On small-time similarity-solution behaviour in the solidification shrinkage of binary alloys

Abstract

In the one-dimensional solidification of a binary alloy undergoing shrinkage, there is a relative motion between solid and liquid phases in the mushy zone, leading to the possibility of macrosegregation; thus, the problem constitutes an invaluable benchmark for the testing of numerical codes that model these phenomena. Here, we revisit an earlier obtained solution for this problem, that was posed on a semi-infinite spatial domain and valid for the case of low superheat, with a view to extending it to the more general situation of a finite spatial domain, arbitrarily large superheat and both eutectic and non-eutectic solidification. We find that a similarity solution is available for short times which contains a boundary layer on the liquid side of the mush–liquid interface; this solution is believed to constitute the correct initial condition for the subsequent numerical solution of the full non-similar problem, which is deferred to future work.

On macrosegregation in a binary alloy undergoing solidification shrinkage

“…The mathematical problem to be solved was recently formulated in [6]; however, only similarity solutions available for short times were computed, although it was suggested that these solutions could be used as initial conditions for solving the problem for all time, when the problem no longer has a similarity solution. Thus, the purpose of this paper is to tackle the remainder of the problem, and thereby solve it up to complete solidification; doing so would then yield the macrosegregation profile over the entire domain.…”

Explicit solutions related to the Rubinstein binary-alloy solidification problem with a heat flux or a convective condition at the fixed face

“…A study of binary-alloy problems can be seen in [2,5,8,12,16,17,24,25]. Recent works on the solidification of a binary alloy are [1,3,5,6,7,9,10,11,12,14,21,22,23,28]. The heat flux condition (1.v) imposed at the fixed face was first considered in [18] for a two-phase Stefan problem for a semi-infinite homogeneous material.…”

“…In Section 2 we obtain the necessary and sufficient condition (2) for problem (P1) in order to obtain the explicit solution (3)- (8). In Section 3 we deduce the inequality (38) for the coefficient µ that characterizes the Rubinstein free boundary x = s(t) for problem (P1) defined in (3). In Section 4 we obtain the necessary and sufficient condition (40) for problem (P2) in order to obtain the explicit solution (41)-(46).…”

Mathematical modelling of macrosegregation in binary alloys induced by convection

“…One of the possible configurations for this event is illustrated in Fig. 1.8, in this case the intermediate secondary dendritic arm has a root that is thinner than normal, thus, the constitutional super cooling 1 which was sufficient at the end of the dendrite to form its secondary arms, is reduced to a lower value such that the smallest radius of curvature can affect the melting point making it smaller in that region resulting in the melting of the lower portion and the separation of the secondary arm from the primary arm.…”

“…Esta tese apresenta: um modelo matemático 2D para a formação da cavidade central utilizando balanço de massa e conservação de energia com calor latente no termo fonte, capaz de verificar numericamente a eficácia de aparatos atualmente usados pela indústria para reduzir a profundidade da cavidade; um modelo de macrosegegação 2D com formulação de domínio único incluindo momento, calor, massa e conservação de soluto, o qual reproduz dados presentes na literatura; e por último, um novo modelo de macrosegreação 1D sujeito à contração, consistindo em três regiões distintas limitadas por três interfaces móveis, para o qual uma solução por similaridade válida para tempos pequenos foi obtida e validada por dados numéricos disponíveis na literatura, servindo como condição inicial para o problema não-similar usado para traçar diretrizes que podem, em certas circunstâncias, reduzir o desvio padrão sobre a concentração inicial de soluto em até 40%. Pr -Prandtl number [1] Ra -Rayleigh number [1] Ga -Galilei number [1] Da -Darcy number [1] St -Stefan number [1] Le -Lewis number [1]…”