Convergence of the solution of the one-phase Stefan problem when the heat transfer coefficient goes to zero (original) (raw)
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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
Applied Mathematics and Computation, 2018
In this paper we consider a one-dimensional one-phase Stefan problem corresponding to the solidification process of a semi-infinite material with a convective boundary condition at the fixed face. The exact solution of this problem, available recently in the literature, enable us to test the accuracy of the approximate solutions obtained by applying the classical technique of the heat balance integral method and the refined integral method, assuming a quadratic temperature profile in space. We develop variations of these methods which turn out to be optimal in some cases. Throughout this paper, a dimensionless analysis is carried out by using the parameters: Stefan number (Ste) and the generalized Biot number (Bi). In addition it is studied the case when Bi goes to infinity, recovering the approximate solutions when a Dirichlet condition is imposed at the fixed face. Some numerical simulations are provided in order to estimate the errors committed by each approach for the corresponding free boundary and temperature profiles.
Mathematical Methods in The Applied Sciences, 2020
We consider a two-phase Stefan problem for a semi-infinite body x > 0, with a convective boundary condition including a density jump at the free boundary with a time-dependent heat transfer coefficient of the type h∕ √ t, h > 0 whose solution was given in D. A. Tarzia, PAMM. Proc. Appl. Math. Mech. 7, 1040307-1040308 (2007). We demonstrate that the solution to this problem converges to the solution to the analogous one with a temperature boundary condition when the heat transfer coefficient h → +∞. Moreover, we analyze the dependence of the free boundary respecting to the jump density. KEYWORDS two-phase Stefan problem, density jump, asymptotic behavior, phase-change process MSC CLASSIFICATION 35R35; 80A22; 35C05 1 (s(t), t) = 2 (s(t), t) = 0 , t > 0 , (3)
Behavior of the solution of a Stefan problem by changing thermal coefficients of the substance
Applied Mathematics and Computation, 2007
We consider a one-dimensional one-phase Stefan problem for a semi-infinite substance. We suppose that there is a transient heat flux at the fixed face and the thermal coefficients are constant. The goal of this paper is to determine the behavior of the free boundary and the temperature by changing the thermal coefficients. We use the maximum principle in order to obtain properties of monotony with respect to the latent heat of fusion, the specific heat and the mass density. We compute approximate solutions through the quasi-stationary, the Goodman's heat-balance integral and the Biot's variational methods and a numerical solution through a finite difference scheme. We show that the solution is not monotone with respect to the thermal conductivity. The results obtained are important in technological applications.
2017
From the one-dimensional consolidation of fine-grained soils with threshold gradient, it can be derived a special type of Stefan problems where the seepage front, due to the presence of this threshold gradient, exhibits the features of a moving boundary. In this kind of problems, in contrast with the classical Stefan problem, the latent heat is considered to depend inversely with the rate of change of the seepage front. In this paper a one-phase Stefan problem with a latent heat that not only depends on the rate of change of the free boundary but also on its position is studied. The aim of this analysis is to extend prior results, finding an analytical solution that recovers, by specifying some parameters, the solutions that have already been examined in the literature regarding Stefan problems with variable latent heat. Computational examples will be presented in order to examine the effect of this parameters on the free boundary.
Communications on Pure and Applied Analysis, 2010
We study a one-phase Stefan problem for a semi-infinite material with temperature-dependent thermal conductivity and a convective term with a convective boundary condition at the fixed face x = 0. We obtain 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 obtain explicit solutions through the unique solution of a Cauchy problem with the time as a parameter and we also give an algorithm in order to compute the explicit solution.
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
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.
Free Boundary Convergence in the Homogenization of the One Phase Stefan Problem
Transactions of The American Mathematical Society, 1982
We consider the one phase Stefan problem in a "granular" medium, i.e., with nonconstant thermal diffusity, and we study the asymptotic behaviour of the free boundary with respect to homogenization. We prove the convergence of the coincidence set in measure and in the Hausdorff metric. We apply this result to the free boundary and we obtain the convergence in mean for the star-shaped case and the uniform convergence for the one-dimensional case, respectively. This gives an answer to a problem posed by J. L. Lions in [L].