Stefan Problem Research Articles

Analysis of the Main Factors Controlling the Formation of Subaerial Taliks, Using a One-Dimensional Mathematical Model. A Case Study for the Shestakovka River Basin (Central Yakutia)

Received March 9, 2023; revised September 7, 2023; accepted October 2, 2023This study presents a mathematical model of heat transfer in a subaerial talik. The model is based on the concepts presented in classical works on permafrost, as well as on the results of geological and geophysical research carried out in the Shestakovka River basin (Central Yakutia). This model is based on the solution of the classical Stefan problem on the moving of the phase transition boundaries for a multilayer and multiphase medium. The solution was calculated on a unstructured mesh. When the phase boundaries move, thawed or frozen layers of soil are formed or wedged out. The layers include: snow cover, seasonally thawed soil, seasonally frozen and frozen sand deposits, as well as soil-vegetative layer. Published empirical relationships were used to calculate thermophysical coefficients, which are presented in this article. Simple variants of the model were considered to clarify the contribution of various factors to the process of formation and evolution of taliks. It has been established that the presence of snow cover and soil-vegetative layer have the most significant effect on the formation of taliks. Calculations show that taliks are formed in the first years of the modeled period, in the presence of snow and the absence of soil-vegetative layer. The soil-vegetative layer, depending on its composition and moisture content (ice content), can prevent the formation and development of taliks. The authors do not consider cases where shrubs contribute to snow accumulation. The humidity and iciness of the layer of sand sediments located in Central Yakutia have practically no effect on this process.

Lattice Boltzmann modelling of salt precipitation during brine evaporation

Salt precipitation during brine evaporation in porous media is an important phenomenon in a variety of natural and engineering scenarios. This work establishes a multiphase multicomponent lattice Boltzmann (LB) method with phase change for simulating salt precipitation during brine evaporation. In the proposed LB models, the gas–brine multiphase flow, brine evaporation, salt concentration evolution, salt precipitate nucleation and growth are simultaneously considered. Simulations of the Stefan problem are first conducted to verify the proposed numerical models and determine the diffusion coefficient of brine vapour. Once the lattice Boltzmann models have been validated, salt precipitation during brine evaporation is simulated to investigate the competition mechanisms between salt precipitate nucleation and growth reaction. The results show that the typical salt precipitation patterns in existing experimental observation can be successfully reproduced, including the ring-like and pancake-like patterns. The difference in the salt precipitation patterns is explained by the competition mechanism between precipitate growth and nucleation according to the present study. Furthermore, the salt precipitation during gas injection into a microfluidic chip is investigated. The evolution of salt and brine saturation shows similar patterns to existing experimental results, and the effects of the gas injection rate on salt precipitation performance are clarified. The LB models in the present work can simulate salt precipitation with comprehensive consideration of multiphase brine evaporation, salt species mass transport, precipitate nucleation and growth, which have not been realized in previous studies. The numerical showcases demonstrate the excellent performance of the proposed models for the simulation of salt precipitation in porous media, which promise to guide practical engineering applications like CO2 sequestration.

Unsteady Stefan problem with kinetic interface conditions for rarefied gas deposition

The Stefan problem is a phase transition problem with a moving interface between two phases, originally established for solid and liquid. The influence of kinetic theory-based non-equilibrium thermodynamic models on the classical Stefan problem for deposition or sublimation processes is investigated. The gas phase is assumed to be in the rarefied slip regime, so default continuum and equilibrium models struggle. The results are dependent on the well-studied Knudsen and Mach numbers, defined at the phase transition interface. The differences from the classical model become significant as the Knudsen number, and correspondingly the level of rarefaction, increases. We observe influences on the temperature, pressure, phase transition speed and deviation from the saturation conditions.

Determination of one unknown coefficient in a two-phase free boundary problem in an angular domain with variable thermal conductivity and specific heat

Two different two-phase free boundary Stefan problems in an angular domain with temperature-dependent thermal coefficients are considered. Analytical similarity solutions are obtained imposing a Dirichlet or Neumann type boundary condition, respectively, by solving functional problems. Moreover, formulas are obtained for the determination of one unknown thermal coefficient in the overspecified problem that consists in adding a Neumann condition to the problem with a Dirichlet one if and only if some restrictions on data are verified.

Survival, extinction, and interface stability in a two-phase moving boundary model of biological invasion

We consider a moving boundary mathematical model of biological invasion. The model describes the spatiotemporal evolution of two adjacent populations: each population undergoes linear diffusion and logistic growth, and the boundary between the two populations evolves according to a two-phase Stefan condition. This mathematical model describes situations where one population invades into regions occupied by the other population, such as the spreading of a malignant tumour into surrounding tissues. Full time-dependent numerical solutions are obtained using a level-set numerical method. We use these numerical solutions to explore several properties of the model including: (i) survival and extinction of one population initially surrounded by the other; and (ii) linear stability of the moving front boundary in the context of a travelling wave solution subjected to transverse perturbations. Overall, we show that many features of the well-studied one-phase single population analogue of this model can be very different in the more realistic two-phase setting. These results are important because realistic examples of biological invasion involve interactions between multiple populations and so great care should be taken when extrapolating predictions from a one-phase single population model to cases for which multiple populations are present. Open source Julia-based software is available on GitHub to replicate all results in this study.

Directional Crystallization in the Presence of a Mushy Layer with Applications to the Earth’s Inner Core Boundary

We formulate the mathematical model of directional crystallization of a binary melt with a mushy layer (region) between purely solid and liquid phases. This model is complicated by melt convection and pressure-dependent phase transition temperature. Approximate analytical solutions to this nonlinear moving-boundary problem are constructed. Namely, the concentration of impurity, fraction of solid phase, mushy region thickness, average fluid velocity, primary interdendritic spacing, mean radius of a chimney, and a characteristic distance between chimneys in a mushy region are found. Using this analytical solution, we describe the mushy region structure near the inner core boundary of the Earth, which is consistent with computer simulations and estimates existing in recent literature. A scheme illustrating the mushy region arrangement with chimneys at the inner core boundary of the Earth is presented. This arrangement based on the developed theory represents the novelty and importance of our study.

Exact controllability to the trajectories of the one-phase Stefan problem

This paper deals with the boundary exact controllability to the trajectories of the one-phase Stefan problem in one spatial dimension. This is a free-boundary problem that models solidification and melting processes. We prove the local exact controllability to (smooth) trajectories. To this purpose, we first reformulate the problem as the local null controllability of a coupled PDE-ODE system with distributed controls. Then, a new Carleman inequality for the adjoint of the linearized PDE-ODE system, coupled on the boundary through nonlocal in space and memory terms, is presented. This leads to the null controllability of an appropriate linear system. Finally, the result is obtained via local inversion, by using Lyusternik-Graves' Theorem. As a byproduct of our approach, we find that some parabolic equations which contains memory terms located on the boundary are null-controllable.

Asymmetric PCM melting and thermal convection in a rectangular enclosure with straight and wavy heat transfer passages

Experimental and numerical investigations of thermal convection in the vicinity of a straight and wavy horizontal heat transfer fluid passage (HTF)s placed in a rectangular enclosure filled with phase change material (PCM) are presented. The enthalpy-porosity-based finite volume solver is used to simulate the spatiotemporal evolution of the thermal convection, which is well-complemented by experimental visualization and thermocouple measurements. Heat transfer from the horizontal HTF exhibits asymmetrical melting. Heat diffusion is the dominating energy transfer mechanism on the bottom part of the HTF passage and is compared with the Stefan problem. The temporal evolution of thermal convection on the top part of HTF passage involves three distinct phases of melting, viz. initial conduction phase, Rayleigh–Bénard (R-B) convection with linear convection, and vortex coalescence effects. The final stage of the melting process is influenced by the topography of a peninsula of unmelt PCM, which is influenced by specific top-boundary conditions. Thermal convection in PCM above the wavy HTF passage exhibits restricted plume development leading to an improved melting rate and a higher average melt height. Augmentation of internal convection characteristics of HTF with the evolution of melting is also explored. A systematic investigation of the convection plume development and heat transfer characteristics in PCM by varying the HTF Reynolds (1700<Re<2750) and Stefan numbers (0.4<Ste<0.6) has been carried out, and analyzed the nature of thermal plumes, melt fraction, and average penetration length with the Fourier number. Additionally, novel correlations are proposed between Nusselt and Rayleigh numbers for straight and wavy tubes in the intermediate melting stage (1.3×104<Ra<4.7×107).

Назначение рациональных режимов электроэрозионной обработки деталей из титанового сплава ВТ5 на основе решения тепловой задачи о перемещении границы фазового превращения материала

The paper presents results of theoretical studies of the VT5 titanium alloy machinability by the electrodischarge machining method based on solving the thermal problem of the material phase transformation boundary displacement (Stefan problem). It proposes a method for determining parameters of electrical pulses for the VT5 alloy electrodischarge machining in order to increase the process productivity; recommendations are provided for that purpose. Density of the heat flux and its duration were determined necessary for implementation of the VT5 alloy electrodischarge machining process. Dependences were established of the minimum value of the heat flux pulse duration, when the VT5 alloy electrodischarge machining process was possible, and of maximum value of the heat flux pulse duration ensuring maximum removal of the VT5 alloy from the workpiece in one pulse, on the heat flux density. It is shown that for the maximum productivity of the VT5 alloy electrodischarge machining at the heat flux density used, it is necessary to assign effective duration of such a flux. Dependences of the heat flux effective duration on its density were established. Besides, to assign rational modes of VT5 ally electrodischarge machining, relationships were established and provided between the VT5 alloy machinability curves (dependences of the material penetration depth on the pulse duration) and other materials, including those for which rational modes are currently defined

Mathematical modeling of the heat process in closure electrical contacts with a heat source

This study presents a mathematical model that describes the heat process during the vaporization of metal at electrical contacts. The arc erosion model for the micro-asperity in the liquid state is established using a one-phase Stefan problem. Two problems involving Joule heating effects on the generalized heat equation are considered. The solution of these two problems is based on the similarity principle, which allows for the reduction of the problem to ordinary differential equations. The existence and uniqueness of the solutions are proven using fixed point theory in a Banach space. Additionally, a solution for the particular case of constant thermal coefficients is provided.

Numerical schemes for a moving-boundary convection-diffusion-reaction model of sequencing batch reactors

Sequencing batch reactors (SBRs) are devices widely used in wastewater treatment, chemical engineering, and other areas. They allow for the sedimentation and compression of solid particles of biomass simultaneously with biochemical reactions with nutrients dissolved in the liquid. The kinetics of these reactions may be given by one of the established activated sludge models (ASMx). An SBR is operated in various stages and is equipped with a movable extraction and fill device and a discharge opening. A one-dimensional model of this unit can be formulated as a moving-boundary problem for a degenerating system of convection-diffusion-reaction equations whose unknowns are the concentrations of the components forming the solid and liquid phases, respectively. This model is transformed to a fixed computational domain and is discretized by an explicit monotone scheme along with an alternative semi-implicit variant. The semi-implicit variant is based on solving, during each time step, a system of nonlinear equations for the total solids concentration followed by the solution of linear systems for the solid component percentages and liquid component concentrations. It is demonstrated that the semi-implicit scheme is well posed and that both variants produce approximations that satisfy an invariant region principle: solids concentrations are nonnegative and less or equal to a set maximal one, percentages are nonnegative and sum up to one, and substrate concentrations are nonnegative. These properties are achieved under a Courant-Friedrichs-Lewy (CFL) condition that is less restrictive for the semi-implicit than for the explicit variant. Numerical examples with realistic parameters illustrate that as a consequence, the semi-implicit variant is more efficient than the explicit one.

The effect of pore-scale contaminant distribution on the reactive decontamination of porous media

AbstractA porous material that has been contaminated with a hazardous chemical agent is typically decontaminated by applying a cleanser solution to the surface and allowing the cleanser to react into the porous material, neutralising the agent. The agent and cleanser are often immiscible fluids and so, if the porous material is initially saturated with agent, a reaction front develops with the decontamination reaction occurring at this interface between the fluids. We investigate the effect of different initial agent configurations within the pore space on the decontamination process. Specifically, we compare the decontamination of a material initially saturated by the agent with the situation when, initially, the agent only coats the walls of the pores (referred to as the ‘agent-on-walls’ case). In previous work (Luckins et al., European Journal of Applied Mathematics, 31(5):782–805, 2020), we derived homogenised models for both of these decontamination scenarios, and in this paper we explore the solutions of these two models. We find that, for an identical initial volume of agent, the decontamination time is generally much faster for the agent-on-walls case compared with the initially saturated case, since the surface area on which the reaction can occur is greater. However for sufficiently deep spills of contaminant, or sufficiently slow reaction rates, decontamination in the agent-on-walls scenario can be slower. We also show that, in the limit of a dilute cleanser with a deep initial agent spill, the agent-on-walls model exhibits behaviour akin to a Stefan problem of the same form as that arising in the initially saturated model. The decontamination time is shown to decrease with both the applied cleanser concentration and the rate of the chemical reaction. However, increasing the cleanser concentration is also shown to result in lower decontamination efficiency, with an increase in the amount of cleanser chemical that is wasted.

On mKdV and associated classes of moving boundary problems: reciprocal connections

AbstractA class of Stefan-type moving boundary problems for the canonical modified Korteweg–de Vries (mKdV) equation of soliton theory is solved via application of a similarity reduction to Painlevé II which involves Airy’s equation. A reciprocal transformation is applied to derive a linked class of solvable moving boundary problems for a basic Casimir member of a compacton hierarchy. Application of a class of involutory transformations with origin in an autonomisation procedure for the Ermakov–Ray–Reid system is then used to isolate novel solvable moving boundary problems for Ermakov-modulated mkdV equations.

Валидация численной модели плавления октадекана с помощью лабораторного эксперимента

The paper presents results of studying the melting process of octadecane, the heat-accumulating substance used in a space heat accumulator. The Stefan problem formulated for the phase transition boundary has an analytical solution only for the one-dimensional case. And in the case of the structure real geometry, it could be solved only using the numerical methods, in particular, by simulation in the ANSYS Fluent universal software system for the finite element analysis. To validate the developed numerical models, model experiments were carried out on octadecane melting in the cylindrical glass flask with heat-insulated side walls, the heater was positioned at the flask lower end. Two-dimensional and three-dimensional numerical models were investigated at the heating powers of 7.1 W and 34.2 W. The molten substance fraction dynamics in the two-dimensional and three-dimensional models was compared to the experimental data. Heat losses in heating the structure and convective heat exchange with the external media were taken into account. Results of numerical and laboratory experiments are presented, which could be used in testing the constant power heating model of a heat-accumulating substance. The results obtained show that with absolute temperature alteration in the heat-accumulating substance by an insignificant value (up to 10°C), the Boussinesq approximation provides results well verified by the experimental data.

Solutions of An Ill-Posed Stefan Problem

Solutions of An Ill-Posed Stefan Problem

On a multiple time-scales perturbation analysis of a Stefan problem with a time-dependent Dirichlet boundary condition

In this paper, a classical Stefan problem with a prescribed and small time-dependent temperature at the boundary is studied. By using a multiple time-scales perturbation method, it is shown analytically how the moving boundary profile is influenced by the prescribed temperature at the boundary and the initial conditions. Only a few exact solutions are available for this type of problems and it turns out that the constructed approximations agree very well with these exact solutions. In particular, approximations of solutions for this type of problems, with periodic and decaying temperatures at the boundary, are constructed. Furthermore, these approximations are valid on a long time scale, and seems to be not available in the literature.

Solving a one-dimensional moving boundary problem based on wave digital principles

We report on a novel method for solving one-dimensional moving boundary problems based on wave digital principles. Here, we exploit multidimensional wave digital algorithms to derive an efficient and robust algorithm for the solution of the considered problem. Our method lets the wave digital model, on which the algorithm is based, expand according to the size of the solution domain. The expanding model introduces new dynamical elements, which must be properly initialized to obtain a calculable algorithm. To deal with this problem, we make use of linear multistep methods to extrapolate the missing values. Our results show the proposed method to indeed be capable of correctly solving a one-dimensional partial differential equation describing a growing biological axon.

Numerical Simulations of Tank Sloshing Problems Based on Moving Pseudo-Boundary Method of Fundamental Solution

The moving pseudo-boundary method of fundamental solutions (MFS) was employed to solve the Laplace equation, which describes the potential flow in a two-dimensional (2D) numerical wave tank. The MFS is known for its ease of programming and the advantage of its high precision. The solution of the boundary value can be expressed by a linear combination of the fundamental solutions. The major issue with such an implementation is the optimal distribution of source nodes in the pseudo-boundary. Traditionally, the positions of the source nodes are assumed to be fixed to keep the set of equations closed. However, in the moving boundary problem, the distribution of source nodes may influence the stability of numerical calculations. Moreover, MFS is unstable in time iterations. Hence, it is necessary to constantly revise the weighting coefficients of fundamental solutions. In this study, the source nodes were free, and their locations were determined by solving a nonlinear least-squares problem using the Levenberg–Marquardt algorithm. To solve the above least-squares problem, the MATLAB© routine lsqnonlin was adopted. Additionally, the weighting coefficients of fundamental solutions were solved as a nonlinear least-squares problem using the aforementioned method. The numerical results indicated that the numerical simulation method adopted in this paper is accurate and reliable in solving the problem of 2D tank sloshing. The main contribution of this study is to expand the application of the MFS in engineering by integrating it with the optimal configuration problem of pseudo-boundaries to solve practical engineering problems.

Invadopodia Formation in Cancer Cell: The Mathematical and Computational Modelling Based on Free Boundary Problem

We present a mathematical model of an individual cell to expand the simulation of invadopodia formation to a three-dimensional (3D) domain for a more realistic complexity. Simulating invadopodia replication in order for it to be biologically relevant is important since it helps us to understand cancer invasion and metastasis better as well as giving some insight into investigating ways to stop the spread of this fatal disease. Invadopodia formation is formulated using the Stefan problem approach, where the free boundary is characterised by the Stefan free boundary condition, in which the boundary membrane is not known in advance. Level set method is proposed to indicate the behaviour of the cell interface and the motion of the plasma membrane. An enthalpy method (phase-transition problem) is used to describe the cell membrane diffusion. In addition to this, we were able to improve the simulation outcome, giving it a more realistic complexity by using a different simulation technique and domain as well as a different data set. Singularities and instabilities were eliminated. The results that were achieved have the potential to be helpful for novel approaches or to be extended to other methods in the development of a more accurate numerical simulation.

The singular set in the Stefan problem

The singular set in the Stefan problem