一个具有内部物质对流的两项连铸Stefan问题

考虑了一个具有内部物质对流和非线性边界热交换的多维连铸Stefan问题,并得到了这个问题整体弱解的存在性、唯一性和对初边界条件的连续依赖性。本项工作改进和推广了J.F.Fodri-gues＆F.Yi的结果,放宽了他们对内部流和边界条件的一些不太符合实际的限制。     

Explicit solutions for a two-phase unidimensional Lamé-Clapeyron-Stefan problem with source terms in both phases

A two-phase Stefan problem with heat source terms of a general similarity type in both liquid and solid phases for a semi-infinite phase-change material is studied. We assume the initial temperature is a negative constant and we consider two different boundary conditions at the fixed face x=0, a constant temperature or a heat flux of the form (q₀>0). The internal heat source functions are given by (j=1 solid phase; j=2 liquid phase) where βj=βj(η) are functions with appropriate regularity properties, ρ is the mass density, l is the fusion latent heat by unit of mass, is the diffusion coefficient, x is the spatial variable and t is the temporal variable. We obtain for both problems explicit solutions with a restriction for data only for the second boundary conditions on x=0. Moreover, the equivalence of the two free boundary problems is also proved. We generalize the solution obtained in [J.L. Menaldi, D.A. Tarzia, Generalized Lamé-Clapeyron solution for a one-phase source Stefan problem, Comput. Appl. Math. 12 (2) (1993) 123-142] for the one-phase Stefan problem. Finally, a particular case where βj (j=1,2) are of exponential type given by βj(x)=exp(−²(x+dj)) with x and dj∈R is also studied in details for both boundary temperature conditions at x=0. This type of heat source terms is important through the use of microwave energy following [E.P. Scott, An analytical solution and sensitivity study of sublimation-dehydration within a porous medium with volumetric heating, J. Heat Transfer 116 (1994) 686-693]. We obtain a unique solution of the similarity type for any data when a temperature boundary condition at the fixed face x=0 is considered; a similar result is obtained for a heat flux condition imposed on x=0 if an inequality for parameter q₀ is satisfied.     

A Stefan problem for a non-classical heat equation with a convective condition

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.     

Determination of the Unknown Boundary Conditions in a Two-phase Stefan Problem

A two-phase Stefan problem with the heat flux boundary conditions, including an unknown function f, is considered. The existence, uniqueness, and continuous dependence upon the initial data of the solution (f, s, u_1, u_2) are proved.     

Convergence to the Stefan problem of the phase relaxation problem with Cattaneo heat flux law

In this paper we show that the solutions of the phase change problem with the Cattaneo-Fourier heat flux law and phase relaxation, converge to the solution of the Stefan problem as the two relaxation parameters go independently to zero.     

A Stefan Problem with Curvature Correction in a Concentrated Capacity

We consider a Stefan problem with the curvature correction in a concentrated capacity. It is a kind of phase transition problems, in which the unknown temperature satisfies both the Stefan condition and the curvature correction on the free boundary, and also fulfills a kind of heat equations with special inner heat source. The inner beat source is related to the derivative of the temperature with respect to the direction vertical to the phase regions. The result established in this paper is the existence of a global weak solution of this problem.     

Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficients

In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component. In particular, the temperature distribution in liquid and solid phases of such kind of body can be modeled by Stefan problem for the generalized heat equation. The method of solution is based on similarity principle, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we determine temperature solution for two phases and free boundaries which describe the position of boiling and melting interfaces. Existence and uniqueness of the similarity type solution is provided by using the fixed point Banach theorem.     

Optimization of heat flux in domains with temperature constraints

In this paper, we deal with a heat flux optimization problem. We maximize the heat output flow on a portion of a domain's boundary, while on the other portion the distribution of the temperature is fixed. The maximization is carried out under the condition that there are no phase changes.The problem is solved using a convex-functional optimization technique, on Banach spaces, within restricted sets, yielding existence and uniqueness of the solution. The explicit form of the solution and the corresponding Lagrange multipliers associated to the problem are also given.In addition, other optimization problems related to the maximum bound of the heat flux with no phase change are solved.This investigation has been supported by the research and development projects Numerical Analysis of Variational Equalities and Inequalities and Free Boundary Problems in Mathematical Physics from CONICET-UNR, Rosario, Argentina.     

The order of convergence in the stefan problem with vanishing specific heat

The paper is concerned with the two-phase Stefan problem with a small parameter ϵ, which coresponds to the specific heat of the material. It is assumed that the initial condition does not coincide with the solution for t = 0 of the limit problem related to ε = 0. To remove this discrepancy, an auxiliary boundary layer type function is introduced. It is proved that the solution to the two-phase Stefan problem with parameter ϵ differs from the sum of the solution to the limit Hele–Shaw problem and a boundary layer type function by quantities of order O(ϵ). The estimates are obtained in H?lder norms. Bibliography: 13 titles.     

Improved Quasi-Steady-State Approximation Analysis of Stefan Problems Under 2nd-Kind Boundary Conditions北大核心CSCD

基于准稳态近似方法,从热平衡角度出发,给出了直角坐标系和圆柱坐标系中恒定热流边界条件下传导型固液相变传热问题的无量纲近似解。对于直角坐标系情形,得到的改进型准稳态近似解精度高,且解的形式为显式表达式,相比于已有的隐式近似解更便于直接使用。对于圆柱坐标系的情形,所得到的近似解是目前文献公开报道的唯一的近似解。此改进型准稳态近似解弥补了传统准稳态近似方法不考虑显热的不足,提高了准稳态近似法的精度,丰富了固液相变传热问题的求解方法,物理意义明确,可用于实际应用问题的初步分析和计算。     

On a Penrose-Fife model with zero interfacial energy leading to a phase-field system of relaxed Stefan type

In this paper we study an initial-boundary value Stefan-type problem with phase relaxation where the heat flux is proportional to the gradient of the inverse absolute temperature. This problem arise naturally as limiting case of the Penrose-Fife model for diffusive phase transitions with nonconserved order parameter if the coefficient of the interfacial energy is taken as zero. It is shown that the relaxed Stefan problem admits a weak solution which is obtained as limit of solutions to the Penrose-Fife phase-field equations. For a special boundary condition involving the heat exchange with the surrounding medium, also uniqueness of the solution is proved.     

On a model of phase relaxation for the hyperbolic Stefan problem

In this paper we study a model of phase relaxation for the Stefan problem with the Cattaneo–Maxwell heat flux law. We prove an existence and uniqueness result for the resulting problem and we show that its solution converges to the solution of the Stefan problem as the two relaxation parameters go to zero, provided a relation between these parameters holds.     

Homotopy perturbation method for a limit case Stefan problem governed by fractional diffusion equation

The work presents a mathematical model describing the time fractional anomalous-diffusion process of a generalized Stefan problem which is a limit case of a shoreline problem. In this model, the governing equations include a fractional time derivative of order 0 < α ? 1 and variable latent heat. The approximate solution of the problem is obtained by homotopy perturbation method. The results thus obtained are compared graphically with the exact solutions. A brief sensitivity study is also performed.     

Continuity of the temperature in boundary heat control problems

Motivated by the boundary heat control problems formulated in the book of Duvaut and Lions, we study a boundary Stefan problem and a boundary porous media problem. We prove continuity of the solution with the appropriate modulus. We also extend the results to the fractional order case and to the anomalous diffusion problems.     

A Quasisteady Stefan Problem with Curvature Correction and Kinetic Undercooling

A quasisteady Stefan problem with curvature correction and kinetic undercooling is considered. It is a problem with phase transition, in which not only the Stefan condition, but also the curvature correction and kinetic undercooling effect hold on the free boundary, and in phase regions elliptic equations are satisfied by the unknown temperature at each time. The existence and uniqueness of a local classical solution of this problem are obtained.     

STEFAN PROBLEM WITH NONLINEAR BOUNDARY FLUX AND INTERNAL CONVECTION OF A MATERIAL

A Stefan problem with nonlinear boundary flux and internal convection of a material are considered. The existence, uniqueness and continuous dependence of globally weak solution of this problem are obtained. This paper extends the results of Fahuai Yi and T.M.Shih, relaxes restrictions that does not be to accord with reality very much on internal convection and boundary conditions in their articles.     

Heat Conduction with Flux Condition on a Free Patch

A new free boundary or free patch problem for the heat equation is presented. In the problem a nonlinear heat flux condition is prescribed on a free portion of the boundary, the patch, the position of which depends on the solution. The existence of a weak solution is established using the theory of set-valued pseudomonotone operators.     

The numerical solution of one-phase classical Stefan problem

In this paper, variable space grid and boundary Immobilisation Techniques based on the explicit finite difference are applied to the one-phase classical Stefan problem. It is shown that all the results obtained by the two methods are in good agreement with the exact solution, and exhibit the expected convergence as the mesh size is refined.     

Global classical solution of quasi-stationary Stefan free boundary problem

We consider an one-phase quasi-stationary Stefan problem (Hele–Shaw problem) in multidimensional case. Under some reasonable conditions we prove that the problem has a classical solution globally in time. The method can be used in two-phase problem as well. We also discuss asymptotic behavior of solution as t→+∞. The method developed here can be extended to a general class of free boundary problems.     

A solvable hyperbolic free boundary problem modelling tumour regrowth

Recently, Tian and Friedman et al. developed a mathematical model on brain tumour recurrence after resection [J.P. Tian, A. Friedman, J. Wang and E.A. Chiocca, Modeling the effects of resection, radiation and chemotherapy in glioblastoma, J. Neuro-Oncol. 91(3) (2009), pp. 287–293]. The model is a free boundary problem with a hyperbolic system of nonlinear partial differential equations. In this article, we conduct a rigorous analysis on this hyperbolic system and prove the local and global existence and uniqueness of the solution. It is well known that most nonlinear free boundary problems are impossible to solve in terms of explicit analytical solutions. In contrast, the free boundary problem in this study is solvable, and the explicit solution is found using the backward characteristic curve method. This explicit solution is then validated by numerical simulation results. An interesting finding in this study is that the problem can be treated as a hyperbolic system defined on an infinite domain where the initial condition has a first-type discontinuity.