Stefan-type Problems Research Articles

Analytical solution of Stefan-type problems

Abstract In this paper, free surface problems of Stefan type for the parabolic heat equation are considered. The analytical solutions of the problems are based on the method of heat polynomials and integral error function in the form of series. Convergence of the series solution is considered and proved. Both one-and two-phase Stefan-type problems are investigated. Numerical results for one-phase inverse Stefan problem are presented and discussed.

Some problems of heat and mass transfer during the operation of engineering systems in multiphase environments

Three types of problems related to problems of heat and mass transfer in the soil are considered. The first class of problems deals with the diagnostics of damage of underground pipelines by thermal fields on the soil surface. The second type studies the dynamics of changes in the temperature of a geothermal reservoir depending on the temperature of the water entering this reservoir and the pressure gap between injection and production wells. The third-type problems consider the propagation of non-stationary thermal fields in the soil from operated engineering systems in the permafrost. The main attention is paid to long-term forecasting of the propagation of non-stationary thermal fields in the frozen soil between operating production wells of northern oil and gas fields. In problems of the first two classes, which served as a basis for the development of problems of the third type, water filtration in the soil is considered, and thermal fields propagate in single-phase media. The third-class problems take into account possible phase transitions in the soil when describing non-stationary thermal fields in permafrost soils, leading to Stefan-type problems. Accounting for water migration for the specific third-type problems on the determination of the radius of frozen soil thawing from production wells in northern oil and gas fields does not significantly affect this process since lateral water migration above the groundwater level is minimal. Therefore, only the latent heat of the initial water content is taken into consideration. This paper discusses a mathematical model containing the most significant physical and climatic data affecting the distribution of thermal fields in permafrost rocks and presents the results of numerical calculations.

On boundary immobilization for one-dimensional Stefan-type problems with a moving boundary having initially parabolic-logarithmic behaviour

In this paper, a recent one-dimensional Stefan-type model for the sorption of a finite amount of swelling solvent in a glassy polymer is revisited, with a view to formalizing the application of the boundary immobilization method to this problem. The key difficulty is that the initial behaviour of the moving boundary is parabolic-logarithmic, rather than algebraic, which has more often than not been the case in similar problems. A small-time analysis of the problem hints at how the usual boundary immobilization formalism can be recovered, and this is subsequently verified through numerical experiments. The relevance of these results to other moving boundary problems from the literature is also discussed.

Moving boundary problems for a canonical member of the WKI inverse scattering scheme: conjugation of a reciprocal and Möbius transformation

Reciprocal links between certain solitonic systems and their hierarchies are well-established. Moreover, the AKNS and WKI inverse scattering schemes are known to be connected by a composition of gauge and reciprocal transformations. Here, a reciprocal transformation allied with a Möbius-type mapping is applied to a class of Stefan-type problems for the solitonic Dym equation to generate a novel exact parametric solution to a class of moving boundary problems for a canonical member of the WKI system.

A non-field analytical method for heat transfer problems through a moving boundary

This paper presents an extension of the non-field analytical method—known as the method of Kulish—to solving heat transfer problems in domains with a moving boundary. This is an important type of problems with various applications in different areas of science. Among these are heat transfer due to chemical reactions, ignition and explosions, combustion, and many others. The general form of the non-field solution has been obtained for the case of an arbitrarily moving boundary. After that some particular cases of the solution are considered. Among them are such cases as the boundary speed changing linearly, parabolically, exponentially, and polynomially. Whenever possible, the solutions thus obtained have been compared with known solutions. The final part of the paper is devoted to determination of the front propagation law in Stefan-type problems at large times. Asymptotic solutions have been found for several important cases of the front propagation.

Approximation of the interface condition for stochastic Stefan-type problems

We consider approximations of the Stefan-type condition by imbalances of volume closely around the inner interface and study convergence of the solutions of the corresponding semilinear stochastic moving boundary problems. After a coordinate transformation, the problems can be reformulated as stochastic evolution equations on fractional power domains of linear operators. Here, the coefficients might fail to have linear growths and might be Lipschitz continuous only on bounded sets. We show continuity properties of the mild solution map in the coefficients and initial data, also incorporating the possibility of explosion of the solutions.

Integral methods of solving boundary-value problems of nonstationary heat conduction and their comparative analysis

The modern state of approximate integral methods used in applications, where the processes of heat conduction and heat and mass transfer are of first importance, is considered. Integral methods have found a wide utility in different fields of knowledge: problems of heat conduction with different heat-exchange conditions, simulation of thermal protection, Stefantype problems, microwave heating of a substance, problems on a boundary layer, simulation of a fluid flow in a channel, thermal explosion, laser and plasma treatment of materials, simulation of the formation and melting of ice, inverse heat problems, temperature and thermal definition of nanoparticles and nanoliquids, and others. Moreover, polynomial solutions are of interest because the determination of a temperature (concentration) field is an intermediate stage in the mathematical description of any other process. The following main methods were investigated on the basis of the error norms: the Tsoi and Postol’nik methods, the method of integral relations, the Gudman integral method of heat balance, the improved Volkov integral method, the matched integral method, the modified Hristov method, the Mayer integral method, the Kudinov method of additional boundary conditions, the Fedorov boundary method, the method of weighted temperature function, the integral method of boundary characteristics. It was established that the two last-mentioned methods are characterized by high convergence and frequently give solutions whose accuracy is not worse that the accuracy of numerical solutions.

Features of the Numerical Solution of Thermal Destruction Fuel Pins Problems in the Fast Reactor

In this paper the description of the basic equations which can be used for calculation of melting of fuel and cladding of the fast reactor, moving of the melt on a fuel pin surface and its solidification is presented. The special attention is given speed of calculation algorithms and fidelity of the phenomena which are observed at a stage of severe accidents in fast reactors.For check of working capacity of initial models, numerical calculations of Stefan-type problems on front movement of melting/solidification in cylindrical geometry are presented. Comparison with the solutions received by known analytical methods is executed. For validation of the numerical realization of calculation algorithms the analysis is carried out and experiments in which melting of the model fuel pins of fast reactors was studied are chosen. On the basis of the chosen experiments calculation schemes taking into account initial and boundary conditions are prepared and modeling is performed. Modeling results are shown in the present paper. Estimation of calculation error of the basic physical parameters is done by results of the modeling and conclusions are drawn on a correctness of algorithms operation.

On a class of moving boundary problems for the potential mkdV equation: conjugation of Bäcklund and reciprocal transformations

A class of moving boundary problems for the solitonic potential mkdV equation is set down which is reciprocal to generalised Stefan-type problems that admit exact solution in terms of a Painleve II symmetry reduction. The latter allows the construction of the exact solution of the moving boundary problems via the iterated action of a Backlund transformation. This results in solution representation in terms of Yablonski–Vorob’ev polynomials. It is indicated how the results may be extended, via a novel reciprocal link, to moving boundary problems for a canonical solitonic equation generated as a member of the WKI inverse scattering system.

Macroscopic models for a mushy region in concrete corrosion

Macroscopic models for the corrosion of concrete due to sulphation, describing the formation of a mushy region, are derived and studied by further expanding previous related studies. These models are derived from averaging using the multiple-scales method applied on microscopic Stefan-type problems to capture the effects of the microscopic transformation of calcite into gypsum. The resulting macroscopic model for the diffusion and production of the sulphate inside the concrete is coupled with a time-dependent Eikonal equation describing the evolution of the reaction at each point of the concrete wall. In certain cases and for specific geometries of the microstructure, the Eikonal equation admits analytical solutions, and the model takes the form of a macroscopic non-local problem. The models derived are solved numerically with the use of a finite-element method, and the results for various microstructure geometries on the microscale are presented.

A level set approach for diffusion and Stefan-type problems with Robin boundary conditions on quadtree/octree adaptive Cartesian grids

We present a numerical method for simulating diffusion dominated phenomena on irregular domains and free moving boundaries with Robin boundary conditions on quadtree/octree adaptive meshes. In particular, we use a hybrid finite-difference and finite-volume framework that combines the level-set finite difference discretization of Min and Gibou (2007) [13] with the treatment of Robin boundary conditions of Papac et al. (2010) [19] on uniform grids. We present numerical results in two and three spatial dimensions on the diffusion equation and on a Stefan-type problem. In addition, we present an application of this method to the case of the simulation of the Ehrlich–Schwoebel barrier in the context of epitaxial growth.

Homogenization of one-phase Stefan-type problems in periodic and random media

We investigate the homogenization of Stefan-type problems with oscillating diffusion coefficients. Both cases of periodic and random (stationary ergodic) mediums are considered. The proof relies on the coincidence of viscosity solutions and weak solutions (which are the time derivatives of the solutions of an obstacle problem) for the Stefan problem. This coincidence result is of independent interest.

Renormalized solution for Stefan type problems: existence and uniqueness

We consider a class of nonlinear degenerate problems of Stefan type: $$u_t- \Delta w -\nabla F(u,w)= g(\cdot,u), w\in \beta(u)$$ where β is a maximal monotone graph in \({\mathbb{R}^2,}\) with homogenous Dirichlet conditions and initial conditions. Under rather general assumptions on F and g, we prove existence and uniqueness of renormalized solutions.

Efficient symmetric discretization for the Poisson, heat and Stefan-type problems with Robin boundary conditions

We present a novel and efficient method for solving the Poisson equation, the heat equation, and Stefan-type problems with Robin boundary conditions over potentially moving, arbitrarily-shaped domains. The method utilizes a level set framework, thus it has all of the benefits of a sharp, implicitly-represented interface such as the ease of handling complex topological changes. This method is straightforward to implement and leads to a linear system that is symmetric and positive definite, which can be inverted efficiently with standard iterative methods. This approach is second-order accurate for both the Poisson and heat equations, and first-order accurate for the Stefan problem. We demonstrate the accuracy in the L 1 and L ∞ norms.

Two algorithms for two-phase Stefan type problems

In this paper, the relaxation algorithm and two Uzawa type algorithms for solving discretized variational inequalities arising from the two-phase Stefan type problem are proposed. An analysis of their convergence is presented and the upper bounds of the convergence rates are derived. Some numerical experiments are shown to demonstrate that for the second Uzawa algorithm which is an improved version of the first Uzawa algorithm, the convergence rate is uniformly bounded away from 1 if τh−2 is kept bounded, where τ is the time step size and h the space mesh size.

Nonlinear dynamics of directional solidification of ternary solutions with mushy layers

We present new analytic results relating to the nonstationary dynamics of Stefan-type problems for the unidirectional solidification of ternary solutions in the presence of two mushy layers. Explicit analytical solutions (thicknesses of mushy layers, growth rates of their boundaries, temperature and impurity concentration fields, solid fractions) of the nonlinear model under consideration are found. Model predictions are in good agreement with existing experimental data.

Solution of a model problem related to singularly perturbed, free boundary, Stefan type problems

A model problem of conjunction of two phases with a small parameter κ in front of the principal term of a free boundary condition is considered. A coercive estimate for the solution with a constant independent of κ is derived in terms of a weighted Holder space. Bibliography: 9 titles.

Guidelines for Poisson Solvers on Irregular Domains with Dirichlet Boundary Conditions Using the Ghost Fluid Method

We consider the variable coefficient Poisson equation with Dirichlet boundary conditions on irregular domains. We present numerical evidence for the accuracy of the solution and its gradients for different treatments at the interface using the Ghost Fluid Method for Poisson problems of Gibou et al. (J. Comput. Phys. 176:205---227, 2002; 202:577---601, 2005). This paper is therefore intended as a guide for those interested in using the GFM for Poisson-type problems (and by consequence diffusion-like problems and Stefan-type problems) by providing the pros and cons of the different choices for defining the ghost values and locating the interface. We found that in order to obtain second-order-accurate gradients, both a quadratic (or higher order) extrapolation for defining the ghost values and a quadratic (or higher order) interpolation for finding the interface location are required. In the case where the ghost values are defined by a linear extrapolation, the gradients of the solution converge slowly (at most first order in average) and the convergence rate oscillates, even when the interface location is defined by a quadratic interpolation. The same conclusions hold true for the combination of a quadratic extrapolation for the ghost cells and a linear interpolation. The solution is second-order accurate in all cases. Defining the ghost values with quadratic extrapolations leads to a non-symmetric linear system with a worse conditioning than that of the linear extrapolation case, for which the linear system is symmetric and better conditioned. We conclude that for problems where only the solution matters, the method described by Gibou, F., Fedkiw, R., Cheng, L.-T. and Kang, M. in (J. Comput. Phys. 176:205---227, 2002) is advantageous since the linear system that needs to be inverted is symmetric. In problems where the solution gradient is needed, such as in Stefan-type problems, higher order extrapolation schemes as described by Gibou, F. and Fedkiw, R. in (J. Comput. Phys. 202:577---601, 2005) are desirable.

A New Look at Equilibria in Stefan-Type Problems in the Plane

We study steady states of Stefan-type problems in the plane with the Gibbs–Thomson correction involving a general anisotropic energy density function. By a local analysis we prove the global result showing that the solution is the Wulff shape. The key element is a stability result which enables us to approximate singular models by regular ones.

Nonlinear dynamics of directional solidification with a mushy layer. Analytic solutions of the problem

We present new analytic results relating to the nonstationary Stefan-type problems for the unidirectional solidification of binary solutions or melts with a mushy layer. Our detailed analysis of the field data is based on the classical model of a mushy layer, which is modified in order to obtain explicit solutions (solid phase thickness and growth rate, temperature distributions, conductive and latent heat fluxes are determined). Predictions for the growth rate and temperature profiles of the mixed-phase and solid regions agree well with existing observations on young sea ice dynamics.