Some criteria for the disappearance of the mushy region in the Stefan problem (original) (raw)
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2016
We study a nonlocal version of the one-phase Stefan problem which develops mushy regions, even if they were not present initially, a model which can be of interest at the mesoscopic scale. The equation involves a convolution with a compactly supported kernel. The created mushy regions have the size of the support of this kernel. If the kernel is suitably rescaled, such regions disappear and the solution converges to the solution of the usual local version of the one-phase Stefan problem. We prove that the model is well posed, and give several qualitative properties. In particular, the longtime behavior is identified by means of a nonlocal mesa solving an obstacle problem.
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For an arbitrary bounded solution of the Stefan problem the mushy region is nonincreasing in time in a sense of the theory of sets. This result takes place for the nonhomogeneous Stefan problem under some conditions on the behavior of a heat source in the mushy region.
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Canadian Mathematical Bulletin, 1992
This paper deals with the Stefan-type problem with a zone of coexistence of both phases. We formulate the problem in the enthalpy form and show that the interfaces between the liquid and the mushy, the mushy and the solid phase are smooth. Our approach is to study the structures of the level sets of the solution via Sard's Lemma and the implicit function theorem.
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We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data.
Computational and Applied Mathematics
A two-phase solidification process for a one-dimensional semi-infinite material is considered. It is assumed that it is ensued from a constant bulk temperature present in the vicinity of the fixed boundary, which it is modelled through a convective condition (Robin condition). The interface between the two phases is idealized as a mushy region and it is represented following the model of Solomon, Wilson, and Alexiades. An exact similarity solution is obtained when a restriction on data is verified, and it is analysed the relation between the problem considered here and the problem with a temperature condition at the fixed boundary. Moreover, it is proved that the solution to the problem with the convective boundary condition converges to the solution to a problem with a temperature condition when the heat transfer coefficient at the fixed boundary goes to infinity, and it is given an estimation of the difference between these two solutions. Results in this article complete and improve the ones obtained in Tarzia (Comput Appl Math 9:201-211, 1990).
ONE PHASE ONE-DIMENSIONAL STEFAN PROBLEM
Article Info ABSTRACT The main purpose of this paper is to introduce a variable time step method to obtain numerical solution to Neumann form of the Stefan problem. Starting the solution for moving boundary problems has been a difficult proposition and this difficulty led to the development of a variety of approximate methods. The iterative scheme to find the subsequent time step sizes for a given space step is developed with assurance of convergence.
Some Considerations Regarding the Exact Solution in the One Phase-Stefan Problem
2006
The one phase Stefan problem in a semi - infinite slab with heat flux boundary ½ condition proportional to t and with constant temperature boundary condition are presented here. In these two cases the exact solution exists, the relation between the two boundary conditions is presented here, and the equivalence between the two problems is demostrated.