Dissertation

Convection in multicomponent systems
Martin Hurban
PhD thesis advisor: Peter Guba

Abstract: In this thesis, two problems involving solidification have been studied. The first is the solidification of a binary alloy pulled horizontally at a prescribed, constant rate. The second is the solidification of a ternary alloy in the vertical directional solidification setting. In the binary alloy case, a configuration in which the solid, liquid and two-phase (solid-liquid) regions are separated by the stationary two-dimensional interfaces is considered. The self-similar solutions of the governing boundary-layer equations are obtained, and their parametric dependence analysed asymptotically. The effect of the boundary-layer flow on the physical characteristics is determined. It is found that the horizontal pulling and the resulting flow in the liquid enhance the formation of the two-phase region. In the ternary alloy case, we identify a steady non-convecting state of during the primary solidification of ternary alloys. A model, which includes the effects thermal and solutal diffusion, segregation effects and finite speed of background solidification is considered. Combinations of various types of boundary conditions have been introduced, namely F-C, C-F and F-F, where C and F refer to the solute concentration and solutal flux fixed at the top or bottom boundary of the system. In the regime of the same Lewis numbers and segregation coefficients of solutes, explicit solution using hypergeometric functions was identified. In the limit of large Lewis and the limit of near constant concentration profile of one solute asymptotic solutions were presented. The behaviour of the concentration profiles was analysed with respect to a static stability scenario, i.e. distribution of a mass within liquid.

References
[1] Kyselica, J., Guba, P., Hurban, M., Solidification and flow of a binary alloy over a moving substrate, Transp. Porous Media, 121 (2) (2018), 419-435.
[2] Hurban, M., Steady non-convection states in ternary alloy solidification, under review, (2019).