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A mathematical model for marangoni flow and mass transfer in electrostatically positioned droplets

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Abstract

A mathematical model is developed for the free surface deformation, full three-dimensional (3-D) Marangoni convection and solute transport phenomena in electrostatically levitated droplets under microgravity. The electric field is calculated by the boundary element method and the shape deformation by the weighted residuals method. The numerical model for the transport phenomena is developed based on the Galerkin finite-element solution of the Navier-Stokes equations, the energy balance equation, and the mass transport equation. Numerical simulations are carried out for droplet deformation by electrostatic forces and 3-D Marangoni convection in droplets heated by three different heating source arrangements. Results show that the electric forces deform a droplet into an oval shape under microgravity by pulling the droplet apart at the two poles. A two-beam heating arrangement results in an axisymmetric flow and temperature distribution in the droplet. Complex 3-D Marangoni flow structure occurs when a tetrahedral or octahedral heating arrangement is applied. The thermal transport in the droplet is conduction dominant for the cases studied. In general, the convection is stronger with higher melting point melts. The internal convection has a strong effect on the concentration distribution in the droplet. For melts with high viscosities, a significant reduction in velocity can be achieved with an appropriate laser beam arrangement, thereby permitting a diffusion-controlled condition to be developed.

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Abbreviations

a :

radius of a sphere

C :

molar concentration

C(r i):

geometric coefficient resulting from boundary integral formulation

C p :

heat capacity

E 0 :

electric field

F :

force vector from numerical formulation

î:

unit vector of ith component

k:

thermal conductivity

Re:

Reynolds number, Re = ρVmax a d/μ

n (n r, nz):

outward normal, its r and z components

Q :

net charge on the droplet

Q c :

critical charge

Q o :

laser beam heat flux constant

r, \(\hat r\), r :

point vector, unit vector, and r coordinate

R :

distance measured from the center of the un-formed droplet

t :

tangential vector

T, T∞, Tr :

temperature, temperature of surroundings, reference temperature

T max, T min :

maximum and minimum temperatures

ΔT :

difference between T max and T min

U max :

maximum velocity

u :

velocity

\(\hat z\) :

unit vector of z direction

z :

z coordinate

z c :

center of mass along the z-axis

β :

thermal expansion coefficient

ɛ 0 :

permittivity of free surface or region designated by Θ2

ɛ :

emissivity

▽:

gradient operator

φ AB :

mass diffusivity coefficient

φ :

shape function of velocity

Φ:

electric potential

γ :

surface tension

κ :

geometric parameter for elliptical functions

η :

molecular viscosity

ρ :

density

θ :

shape function of temperature

ω :

shape function of pressure

ξ :

shape function of molar concentration

σ :

electrical conductivity

σ e :

surface charge distribution

σ s :

Stefan-Boltzmann constant

\(\bar \sigma \) :

stress tensor

Θ:

computational domain

d :

droplet

i :

the ith point

l :

laser beam

1:

region inside the droplet

2:

region outside the droplet

3:

region of the solute metal outside the droplet

i :

the ith component

T :

matrix transpose

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Authors and Affiliations

  1. the School of Mechanical and Materials Engineering, Washington State University, 99164, Pullman, WA

    Y. Huo (Graduate Student) & B. Q. Li (Professor)

Authors
  1. Y. Huo
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  2. B. Q. Li
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Huo, Y., Li, B.Q. A mathematical model for marangoni flow and mass transfer in electrostatically positioned droplets. Metall Mater Trans B 36, 271–281 (2005). https://doi.org/10.1007/s11663-005-0029-9

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  • DOI: https://doi.org/10.1007/s11663-005-0029-9

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