Micro/nanoparticle melting with spherical symmetry and surface tension

Numerical study of two ill-posed one phase Stefan problems

“…For a given set of parameter values, there is a point in time after which the solid is locally superheated, with the temperature everywhere in the solid greater than the melting temperature (although less than the bulk melting temperature for a flat interface). This behaviour is illustrated in Figure 1 [12].…”

“…In Section 3 of the present study, we consider the one dimensional ('toy') version of this novel moving boundary problem, and note that when the surface tension parameter vanishes, the problem reduces to the ill-posed superheated Stefan problem (1)-(5). Indeed, McCue et al [12] observe that blow-up for (8)- (14) may be of the same form as that for (1)- (5). Section 3's brief numerical study of the toy problem supports this view, and suggests the matter is worth We first introduce the transformation ξ = x/s(t) due to Landau [14].…”

“…The two phase problem (8)- (14) is treated in [12,13]. Numerical solutions show that as melting proceeds, the interface r = s(t) moves towards the centre of the ball, and the melting temperature decreases according to (11).…”

“…In this case the problem without surface tension (σ = 0) is known to be well-posed, and it is the addition of surface tension that appears to lead to blow-up. [12] use this observation to suggest that u ∼ σ(1 − 1/r) for r − s(t) 1 as t → t − c , which leads to a novel one phase problem that they leave for further research. In Section 3 of the present study, we consider the one dimensional ('toy') version of this novel moving boundary problem, and note that when the surface tension parameter vanishes, the problem reduces to the ill-posed superheated Stefan problem (1)-(5).…”

“…As mentioned in Section 1, this is the one dimensional version of a problem suggested by McCue et al [12,Discussion]. The two parameters are σ, a surface tension parameter, and β, the Stefan number.…”