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

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.

[1]  B. Li,et al.  Free surface profiles and thermal convection in electrostatically levitated droplets , 2000 .

[2]  K. Beard,et al.  Small-amplitude oscillations of electrostatically levitated drops , 1990, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[3]  Won-Kyu Rhim,et al.  An electrostatic levitator for high-temperature containerless materials processing in 1-g , 1993 .

[4]  A Hybrid Boundary/Finite Element Method for Simulating Viscous Flows and Shapes of Droplets in Electric Fields , 2001 .

[5]  B. Li,et al.  A Boundary/Finite Element Analysis of Magnetic Levitation Systems: Surface Deformation and Thermal Phenomena , 1998 .

[6]  L. Rayleigh.  XX. On the equilibrium of liquid conducting masses charged with electricity , 1882 .

[7]  Shape and stability of electrostatically levitated drops , 1983, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[8]  R. G. Cox,et al.  Electrohydrodynamic deformation and bursts of liquid drops , 1971, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[9]  B. Li,et al.  A coupled boundary/finite element method for the computation of magnetically and electrostatically levitated droplet shapes , 1999 .

[10]  D. A. Dunnett.  Classical Electrodynamics , 2020, Nature.

[11]  M. Bastian,et al.  An efficient automatic mesh generator for quadrilateral elements implemented using C , 2003 .

[12]  L. Ratke,et al.  Liquid immiscible alloys , 1995 .

[13]  W. Rhim,et al.  Non-Contact Measurements of Thermophysical Properties of Titanium at High Temperature , 2000 .

[14]  B. Li,et al.  Three-dimensional Marangoni convection in electrostatically positioned droplets under microgravity , 2004 .