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.

Macroscopic models for melting derived from averaging microscopic Stefan problems II: Effect of varying geometry and composition

2002 ◽  
Vol 13 (3) ◽  
pp. 261-282 ◽  
Author(s):  
A. A. LACEY ◽  
L. A. HERRAIZ
Keyword(s):  
Alloy Composition ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Macroscopic Model ◽  
Mushy Region ◽  
Free Boundaries ◽  
Final Model ◽  
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. A method of multiple scales, with restrictions on the form of the microscopic free boundaries, is used to derive a macroscopic model for the mushy region. The final model depends both on the microscopic structure and on how the free-boundary temperature varies with curvature (Gibbs–Thomson effect), kinetic undercooling, or, for an alloy, composition.

Macroscopic models for calcium carbonate corrosion due to sulfation. Variation of diffusion and volume expansion

2018 ◽  
Vol 30 (3) ◽  
pp. 529-556
Author(s):  
CHRISTOS V. NIKOLOPOULOS
Keyword(s):  
Calcium Carbonate ◽  
Volume Expansion ◽  
Moving Boundary ◽  
Multiple Scales ◽  
Corrosion Process ◽  
Mushy Region ◽  
Moving Boundary Problems ◽  
Boundary Problems ◽  
Macroscopic Models ◽  
The Subject

The subject of the present paper is the derivation and analysis of mathematical models for the formation of a mushy region during calcium carbonate corrosion. More specifically there is emphasis on the variation of the overall diffusion resulting from the changing shape of a single pore due to corrosion process and on the resulting volume expansion of the material as the outcome of the transformation of calcium carbonate to gypsum. These models are derived by averaging, with the use of the multiple scales method applied on microscopic moving-boundary problems. The latter problems describe the transformation of calcium carbonate into gypsum in the microscopic scale. The derived macroscopic models are solved numerically with the use of an implicit in time, finite element method. The results of the simulations for various microstructure geometries in the micro-scale and a discussion are also presented.

scholarly journals Two-Phase Free Boundary Problems: From Existence to Smoothness

2017 ◽  
Vol 17 (2) ◽  
Author(s):  
Daniela De Silva ◽  
Fausto Ferrari ◽  
Sandro Salsa
Keyword(s):  
Free Boundary ◽  
Viscosity Solutions ◽  
Free Boundary Problems ◽  
Elliptic Operators ◽  
Free Boundaries ◽  
Two Phase ◽  
Boundary Problems ◽  
Open Questions

AbstractWe describe the theory we developed in recent times concerning two-phase free boundary problems governed by elliptic operators with forcing terms. Our results range from existence of viscosity solutions to smoothness of both solutions and free boundaries. We also discuss some open questions, possible object of future investigation.

scholarly journals A Conservative Macroscopic Model for Binary-mixture Fluidized Beds

2021 ◽  
Author(s):  
Mohamed Sobhi Alagha ◽  
Pal Szentannai
Keyword(s):  
Binary Mixture ◽  
Fluidized Beds ◽  
Vertical Mixing ◽  
Computational Cost ◽  
Mass Conservation ◽  
Macroscopic Model ◽  
Two Phase ◽  
Phase Theory ◽  
Macroscopic Modeling ◽  
Macroscopic Models

Two approaches are commonly used for modeling the vertical mixing of binary-mixture fluidized beds, Computational Fluid Dynamics (CFD) and macroscopic modeling. A common realization of the latter one is the Gibiralo–Rowe (G-R) model, which uses the Two-Phase Theory. This macroscopic model obviously overperforms CFDs regarding computational cost; however, determining its coefficients is a still challenging issue. Although several methods were published for solving this, the general problem with most of them remains their neglecting the conservation of mass. In the present new procedure, the mass conservation is applied to correct the values of the G-R model coefficients estimated from known equations. The present model was validated on a wide variety of fluidized bed systems. The results show that this conservative and macroscopic model gives more accurate predictions than the recently published other macroscopic models, and this one is, in general, better than the CFD model from the perspective of prediction accuracy as well.

A rapid method for the identification of the free boundary in two-phase Stefan problems

1994 ◽  
Vol 14 (3) ◽  
pp. 411-420 ◽  
Author(s):  
T. MÄNNIKKÖ ◽  
P. NEITTAANMÄKI ◽  
D. TIBA
Keyword(s):  
Free Boundary ◽  
Rapid Method ◽  
Two Phase ◽  
Stefan Problems

scholarly journals Two-phase problem with a free boundary for systems of parabolic equations with a nonlinear term of convection

2021 ◽  
pp. 110-122
Author(s):  
А.Н. Элмуродов
Keyword(s):  
Free Boundary ◽  
Parabolic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Reaction Diffusion ◽  
Free Boundaries ◽  
Uniqueness Of Solutions ◽  
Two Phase ◽  
Diffusion Type ◽  
A Free Boundary Problem

Эта статья посвящена задаче со свободной границей для полулинейных параболических уравнений, в которой описывается феномен сегрегации местообитаний в популяционной экологии. Основная цель — показать глобальное существование, единственность решений проблемы. Предлагается двухфазная математическая модель со свободными границами для параболических уравнений типа реакция-диффузия. Установлены априорные оценки щаудеровского типа, на основе которых доказана однозначная разрешимость задачи. Неустойчивость каждого решения полностью определяется с помощью теоремы сравнения. This article is concerned with a free boundary problem for semilinear parabolic equations, wbich describes the habitat segregation phenomenon in population ecology. The main goal is to show global existence, the uniqueness of solutions to the problem. A two-phase mathematical model with free boundaries for parabolic equations of the reaction-diffusion type is proposed. A priori estimates of Schauder type are established, on the basis of which the unique solvability of the problem is proved. The instability of each solution is fully determined using the comparison theorem.

scholarly journals On the Two-phase Fractional Stefan Problem

2020 ◽  
Vol 20 (2) ◽  
pp. 437-458 ◽  
Author(s):  
Félix del Teso ◽  
Jørgen Endal ◽  
Juan Luis Vázquez
Keyword(s):  
Free Boundary ◽  
Stefan Problem ◽  
Free Boundary Problems ◽  
Classical Problem ◽  
Nonlocal Diffusion ◽  
Two Phase ◽  
Boundary Problems ◽  
Numerical Studies ◽  
The One ◽  
Evolution Type

AbstractThe classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing the main properties of the classical problem that are of interest to us. Then we introduce the fractional Stefan problem and develop the basic theory. After that we center our attention on selfsimilar solutions, their properties and consequences. We first discuss the results of the one-phase fractional Stefan problem, which have recently been studied by the authors. Finally, we address the theory of the two-phase fractional Stefan problem, which contains the main original contributions of this paper. Rigorous numerical studies support our results and claims.

scholarly journals Investigation of representing hysteresis in macroscopic models of two-phase flow in porous media using intermediate scale experimental data

2017 ◽  
Vol 53 (1) ◽  
pp. 199-221 ◽  
Author(s):  
Abdullah Cihan ◽  
Jens Birkholzer ◽  
Luca Trevisan ◽  
Ana Gonzalez-Nicolas ◽  
Tissa Illangasekare
Keyword(s):  
Experimental Data ◽  
Porous Media ◽  
Two Phase Flow ◽  
Flow In Porous Media ◽  
Phase Flow ◽  
Two Phase ◽  
Intermediate Scale ◽  
Macroscopic Models
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