scholarly journals Solving two‐phase freezing Stefan problems: Stability and monotonicity

2019 ◽  
Vol 43 (14) ◽  
pp. 7948-7960
Author(s):  
Miguel A. Piqueras ◽  
Rafael Company ◽  
Lucas Jódar
Keyword(s):  
Two Phase ◽  
Stefan Problems

Exact Solutions of a Class of Two-Phase Stefan Problems in Heterogeneous Cylinders and Spheres

2014 ◽  
Author(s):  
Olawanle P. Layeni ◽  
Olusoji Ilori ◽  
Ebenezer O. B. Ajayi
Keyword(s):  
Difference Equations ◽  
Spherical Symmetry ◽  
Spatial Dimension ◽  
Similarity Solutions ◽  
Suitable Model ◽  
Two Phase ◽  
Stefan Problems ◽  
Temperature Regimes ◽  
Arduous Task ◽  
Cylinders And Spheres

The classical Stefan problem proffers a suitable model for determining the temperature regimes as well as conjugate interfacial positions for multiphase problems. Obtaining the solutions to these problems exactly, especially in systems with cylindrical or spherical symmetry, is often an arduous task. This is largely due to inherent nonlinearities in the mathematical statements of Stefan problems. In this paper, a tractable and effective approach is proposed. Subsequent to a recast as a system of differential-difference equations, and a methodical reduction to constant coefficient difference equations, exact similarity solutions are derived for a class of heterogeneous two-phase Stefan problems with cylindrical or spherical symmetry in one spatial dimension, under either Gaussian or hypergeometric perturbations.

Macroscopic models for melting derived from averaging microscopic Stefan problems I: Simple geometries with kinetic undercooling or surface tension

2000 ◽  
Vol 11 (2) ◽  
pp. 153-169 ◽  
Author(s):  
A. A. LACEY ◽  
L. A. HERRAIZ
Keyword(s):  
Free Boundary ◽  
Multiple Scales ◽  
Two Dimensions ◽  
Macroscopic Model ◽  
Mushy Region ◽  
Free Boundaries ◽  
Two Phase ◽  
Macroscopic Models ◽  
Stefan Problems ◽  
Kinetic Undercooling

A mushy region is assumed to consist of a fine mixture of two distinct phases separated by free boundaries. For simplicity, the fine structure is here taken to be periodic, first in one dimension, and then a lattice of squares in two dimensions. A method of multiple scales is employed, with a classical free-boundary problem being used to model the evolution of the two-phase microstructure. Then a macroscopic model for the mush is obtained by an averaging procedure. The free-boundary temperature is taken to vary according to Gibbs–Thomson and/or kinetic-undercooling effects.

Method of the integral error functions for the solution of the one- and two-phase Stefan problems and its application

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1017-1029 ◽  
Author(s):  
Merey Sarsengeldin ◽  
Stanislav Kharina
Keyword(s):  
A Priori ◽  
Linear Combinations ◽  
Two Phase ◽  
Electrical Arc ◽  
Error Functions ◽  
Stefan Problems ◽  
Finite Sums ◽  
Integral Error ◽  
Faà Di Bruno Formula ◽  
The One

The analytical solutions of the one- and two-phase Stefan problems are found in the form of series containing linear combinations of the integral error functions which satisfy a priori the heat equation. The unknown coefficients are defined from the initial and boundary conditions by the comparison of the like power terms of the series using the Faa di Bruno formula. The convergence of the series for the temperature and for the free boundary is proved. The approximate solution is found using the replacement of series by the corresponding finite sums and the collocation method. The presented test examples confirm a good approximation of such approach. This method is applied for the solution of the Stefan problem describing the dynamics of the interaction of the electrical arc with electrodes and corresponding erosion.

scholarly journals Some recent results for heat-diffusion moving boundary problems

1985 ◽  
Vol 32 (3) ◽  
pp. 437-460
Author(s):  
James M. Hall ◽  
Jeffrey N. Dewynne
Keyword(s):  
Stefan Problem ◽  
Moving Boundary ◽  
Formal Series ◽  
Heat Diffusion ◽  
Upper And Lower Bounds ◽  
Two Phase ◽  
Boundary Problems ◽  
Stefan Problems ◽  
Analytical Approximations ◽  
Cylinders And Spheres

Integral formulations for the three classical single phase Stefan problems involving the infinite slab and inward solidifying cylinders and spheres are utilized to generate standard analytical approximations. These approximations include the pseudo steady state estimate, large Stefan number expansions, upper and lower bounds, approximations based on integral iteration and related results such as formal series solutions. In order to demonstrate the applicability and limitations of the integral formulations three generalizations of the classical stefan problem are considered briefly. These problems are diffusion with two simultaneous chemical reactions, a Stefan problem with two moving boundaries and the genuine two phase Stefan problem.

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