Stefan-type Problem Research Articles

Inversion problem for Radon transforms defined on pseudoconvex sets

This paper is devoted to some questions of inversion for the classical and generalized integral Radon transform. The main question is to determine information about the integrand functions if the values of some integrals are known. A feature of the work of the authors of this message is an analysis of the case when the function is integrated according to hyperplanes in finite-dimensional Euclidean space, and the integrands depend not only on the variables of integration, but also on some of the variables characterizing the hyperplanes. At the same time, the number of independent variables describing known integrals are smaller than those of the unknown integrand. We consider discontinuous integrands defined specifically introduced pseudo-convex sets. A Stefan-type problem is posed about finding surfaces discontinuities of the integrand function. The work provides formulas based on the application special integro-differential operators to known data and allowing you to solve the assigned tasks.

Mathematical model of thermal phenomena of closure electrical contact with Joule heat source and nonlinear thermal coefficients

The mathematical model describing the dynamics of closed contact heating which involves vaporization of the metal when instantaneous explosion of micro-asperity occurs is presented through a Stefan type problem. The temperature field for metallic vaporization zone is introduced as heat resistance that decreases linearity. Temperature fields for liquid and solid phases of the metal described by spherical heat equations with nonlinear thermal coefficients and Joule heat source have to be determined as well as the free boundaries. Joule heating component depends on space and time variable when alternating current is considered. Solution method of the problem based on similarity variable transformation is applied which enables us to reduce the problem to an ordinary differential equations and nonlinear integral equations. Existence and uniqueness of the integral equations are proved by using fixed point theorem in Banach space.

Two Free Boundaries Problem for a Parabolic Equation

В данной работе рассматривается задача типа Стефана с двумя свободными границами для квазилинейного параболического уравнения в одномерном случае. Исследование нелинейных задач со свободными границами методом, основанным на построении априорных оценок. Поэтому сначала устанавливаются некоторые первоначальные априорные оценки для решения рассматриваемой задачи. Основной трудностью при построении теории для задач квазилинейных параболических уравнений второго порядка является получение априорной оценки модуля производной решение, а также в задачах со свободной границей требуются дополнительные рассуждения. Для этого задача сводится к задаче с фиксированной границей через замену переменных. Полученная задача имеет зависящие от времени и положения в пространстве коэффициенты с нелинейными слагаемыми. Далее построены априорных оценок типа Шаудера для решения уравнения с нелинейными слагаемыми и закрепленной границей. На основе полученных оценок доказана единственность решения задачи. Затем мы доказываем глобальное существование решения задачи с помощью теоремы Лерэ-Шаудера о неподвижной точке. This paper considers a two-free-boundary Stefan-type problem for a quasi-linear parabolic equation in one dimension. Nonlinear problems with free boundaries are studied using a method based on constructing a priori estimates. Therefore, some initial a priori estimates for the solution to the problem under consideration are first established. The main difficulty in constructing a theory for second-order quasi-linear parabolic equations is obtaining an a priori estimate for the solution’s derivative module, and additional arguments are required in problems with a free boundary. To address this, the problem is reduced to a fixed-boundary problem through a change of variables. The resulting problem has time- and spacedependent coefficients with nonlinear terms. Next, Schauder-type a priori estimates are constructed for the equation with nonlinear terms and a fixed boundary. Based on these estimates, the uniqueness of the solution to the problem is proven. Then, the global existence of the solution to the problem is demonstrated using the Leray-Schauder fixed-point theorem.

Analytical solution for a cylinder glaciation model with variable latent heat and thermal diffusivity

We consider a Stefan type problem that models the glaciation of the outer surface of a cylindrical gas pipeline in sea water. An equivalent integral formulation is obtained, considering that the latent heat and the thermal diffusivity depend on the radial distance. When the radial component of the heat flux vector from sea water to glaciation front is inversely proportional to the radial distance, local existence and uniqueness of solution is shown. In the general case, existence is ensured but uniqueness of solution is not guaranteed.

A class of moving boundary problems with a source term: application of a reciprocal transformation

We consider a new Stefan-type problem for the classical heat equation with a latent heat and phase-change temperature depending of the variable time. We prove the equivalence of this Stefan problem with a class of boundary value problems for the nonlinear canonical evolution equation involving a source term with two free boundaries. This equivalence is obtained by applying a reduction to a Burgers equation and a reciprocal-type transformations. Moreover, for a particular case, we obtain a unique explicit solution for the two different problems.

On an anisotropic fractional Stefan-type problem with Dirichlet boundary conditions

<abstract><p>In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain $ \Omega\subset\mathbb{R}^d $ with time-dependent Dirichlet boundary condition for the temperature $ \vartheta = \vartheta(x, t) $, $ \vartheta = g $ on $ \Omega^c\times]0, T[$, and initial condition $ \eta_0 $ for the enthalpy $ \eta = \eta(x, t) $, given in $ \Omega\times]0, T[$ by</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \frac{\partial \eta}{\partial t} +\mathcal{L}_A^s \vartheta = f\quad\text{ with }\eta\in \beta(\vartheta), $\end{document} </tex-math></disp-formula></p> <p>where $ \mathcal{L}_A^s $ is an anisotropic fractional operator defined in the distributional sense by</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \langle\mathcal{L}_A^su, v\rangle = \int_{\mathbb{R}^d}AD^su\cdot D^sv\, dx, $\end{document} </tex-math></disp-formula></p> <p>$ \beta $ is a maximal monotone graph, $ A(x) $ is a symmetric, strictly elliptic and uniformly bounded matrix, and $ D^s $ is the distributional Riesz fractional gradient for $ 0 < s < 1 $. We show the existence of a unique weak solution with its corresponding weak regularity. We also consider the convergence as $ s\nearrow 1 $ towards the classical local problem, the asymptotic behaviour as $ t\to\infty $, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph $ \beta $.</p></abstract>

Slow Migration of Brine Inclusions in First-Year Sea Ice

We derive a thermodynamically consistent model for phase change in sea ice by adding salt to the framework introduced by Penrose and Fife. Taking the salt entropy relative to the liquid water molar fraction provides a transparent mechanism for salt rejection under ice formation. We identify slow varying coordinates, including salt density relative to liquid water molarity weighted by latent heat, and use multiscale analysis to derive a quasi-equilibrium Stefan-type problem via a sharp interface scaling. The singular limit is under-determined and the leading order system is closed by imposing local conservation of salt under interface perturbation. The quasi-steady system determines interface motion as balance of curvature, temperature gradient, and salt density. We resolve this numerically for axisymmetric surfaces and show that the thermal gradients typical of arctic sea ice can have a decisive impact on the mode of pinch-off of cylindrical brine inclusions and on the size distribution of the resultant spherical shapes. The density and distribution of inclusion sizes is a key component of sea ice albedo which factors into global climate models.

A Mathematical Model and Simulations of Low Temperature Nitriding

Low-temperature nitriding of steel or iron can produce an expanded austenite phase, which is a solid solution of a large amount of nitrogen dissolved interstitially in fcc lattice. It is characteristic that the nitogen depth profiles in expanded austenite exhibit plateau-type shapes. Such behavior cannot be considered with a standard analytic solution for diffusion in a semi-infinite solid and a new approach is necessary. We formulate a model of interdiffusion in viscoelastic solid (Maxwell model) during the nitriding process. It combines the mass conservation and Vegard’s rule with the Darken <i>bi</i>-velocity method. The model is formulated in any dimension, <i>i.e.</i>, a mixture is included in , <i>n</i> = 1, 2, 3. For the system in one dimension, <i>n</i> = 1, we transform a differential-algebraic system of 5 equations to a differential system of 2 equations only, which is better to study numerically and analytically. Such modification allows the formulation of effective mixed-type boundary conditions. The resulting nonlinear strongly coupled parabolic-elliptic differential initial-boundary Stefan type problem is solved numerically and a series of simulations is made.

On a Two-Dimensional Boundary-Value Stefan-Type Problem Arising in Cryosurgery

In this paper, we present the formulation and a method for solving the problem of freezing living biological tissue with a flat circular cryoapplicator. The model is a two-dimensional boundary-value problem of Stefan type with nonlinear heat sources of a special type that provide the actually observed spatial localization of the temperature field. Some results of computer simulation are presented.

Modeling of Heat and Mass Transfer in Heat-Shielding Composite Materials Based on the Universal Law of Binder Decomposition

A physical and mathematical model of heat and mass transfer under phase transformations during the high-temperature, aerodynamic heating of high-speed aircraft is constructed in this work based on the identified law of the decomposition of binders of most heat-shielding composite materials. The mathematical model includes a description of the occurrence and advancement of the binder-decomposition zone (pyrolysis) limited by two moving boundaries of phase transformations, the heat transfer and filtration of pyrolysis gases in the porous coke residue, their injection into the gasdynamic boundary layer, and the distribution of the temperature and density of the composite material in the zone pyrolysis. The mathematical model and analytical method for the solution of the nonlinear heat and mass transfer problem allow the Stefan-type problem to be reduced to the solution of the transcendental equation with respect to the mass velocity of the pyrolysis zone. The results were obtained for temperature fields with allowance for the filtration in the porous residue and the pressure distribution of pyrolysis gases and for temperature fields in the pyrolysis zone that were not affected by binder decomposition, as well as the distribution of pyrolysis gas density in the pyrolysis zone.

The research of a Stefan problem with unknown pressure in a liquid phase

We consider mathematical modeling of problems with free boundaries, namely, the investigation of the Stefan type problem with a non-constant melting point, taking into account the pressure in the liquid phase. A mathematical model has been constructed that describes the melting process of heavy paraffin-base crude oil. We prove an existence and uniqueness of a solution of a Stefan-type problem in the Holder spaces with a melting point that depends on the unknown pressure are considered. We study the solvability in Holder spaces and the asymptotic behavior for t, which tends to ∞, solving a single-phase Stefan-type problem for a nonlinear parabolic equation.

Florin-Type Problem for the Parabolic Equation with Power Nonlinearity

We consider a problem without initial condition with free boundary for a parabolic equation with power nonlinearity and prove the uniqueness and existence theorems. The problem is reduced to a Stefan-type problem with initial condition. The equivalence of the problems and bilateral a priori estimates for the required functions are established. The behavior of the free boundary is investigated.

Reactive instabilities in linear acidizing on carbonates

Acid injection, reactive instabilities and wormholing in carbonate reservoirs is investigated through the analysis of the linear acidizing problem theoretically and experimentally. Theoretically acidizing was analyzed by formulating the problem as a reactive moving boundary Stefan type problem. Wormholing is viewed as a reactive infiltration instability to the trivial solution of uniform dissolution. A linear stability analysis from the equilibrium state is performed and the critical wavenumber below which instabilities have a positive growth rate is identified. When applied to the scale of the experiment, an optimum injection velocity is identified for a given formation and injection concentration, for the growth for a single wormhole. This optimum injection velocity scales with the inverse of the specimen diameter. Experimentally, linear acidizing tests were performed in Mons chalk, a high porosity analogue of North Sea reservoir chalk. In the experiments the critical injection velocity for wormhole formation at minimum acid injection was obtained and the results were compared with the theoretical predictions.

One-Phase Stefan-Like Problems with Latent Heat Depending on the Position and Velocity of the Free Boundary and with Neumann or Robin Boundary Conditions at the Fixed Face

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.

A stochastic Stefan-type problem under first-order boundary conditions

Moving boundary problems allow to model systems with phase transition at an inner boundary. Motivated by problems in economics and finance, we set up a price-time continuous model for the limit order book and consider a stochastic and nonlinear extension of the classical Stefan-problem in one space dimension. Here, the paths of the moving interface might have unbounded variation, which introduces additional challenges in the analysis. Working on the distribution space, the Itô–Wentzell formula for SPDEs allows to transform these moving boundary problems into partial differential equations on fixed domains. Rewriting the equations into the framework of stochastic evolution equations and stochastic maximal $L^{p}$-regularity, we get existence, uniqueness and regularity of local solutions. Moreover, we observe that explosion might take place due to the boundary interaction even when the coefficients of the original problem have linear growths.

On the analytical modelling of the initial ice growth in a supercooled liquid droplet

Ice formation in metastable, super-cooled droplets, which are frequently found in the atmosphere, influences the appearance and characteristics of atmospheric clouds significantly, for example regarding precipitation. Its numerical investigation can provide deep insight into the underlying physical mechanisms and supports the deduction of models that describe these processes on the microscale; those models are required for a description of the macrophysical system. However, even the processes on the microscale span about four orders of magnitude. A semi-analytical sub-scale model based on similarity solutions is thus deduced in order to narrow the gap between the different scales describing the initially spherical ice growth in a super-cooled droplet, which can be reduced to a radially symmetric, but highly non-linear Stefan-type problem. All relevant physical effects, e.g. the reduction of the melting temperature, the expansion of the water phase due to the decrease of density upon solidification and high degrees of supercooling, are taken into account in contrast to classical approaches. The maximum relative error in terms of the freezing time, which is given explicitly as well as the temperature fields, is less than 10% at a degree of supercooling of 35 K and decreases rapidly as the ambient temperature increases.

A two-scale Stefan problem arising in a model for tree sap exudation

The study of tree sap exudation, in which a (leafless) tree generates elevated stem pressure in response to repeated daily freeze-thaw cycles, gives rise to an interesting multi-scale problem involving heat and multiphase liquid/gas transport. The pressure generation mechanism is a cellular-level process that is governed by differential equations for sap transport through porous cell membranes, phase change, heat transport, and generation of osmotic pressure. By assuming a periodic cellular structure based on an appropriate reference cell, we derive an homogenized heat equation governing the global temperature on the scale of the tree stem, with all the remaining physics relegated to equations defined on the reference cell. We derive a corresponding strong formulation of the limit problem and use it to design an efficient numerical solution algorithm. Numerical simulations are then performed to validate the results and draw conclusions regarding the phenomenon of sap exudation, which is of great importance in trees such as sugar maple and a few other related species. The particular form of our homogenized temperature equation is obtained using periodic homogenization techniques with two-scale convergence, which we investigate theoretically in the context of a simpler two-phase Stefan-type problem corresponding to a periodic array of melting cylindrical ice bars with a constant thermal diffusion coefficient. For this reduced model, we prove results on existence, uniqueness and convergence of the two-scale limit solution in the weak form, clearly identifying the missing pieces required to extend the proofs to the fully nonlinear sap exudation model. Numerical simulations of the reduced equations are then compared with results from the complete sap exudation model.

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.

A NUMERICAL METHOD FOR A STEFAN-TYPE PROBLEM

A Stefan-type problem is considered. This is an initial-boundary value problem on a composite domain for a parabolic reaction-diffusion equation with a moving interface boundary. At the moving boundary between the two subdomains, an interface condition is prescribed for the solution of the problem and its derivatives. A finite difference scheme is constructed that approximates the initial-boundary value problem. An iterative Newton-type method for the solution of the difference scheme and a numerical method for the analysis of the errors of the computed discrete solutions are both developed.

The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction

We consider reaction-diffusion systems which, in addition to certain slow reactions, contain a fast irreversible reaction in which chemical components A and B form a product P. In this situation and under natural assumptions on the RD-system we prove the convergence of weak solutions, as the reaction speed of the irreversible reaction tends to infinity, to a weak solution of a limiting system. The limiting system is a Stefan-type problem with a moving interface at which the chemical reaction front is localized.