The mathematical model of the process of the dissolution of the solid particle in the liquid metal under the conducting of the electric current

Abstract
Dissolution process for solid particles surrounded by liquid metal at conducting of electric current essentially differs from dissolution process according to atomic diffusion mechanism. Heterogeneities in liquid conductor at presence of the external physical fields cause a perturbation of this medium's field of velocities. An example of this heterogeneity is solid inclusion. In liquid conductor electric current under influence of some perturbation as a result of interaction with magnetic field raised by this current provokes a motion of fluid which was immovable when an electric current was absent. These motions of conducting fluid will be the more, the more of perturbation of electric current. The spherical solid metal inclusion, immersed in the conducting liquid, represents the given model. The electric current is conducted through the all system. On the basis of magnetohydrodynamics' laws the expression for Stoke's stream function are derived which depended on magnetic permeability and viscosity of liquid, electric conductivity of liquid and solid inclusion, radius of this inclusion and electric current's density. The all variants of the inclusion's conduction were considered : from the until the state. From the developed theory concludes, that the biggest velocity occurs under lowest (dielectric) and the highest (superconductor) inclusion's conductivity. The liquid flow directions for the cases of the higher and lower inclusion's conductivity in comparison with the liquid's conductivity are opposite. Using the derived formula for stream function stationary diffusion equation is solved, and expression for full diffusional flow from inclusion's surface is obtained. The inclusion's dissolution in the molten metal occurs, providing that the inclusion's components dissolve in the molten metal and diffusion coefficients are not equaled zero. The mechanism of the inclusion's dissolution is the convective diffusion. Dividing the full diffusional flow from spherical inclusion's surface by the full diffusional flow in the case of stationary atomic diffusion, the value named by Sherwood's criterion Sh (or Nusselt's diffusion criterion) may be obtained. The dissolution occurs faster than this process runs by the atomic diffusion with the same temperature.The developed mathematical formalism may be used for analyzing of the processes of the mass transfer in the hard alloys (e.g., WC-Co,Ti-Fe-Ni) and others heterogeneous compositions under electric treatment.