Metallurgical and Materials Transactions A

, Volume 34, Issue 12, pp 3009–3019

Modeling of marangoni-induced droplet motion and melt convection during solidification of hypermonotectic alloys

  • Menghuai Wu
  • Andreas Ludwig
  • Lorenz Ratke


A two-phase volume averaging approach to model Marangoni-induced droplet motion of the minority liquid phase and the convection in the parent melt during solidification of the hypermonotectic alloys is presented. The minority liquid phase decomposed from the parent melt as droplets in the miscibility gap was treated as the second-phase L2. The parent melt including the solidified monotectic matrix was treated as the first phase L1. Both phases were considered as different and spatially interpenetrating continua. The conservation equations of mass, momentum, solute, and enthalpy for both phases, and an additional transport equation for the droplet density, were solved. Nucleation of the L2 droplets, diffusion-controlled growth, interphase interactions such as Marangoni force at the L1-L2 interface, Stokes force, solute partitioning, and heat release of decomposition were taken into account by corresponding source and exchange terms in the conservation equations. The monotectic reaction was modeled by adding the latent heat on the L1 phase during monotectic reaction, and applying an enlarged viscosity to the solidified monotectic matrix. A two-dimensional (2-D) square casting with hypermonotectic composition (Al-10 wt pct Bi) was simulated. This paper focused on Marangoni motion, hence gravity was not included. Results with nucleation, droplet evolution, Marangoni-induced droplet motion, solute transport, and macrosegregation formation were obtained and discussed.



alloy concentration


critical concentration

c1, c2

volume-averaged species concentration

c*1, c*2

interface concentration under thermal equilibrium




L2 monotectic concentration


monotectic concentration


interface species



C12(= −C21)

species exchange rate

C12d(= −C21d)

species transfer at L1-L2 interface

C12p(= −C21p)

solute partitioning due to phase change


mix concentration

cp(1), cp(2)

specific heat

D1, D2

diffusion coefficient


droplet diameter

f1, f2

volume fraction


Marangoni force on single droplet


Stokes force on a single droplet


volume-averaged Marangoni force




heat-transfer coefficient at casting-mold interface


volume heat-transfer coefficient between two liquid phases

h1, h2


h1ref, h2ref

enthalpy at Tref


interface enthalpy


heat of decomposition


latent heat of monotectic reaction


momentum exchange coefficient


solute partitioning coefficient

k1, k2

thermal conductivity

L1, L2

two liquid phases

M12 (= −M21)

mass-transfer rate per volume


slope of liquidus in phase diagram at c0


mass-transfer rate for a single droplet


droplet nucleation rate


droplet density


maximum droplet density



Q12(= −Q21)

energy exchange rate

Q12d(= −Q21d)

energy exchange by heat transfer

Q12p(= −Q21p)

energy exchange due to phase change


droplet radio


solid-phase Al


solid-phase Bi


critical temperature

T, T1, T2



melting point of pure metal (Al)


melting point of pure metal (Bi)


monotectic temperature


reference temperature for enthalpy definition


temperature gradient




Gaussian distribution width of droplet nucleation law


undercooling for maximum droplet nucleation rate



U12 (= −U21)

momentum exchange rate

U12d (= −U21d)

momentum exchange due to Stokes force

U12p (= −U21p)

momentum exchange due to phase change

u1, u2

velocity component in x direction

u1, u2

velocity vector

u12, u21

interphase velocity


interface velocity

v1, v2

velocity component in y direction

ρ1, ρ2



surface tension at liquid-liquid interface


experimental parameter in Eq. [19]

μ1, μ2


τ1, τ2

stress-strain tensors

Subscripts 1, and 2

indicate first and second liquid phases.


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Copyright information

© ASM International & TMS-The Minerals, Metals and Materials Society 2003

Authors and Affiliations

  • Menghuai Wu
    • 1
  • Andreas Ludwig
    • 1
  • Lorenz Ratke
    • 2
  1. 1.Simulation and Modeling of Metallurgical ProcessesUniversity of LeobenLeobenAustria
  2. 2.the Institute for Space SimulationGerman Aerospace Research Establishment DLRCologneGermany

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