On a two‐phase Stefan problem with convective boundary condition including a density jump at the free boundary (original) (raw)
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Journal of Mathematical Analysis and Applications, 2012
Stefan problem Free boundary problems Heat transfer coefficient Asymptotic behavior Order of convergence We consider the one-phase unidimensional Stefan problem with a convective boundary condition at the fixed face, with a heat transfer coefficient (proportional to the Biot number) h > 0. We study the limit of the temperature θ h and the free boundary s h when h goes to zero, and we also obtain an order of convergence. The goal of this paper is to do the mathematical analysis of the physical behavior given in [C. Naaktgeboren, The zero-phase Stefan problem, Int. J. Heat Mass Transfer 50 4614-4622].
MAT Serie A
We study a one-phase Stefan problem for a semi-infinite material with temperaturedependent thermal conductivity and a convective term with a constant temperature boundary condition or a heat flux boundary condition of the type −q 0 / √ t (q 0 > 0) at the fixed face x = 0. We obtain in both cases sufficient conditions for data in order to have a parametric representation of the solution of the similarity type for t ≥ t 0 > 0 with t 0 an arbitrary positive time. We improve the results given in
A Stefan problem for a non-classical heat equation with a convective condition
Applied Mathematics and Computation, 2010
We prove the existence and uniqueness, local in time, of the solution of a one-phase Stefan problem for a non-classical heat equation for a semi-infinite material with a convective boundary condition at the fixed face x = 0. Here the heat source depends on the temperature at the fixed face x = 0 that provides a heating or cooling effect depending on the properties of the source term. We use the Friedman-Rubinstein integral representation method and the Banach contraction theorem in order to solve an equivalent system of two Volterra integral equations. We also obtain a comparison result of the solution (the temperature and the free boundary) with respect to the one corresponding with null source term.
Zeitschrift für angewandte Mathematik und Physik
Recently it was obtained in [Tarzia, Thermal Sci. 21A (2017) 1-11] for the classical twophase Lamé-Clapeyron-Stefan problem an equivalence between the temperature and convective boundary conditions at the fixed face under a certain restriction. Motivated by this article we study the two-phase Stefan problem for a semi-infinite material with a latent heat defined as a power function of the position and a convective boundary condition at the fixed face. An exact solution is constructed using Kummer functions in case that an inequality for the convective transfer coefficient is satisfied generalizing recent works for the corresponding one-phase free boundary problem. We also consider the limit to our problem when that coefficient goes to infinity obtaining a new free boundary problem, which has been recently studied in [Zhou-Shi
2020
Different one-phase Stefan problems for a semi-infinite slab are considered, involving a moving phase change material as well as temperature dependent thermal coefficients. Existence of at least one similarity solution is proved imposing a Dirichlet, Neumann, Robin or radiative-convective boundary condition at the fixed face. The velocity that arises in the convective term of the diffusion-convection equation is assumed to depend on temperature and time. In each case, an equivalent ordinary differential problem is obtained giving rise to a system of an integral equation coupled with a condition for the parameter that characterizes the free boundary, which is solved though a double-fixed point analysis. Some solutions for particular thermal coefficients are provided.
2011
An analytical solution for a Stefan problem with variable latent heat, Int. J. Heat Mass Transfer 47 (2004) 5387-5390], and Lorenzo-Trueba and Voller [J. Lorenzo-Trueba, V.R. Voller, Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation, J. Math. Anal. Appl. 366 -549], a model associated with the formation of sedimentary ocean deltas is studied through a one-phase Stefan-like problem with variable latent heat. Motivated by these works, we consider a two-phase Stefan problem with variable latent of fusion and initial temperature, and constant heat flux boundary conditions. We obtain the sufficient condition on the data in order to have an explicit solution of a similarity type of the corresponding free boundary problem for a semi-infinite material. Moreover, the explicit solution given in the first quoted paper can be recovered for a particular case by taking a null heat flux condition at the infinity.
Convergence of the solution of the one-phase Stefan problem with respect two parameters
2015
A one-phase unidimensional Stefan problem with a convective boundary condition at the fixed face x = 0, with a heat transfer coefficient h > 0 (proportional to the Biot number) and an initial position of the free boundary b = s(0) > 0 is considered. We study the limit of the temperature θ = θ b,h and the free boundary s = s b,h when b → 0 + (for all h > 0) and we also obtain an order of convergence. Moreover, we study the limit of the temperature θ b,h and the free boundary s b,h when (b, h) → (0 + , 0 +).
Existence and Uniqueness of Solution to the Two-Phase Stefan Problem with Convection
Applied Mathematics & Optimization, 2021
The well posedness of the two-phase Stefan problem with convection is established in L 1. First we consider the case with a singular enthalpy and we fix the convection velocity. In the second part of the paper we study the case of a smoothed enthalpy, but the convection velocity is the solution to a Navier-Stokes equation. In the last section we give some numerical illustrations of a physical case simulated using the models studied in the paper.
Mathematical Problems in Engineering
A one-phase Stefan-type problem for a semi-infinite material which has as its main feature a variable latent heat that depends on the power of the position and the velocity of the moving boundary is studied. Exact solutions of similarity type are obtained for the cases when Neumann or Robin boundary conditions are imposed at the fixed face. Required relationships between data are presented in order that these problems become equivalent to the problem where a Dirichlet condition at the fixed face is considered. Moreover, in the case where a Robin condition is prescribed, the limit behaviour is studied when the heat transfer coefficient at the fixed face goes to infinity.