Three-phase boundary motions under constant velocities.
journal contributionposted on 01.01.1994 by Fernando Reitich, H. Mete. Soner
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Abstract: "In this paper we deal with the dynamics of material interfaces such as solid-liquid, grain or antiphase boundaries. We concentrate on the situation in which these internal surfaces separate three regions in the material with different physical attributes (e.g. grain boundaries in a polycrystal). The basic two-dimensional model proposes that the motion of an interface ╬│[subscript ij] between regions i and j (i,j = 1,2,3, i [not =] j) is governed by the equation V[subscript ij] = [mu subscript ij] (f[subscript ij]k[subscript ij] + F[subscript ij]). (0.1) Here V[subscript ij], k[subscript ij], [mu subscript ij] and F[subscript ij] denote, respectively, the normal velocity, the curvature, the mobility and the surface tension of the interface and the numbers F[subscript ij] stand for the (constant) difference in bulk energies. At the point where the three phases coexist, local equilibrium requires that the curves meet at prescribed angles. (0.2) In case the material constants f[subscript ij] are small, f[subscript ij] = [bracket over epsilon f subscript ij] and [epsilon] << 1, previous analyses based on the parabolic nature of the equations (0.1) do not provide good qualitative information on the behavior of solutions. In this case it is more appropriate to consider the singular case with f[subscript ij] = 0. It turns out that this problem, (0.1) with f[subscript ij] = 0, admits infinitely many solutions. Here, we show that a unique solution, 'the vanishing surface tension (VST) solution', is selected by letting [epsilon] -> 0. Furthermore, we introduce the concept of weak viscosity solution for the problem with [epsilon] = 0 and show that the VST solution coincides with the unique weak solution. Finally, we give examples showing that, in several cases of physical relevance, the VST solution differs from results proposed previously."