A mathematical model for marangoni flow and mass transfer in electrostatically positioned droplets
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I. INTRODUCTION
DURING electrostatic levitation, the gravitational force acting on a sample is balanced by the Coulomb force that results from the interaction between an applied electric field and the electrical charges pressed on the sample. For a liquid droplet, Rayleigh’s linear stability analysis shows that there exists a limit for the amount of charges that can be applied on the droplet and that the droplet will disintegrate if the charges exceed the limit.[1] This has limited the size of the sample to be levitated using the electrostatic forces in terrestrial environment. In microgravity, the Coulomb forces are exploited to position the sample at a desired position and thus a free sample of a much larger size is possible. Since it is not in contact with container walls, a levitated droplet is free from unwanted impurity contamination. This distinctive advantage, combined with a microgravity environment, has been exploited for the purpose of measuring the thermophysical properties, such as viscosity, heat capacity, surface tension, conductivity, emissivity, and diffusivity, of highly corrosive, high melting point metals and undercooled melts, which would otherwise be either difficult or impossible with conventional measuring techniques under earthbound conditions.[2,3,4] Compared with the wellknown electromagnetic levitation processes by which a droplet is levitated by Faraday’s principle and which are applicable to electrically conducting materials, electrostatically levitated droplets enjoy additional advantages of being able to be applied to virtually any kind of materials, regardless of whether they are electrically conducting.
Y. HUO, Graduate Student, and B.Q. LI, Professor, are with the School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164. Contact e-mail: [email protected] Manuscript submitted March 11, 2004. METALLURGICAL AND MATERIALS TRANSACTIONS B
In planning experiments for electrostatically levitated droplets, information on the internal fluid flow, heat and mass transfer, and surface deformation of a droplet in electric fields is needed. Unlike the electromagnetic levitation systems where strong stirring is generated by induced Lorentz forces, electrostatic forces are not vorticial and thus do not induce an internal flow in droplets that are electrically conducting such as metals and semiconductor melts. Internal flows do occur, however, in these droplets from different origins. One important contribution to the internal flow in these droplets is the surface tension gradient along the droplet-free surface. The imbalance in the surface shear force results from the surface thermal gradient and drives the Marangoni convection. There are both advantages and disadvantages associated with internal flows in a free droplet. On the one hand, internal flow should be avoided if properties such as surface tension, viscosity, and diffusivities are to be measured. On the other hand, a strong internal flow may be desirable if a certain solidification microstructure is to
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