Stefan Type 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.

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.

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 delay induced nonlocal free boundary problem

We study the dynamics of a population with an age structure whose population range expands with time, where the adult population is assumed to satisfy a reaction–diffusion equation over a changing interval determined by a Stefan type free boundary condition, while the juvenile population satisfies a reaction–diffusion equation whose evolving domain is determined by the adult population. The interactions between the adult and juvenile populations involve a fixed time-delay, which renders the model nonlocal in nature. After establishing the well-posedness of the model, we obtain a rather complete description of its long-time dynamical behaviour, which is shown to follow a spreading–vanishing dichotomy. When spreading persists, we show that the population range expands with an asymptotic speed, which is uniquely determined by an associated nonlocal elliptic problem over the half line. We hope this work will inspire further research on age-structured population models with an evolving population range.

Approximation of random diffusion equations by nonlocal diffusion equations in free boundary problems of one space dimension

We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the nonlocal diffusion problem with free boundaries considered in [J. Cao, Y. Du, F. Li and W.-T. Li, The dynamics of a Fisher–KPP nonlocal diffusion model with free boundaries, J. Functional Anal. 277 (2019) 2772–2814; C. Cortazar, F. Quiros and N. Wolanski, A nonlocal diffusion problem with a sharp free boundary, Interfaces Free Bound. 21 (2019) 441–462]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form [Formula: see text] for small [Formula: see text], where [Formula: see text] has compact support. We also give an estimate of the error term of the approximation by some positive power of [Formula: see text].

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.

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.

Numerical Solution to Recover Time-dependent Coefficient and Free Boundary from Nonlocal and Stefan Type Overdetermination Conditions in Heat Equation

This paper investigates the recovery for time-dependent coefficient and free boundary for heat equation. They are considered under mass/energy specification and Stefan conditions. The main issue with this problem is that the solution is unstable and sensitive to small contamination of noise in the input data. The Crank-Nicolson finite difference method (FDM) is utilized to solve the direct problem, whilst the inverse problem is viewed as a nonlinear optimization problem. The latter problem is solved numerically using the routine optimization toolbox lsqnonlin from MATLAB. Consequently, the Tikhonov regularization method is used in order to gain stable solutions. The results were compared with their exact solution and tested via root mean squares error (RMSE). We found that the numerical results are accurate and stable.

On One Mathematical Model of Cooling Living Biological Tissue

When cooling living biological tissue (active, non-inert medium), cryomedicine uses cryo-instruments with various forms of cooling surface. Cryoinstruments are located on the surface of biological tissue or completely penetrate into it. With a decrease in the temperature of the cooling surface, an unsteady temperature field appears in the tissue, which in the general case depends on three spatial coordinates and time. To date, there are a large number of scientific publications that consider mathematical models of cryodestruction of biological tissue. However, in the overwhelming majority of them, the Pennes equation (or some of its modifications) is taken as the basis of the mathematical model, from which the linear nature of the dependence of heat sources of biological tissue on the desired temperature field is visible. This character of the dependence does not allow one to describe the actually observed spatial localization of heat. In addition, Pennes' model does not take into account the fact that the freezing of the intercellular fluid occurs much earlier than the freezing of the intracellular fluid and the heat corresponding to these two processes is released at different times. In the proposed work, a new mathematical model of cooling and freezing of living biological tissue are built with a flat rectangular applicator located on its surface. The model takes into account the above features and is a three-dimensional boundary-value problem of the Stefan type with nonlinear heat sources of a special type and has applications in cryosurgery. A method is proposed for the numerical study of the problem posed, based on the use of locally one-dimensional difference schemes without explicitly separating the boundary of the influence of cold and the boundaries of the phase transition. The method was previously successfully tested by the author in solving other two-dimensional problems arising in cryomedicine.

Exact sharp-fronted travelling wave solutions of the Fisher–KPP equation

A family of travelling wave solutions to the Fisher–KPP equation with speeds c=±5∕6 can be expressed exactly using Weierstraß elliptic functions. The well-known solution for c=5∕6, which decays to zero in the far-field, is exceptional in the sense that it can be written simply in terms of an exponential function. This solution has the property that the phase-plane trajectory is a heteroclinic orbit beginning at a saddle point and ending at the origin. For c=−5∕6, there is also a trajectory that begins at the saddle point, but this solution is normally disregarded as being unphysical as it blows up for finite z. We reinterpret this special trajectory as an exact sharp-fronted travelling solution to a Fisher–Stefan type moving boundary problem, where the population is receding from, instead of advancing into, an empty space. By simulating the full moving boundary problem numerically, we demonstrate how time-dependent solutions evolve to this exact travelling solution for large time. The relevance of such receding travelling waves to mathematical models for cell migration and cell proliferation is also discussed.

On the correctness of one model of the Stefan-Type filtration theory

In this paper a mathematical model of filtration theory with phase transitions is investigated. When using surface-active substances (surfactants) for the development of oil and gas fields in the reservoir occur sorption processes at the interfaces of individual phases (surfactants and oil, or surfactants and soil). In real processes, a finite time is required for achievement equilibrium. Therefore considering the mathematical model was called the mathematical model with phase relaxation. The solvability of the mathematical model, the limiting transition in relaxation time are investigated. It is proved that in the limiting case, the original problem is a problem of Stefan type.

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.

A Note on Models for Anomalous Phase-Change Processes

Abstract We review some fractional free boundary problems that were recently considered for modeling anomalous phase-transitions. All problems are of Stefan type and involve fractional derivatives in time according to Caputo’s definition. We survey the assumptions from which they are obtained and observe that the problems are nonequivalent though all of them reduce to a classical Stefan problem when the order of the fractional derivatives is replaced by one. We further show that a simple heuristic approach built upon a fractional version of the energy balance and the classical Fourier’s law leads to a natural generalization of the classical Stefan problem in which time derivatives are replaced by fractional ones.

Optimal Control of Multiphase Free Boundary Problems for Nonlinear Parabolic Equations

We consider the optimal control of singular nonlinear partial differential equation which is the distributional formulation of the multiphase Stefan type free boundary problem for the general second order parabolic equation. Boundary heat flux is the control parameter, and the optimality criteria consist of the minimization of the $L_2$-norm declination of the trace of the solution to the PDE problem at the final moment from the given measurement. Sequence of finite-dimensional optimal control problems is introduced through finite differences. We establish existence of the optimal control and prove the convergence of the sequence of discrete optimal control problems to the original problem both with respect to functional and control. Proofs rely on establishing a uniform $L_{\infty}$ bound, and $W_2^{1,1}$-energy estimate for the discrete nonlinear PDE problem with discontinuous coefficient.

Optimal control of coefficients in parabolic free boundary problems modeling laser ablation

Inverse Stefan type free boundary problem for the second order parabolic equation arising in modeling of laser ablation of biomedical tissues is analyzed, where information on the coefficients, heat flux on the fixed boundary, and density of heat sources are missing and must be found along with the temperature and free boundary. New PDE constrained optimal control framework in Hilbert–Besov spaces introduced in Abdulla (2013) is employed, where the missing data and the free boundary are components of the control vector, and optimality criteria are based on the final moment measurement of the temperature and position of the free boundary. Discretization by finite differences is pursued, and convergence of the discrete optimal control problems to the original problem is proved. Numerical algorithm based on the Fréchet differentiability and the gradient method in Hilbert–Besov spaces is implemented. The results of the numerical experiments for the identification of the free boundary and diffusion coefficient are presented.

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.

High-friction limits of Euler flows for multicomponent systems

The high-friction limit in Euler–Korteweg equations for fluid mixtures is analyzed. The convergence of the solutions towards the zeroth-order limiting system and the first-order correction is shown, assuming suitable uniform bounds. Three results are proved: the first-order correction system is shown to be of Maxwell–Stefan type and its diffusive part is parabolic in the sense of Petrovskii. The high-friction limit towards the first-order Chapman–Enskog approximate system is proved in the weak-strong solution context for general Euler–Korteweg systems. Finally, the limit towards the zeroth-order system is shown for smooth solutions in the isentropic case and for weak-strong solutions in the Euler–Korteweg case. These results include the case of constant capillarities and multicomponent quantum hydrodynamic models.

From the simple reacting sphere kinetic model to the reaction-diffusion system of Maxwell–Stefan type

In this article we deduce a mathematical model of Maxwell-Stefan type for a reactive mixture of polyatomic gases with a continuous structure of internal energy. The equations of the model are derived in the diffusive limit of a kinetic system of Boltzmann equations for the considered mixture, in the general non-isothermal setting. The asymptotic analysis of the kinetic system is performed under a reactive-diffusive scaling for which mechanical collisions are dominant with respect to chemical reactions. The resulting system couples the Maxwell-Stefan equations for the diffusive fluxes with the evolution equations for the number densities of the chemical species and the evolution equation for the temperature of the mixture. The production terms due to the chemical reaction and the Maxwell-Stefan diffusion coefficients are moreover obtained in terms of the collisional kernels and parameters of the kinetic model.

Fluid velocity based simulation of hydraulic fracture: a penny shaped model\u2014part I: the numerical algorithm

In the first part of this paper, a universal fluid velocity based algorithm for simulating hydraulic fracture with leak-off, previously demonstrated for the PKN and KGD models, is extended to obtain solutions for a penny-shaped crack. The numerical scheme is capable of dealing with both the viscosity and toughness dominated regimes, with the fracture being driven by a power-law fluid. The computational approach utilizes two dependent variables; the fracture aperture and the reduced fluid velocity. The latter allows for the application of a local condition of the Stefan type (the speed equation) to trace the fracture front. The obtained numerical solutions are carefully tested using various methods, and are shown to achieve a high level of accuracy.

Brash ice growth model – development and validation

Brash ice growth in frequently navigated areas like fairways or ports is quick due to the ‘freezing – breaking’ cycle induced by sub-zero temperatures and ship traffic. This problem is very acute in ports in Arctic areas where the temperatures are very low for long durations and the ship traffic is frequent. In order to take adequate action in managing the brash ice, the forecasts of the amount of brash ice expected should be reliable. The aim of this work is to develop and validate these prediction methods.The growth model developed is based on extension of earlier growth models which modify the Stefan type growth modelling. The improvement on the earlier models is that the brash ice layer is divided into three layers (instead of two in earlier models): The consolidated layer just below the water level, the brash ice over the water level and the unfrozen brash ice below the consolidated layer. The thermodynamic model follows the Stefan formulation including only the heat flux from latent heat release upon freezing (Stefan, 1891 and e.g. Anderson, 1961). The modelling includes the cyclic breaking and refreezing.The validation of the model is made using measurements carried out in winter 2013 in Luleå port and in winter 2015 in Sabetta in the Yamal peninsula. Luleå data suggests that the sideways motion of brash ice due to ship motion and wake should be taken into account when assessing the brash ice thickness. The analytical calculation over-estimates the brash ice thickness in the actual channel but under-estimates the total amount of broken ice. When applied to Sabetta data, the analytical calculation predicts well the observed brash ice thickness. It can be concluded that the analytical method that does not take into account any radiation heat fluxes can be applied in the high Arctic where solar radiation plays a minor role and ice surface is clearly below zero.