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

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## 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## Nomenclature

*C*_{A}coefficient for the virtual added mass force

*F*_{D}drag force

*F*_{g}gravity force

*F*_{I}pushing force due to surface energy interactions

*F*_{M}Magnus force

*F*_{S}Saffman force

*F*_{VM}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

*L*_{ref}reference length

*RI*_{t}position of the tip of the SL interface

*RI*_{t}*position of the tip of the SL interface in nondimensional form

*R*_{P}radius of the particle

- Re
flow Reynolds number

*V*_{0}far-field convection velocity

*V*_{100}convection velocity at 100

*µ*m from the interface*V*_{Lx}liquid velocity in the

*x*direction*V*_{P}particle velocity

*V*_{ref}reference velocity

*V*_{rel}velocity of the particle relative to the liquid

*V*_{SL}solidification velocity

- We
Weber number

*a*_{0}atomic diameter

*d*distance between the particle and the solid/liquid interface

*g*gravitational acceleration

*k*_{L}thermal conductivity of liquid

*k*_{P}thermal conductivity of particle

*k**ratio of

*k*_{ P }by*k*_{ L }*m*_{P}mass of the particle

*t*time

*t*_{ref}reference time

*x, y, z*coordinate axes

*α*switching variable, angle between the gravity vector and SL interface

*β*switching variable

- Δ
*γ*_{0} surface energy difference

*δ*boundary layer width

*η*dynamic viscosity of the melt

*v*kinematic viscosity of the melt

*ρ*density

*ω*rotational velocity

- Δ
*ρ* 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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