JCJC‐0084‐01 (Dyn‐BMO).J.V. acknowledges the support of theGrant Agency of the Czech

Republic, project P210/11/1366. Numerical models where per-formed using the free FEM software Elmer, from CSC. Thispaper also benefited from LaTeX, GMT and Gnuplot magic.We thank J. Rudge and P. J. Tackley for constructive reviews.

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