Numerical solution of differential and integral equations
The thermo- and hydrodynamic processes of impacting fluid metal droplet on solid surface are considered. The mathematical model is based on Navier-Stokes equations for incompressible fluid, and heat transmission equations at the substrate and droplet with due regard to the forces of surface tension and phase changing during freezing. It is supposed that the spherical - symmetrical droplet falls perpendicularly with some velocity on a wettable, non-deformable substrate, its initial temperature less than the crystallization temperature of a particle substance.
Droplet surface deforms after the impact, and the liquid inside particle the cools and solidifies. The displacement velocity of crystallization front and the dynamics fluid area of particle determine the morphology of solidified droplet. In order to describe the crystallization process of solder eutectic structure, Stephan problem approach is used. The heat from the droplet is transmitted only into the substrate covered by thin layers considered as a thermistor. Particle diameter is of about 40 to 100 mm, and typical velocity of precipitating is 1 to 10 m/s.
For the numerical realization, a finite difference algorithm is used. Navier-Stokes and heat transmission equations are preliminary conversed to self-conjugate form. After that, the exponential schemes on a regular grid, used for the equation approximation, have second order of accuracy and are monotonous under any proportion of space and time steps. The free surface of a droplet is marked by a system of sequentially enumerated particles.
The correctness of model and used algorithm have been verified via comparing the results and experimental data. The morphology of the solidified particle influenced by impact velocity, droplet dimension, metal overheating, temperature and thermo - physical properties of substrate has been investigated.
Note. Abstracts are published in author's edition
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