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Liquid convection effects on the pushing-engulfment transition of insoluble particles by a solidifying interface: Part I. Analytical calculation of the lift forces

  • Sundeep Mukherjee
  • Doru M. Stefanescu
Article

Abstract

During the solidification of a liquid containing insoluble particles, the particles can be instantaneously engulfed, or continuously pushed, or pushed and subsequently engulfed. A critical velocity for the pushing-engulfment transition is observed experimentally. Most models proposed to date ignore the complications arising from the liquid convection ahead of the solid-liquid interface. They simply solve the balance between the attractive drag force exercised by the liquid on the particle and the repulsive interfacial force. This work is an effort to calculate analytically the lift forces (Saffman and Magnus forces) under certain assumptions regarding the nature of fluid flow ahead of the solid/liquid interface. This makes possible the quantitative evaluation of the three experimentally observed regimes occurring during particle-interface interaction: (1) at low convection—no effect on the critical velocity for the particle engulfment transition; (2) at intermediate convection—increased critical velocity; (3) at high convection—no particle-interface interaction.

The model was applied to evaluate the gravity level required for microgravity experimental work on particle pushing where the effect of liquid convection during solidification is negligible. This is necessary to validate existing theoretical models that do not take into account fluid flow parallel to the solidification interface.

Keywords

Material Transaction Critical Velocity Lift Force Convection Regime Gravity Level 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

CA

coefficient for the virtual added mass force

FD

drag force

Fg

gravity force

FI

pushing force due to surface energy interactions

FM

Magnus force

FS

Saffman force

FVM

force to accelerate virtual added mass of the particle

L

distance between the center of the particle and the unperturbed solid/liquid interface, characteristic length

L*

nondimensional distance between the center of the particle and the unperturbed solid/liquid interface

Lref

reference length

RIt

position of the tip of the SL interface

RIt*

position of the tip of the SL interface in nondimensional form

RP

radius of the particle

Re

flow Reynolds number

V0

far-field convection velocity

V100

convection velocity at 100 µm from the interface

VLx

liquid velocity in the x direction

VP

particle velocity

Vref

reference velocity

Vrel

velocity of the particle relative to the liquid

VSL

solidification velocity

We

Weber number

a0

atomic diameter

d

distance between the particle and the solid/liquid interface

g

gravitational acceleration

kL

thermal conductivity of liquid

kP

thermal conductivity of particle

k*

ratio of k P by k L

mP

mass of the particle

t

time

tref

reference time

x, y, z

coordinate axes

α

switching variable, angle between the gravity vector and SL interface

β

switching variable

Δγ0

surface energy difference

δ

boundary layer width

η

dynamic viscosity of the melt

v

kinematic viscosity of the melt

ρ

density

ω

rotational velocity

Δρ

density difference

Subscripts

I

interface

L

liquid

P

particle

S

solid

ref

reference

rel

relative

t

at the tip of interface perturbation

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Copyright information

© ASM International & TMS-The Minerals, Metals and Materials Society 2004

Authors and Affiliations

  • Sundeep Mukherjee
    • 1
  • Doru M. Stefanescu
    • 2
  1. 1.Metallurgical and Materials EngineeringThe University of AlabamaTuscaloosa
  2. 2.Solidification LaboratoryThe University of AlabamaTuscaloosa

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