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Relationship among solutions for three-phase change problems with Robin, Dirichlet, and Neumann boundary conditions

Julieta Bollati, María Fernanda Natale, José Abel Semitiel, Domingo Alberto Tarzia

TL;DR

This work addresses a three-phase Stefan melting problem in a semi-infinite domain with a convective boundary at $x=0$ and delivers a unique explicit solution via a similarity transformation. It establishes rigorous equivalence between Robin (convective), Dirichlet, and Neumann boundary-condition formulations under precise data relations, enabling translation between boundary-condition configurations. The paper provides existence/uniqueness results for the convective case and explicit similarity solutions for the Dirichlet and Neumann cases, with the moving interfaces characterized by dimensionless parameters $\xi_i$, $\mu_i$, and $\lambda_i$. These findings enhance understanding of boundary-condition effects in multiphase heat transfer and offer practical tools for cross-formulation analyses in phase-change problems.

Abstract

This study investigates the melting process of a three-phase Stefan problem in a semi-infinite material, imposing a convective boundary condition at the fixed face. By employing a similarity-type transformation, the problem is reduced to a solvable form, yielding a unique explicit solution. The analysis uncovers significant equivalences among the solutions of three different three-phase Stefan problems: one with a Robin boundary condition, another with a Dirichlet boundary condition, and a third one with a Neumann boundary condition at the fixed face. These equivalences are established under the condition that the problem data satisfy a specific relationship, providing new insights into the behaviour of phase change problems under varying boundary conditions.

Relationship among solutions for three-phase change problems with Robin, Dirichlet, and Neumann boundary conditions

TL;DR

This work addresses a three-phase Stefan melting problem in a semi-infinite domain with a convective boundary at and delivers a unique explicit solution via a similarity transformation. It establishes rigorous equivalence between Robin (convective), Dirichlet, and Neumann boundary-condition formulations under precise data relations, enabling translation between boundary-condition configurations. The paper provides existence/uniqueness results for the convective case and explicit similarity solutions for the Dirichlet and Neumann cases, with the moving interfaces characterized by dimensionless parameters , , and . These findings enhance understanding of boundary-condition effects in multiphase heat transfer and offer practical tools for cross-formulation analyses in phase-change problems.

Abstract

This study investigates the melting process of a three-phase Stefan problem in a semi-infinite material, imposing a convective boundary condition at the fixed face. By employing a similarity-type transformation, the problem is reduced to a solvable form, yielding a unique explicit solution. The analysis uncovers significant equivalences among the solutions of three different three-phase Stefan problems: one with a Robin boundary condition, another with a Dirichlet boundary condition, and a third one with a Neumann boundary condition at the fixed face. These equivalences are established under the condition that the problem data satisfy a specific relationship, providing new insights into the behaviour of phase change problems under varying boundary conditions.

Paper Structure

This paper contains 8 sections, 11 theorems, 52 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Three-phase Stefan problems with different boundary conditions
  3. Existence and uniqueness of solution by imposing a convective condition at the fixed face
  4. Similarity-type solutions with a temperature and a flux condition at the fixed face
  5. Relationship among problems
  6. Equivalence between problems (P1) and (P2)
  7. Equivalence between problems (P2) and (P3)
  8. Equivalence between problems (P1) and (P3)

Key Result

Theorem 2.1

Assuming $h_0>h_2$ with there exists a unique solution to the problem EcCalor-Fase3--w_1 y w_2 0 and convectiva. The temperature $v_i$ in each phase, for $i=1,2,3$, is described by v3bis, v2bis and v1bis, respectively. The free boundaries $w_2$ and $w_1$, given by w2 and w1, are characterized by the dimensionless parameter

Figures (5)

  • Figure 1: Colour map of the temperature and the free boundaries of the three phase Stefan problem with a Robin type condition defined by \ref{['EcCalor-Fase3']}--\ref{['w_1 y w_2 0']} and \ref{['convectiva']}.
  • Figure 2: Colour map of the temperature and the free boundaries of the three phase Stefan problem with a Dirichlet type condition defined by \ref{['EcCalor-Fase3']}-\ref{['w_1 y w_2 0']} and \ref{['dirichlet']}.
  • Figure 3: Colour map of the temperature and the free boundaries of the three phase Stefan problem with a Neumann type condition defined by \ref{['EcCalor-Fase3']}-\ref{['w_1 y w_2 0']} and \ref{['neumann']}.
  • Figure 4: Coefficient $h_0$ characterizing heat transfer under convective conditions as a function of $A$.
  • Figure 5: Coefficient $q_0$ characterizing the flux condition as a function of $A$.

Theorems & Definitions (30)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 3.1
  • proof
  • ...and 20 more