Classical Stefan Problem Research Articles

An alternative two-phase formulation and analytical solution for a Stefan problem in one-dimension accounting for volume change and sensible heat

Stefan problems, or free boundary problems, in the phase change process occur in many engineering problems. This paper develops a compact formula for a two-phase melting process in a one-dimensional semi-infinite slab with a novel approach. Unlike the classic Stefan problem, the novel approach considers volume change (or density difference) and the net-sensible-heat of the two phases. Furthermore, an analytical solution is proposed. To illustrate, the melting process of ice-water in a pipe is investigated. The results have been validated through an exact solution and comparisons with other relevant models. A special case of a one-phase problem is also proposed. The proposed model can be used as an alternative tool for engineers in estimating the position of the melt-front, depending on the dominant effect on the phase change process.

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.

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.

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.

A phase-field Lattice Boltzmann method for liquid-vapor phase change problems based on conservative Allen-Cahn equation and adaptive treegrid

A phase-field Lattice Boltzmann method (LBM) based on Allen-Cahn (A-C) equation with a modified adaptive mesh refinement (AMR) method is presented to simulate the liquid-vapor phase change problem. Three Lattice Boltzmann equations are solved: one for the conservative Allen-Cahn equation, one for the incompressible Navier-Stokes equations, and one for the energy equation. These three equations are written in the form of multiple-relaxation-time, together with the equilibrium and forcing terms using the set of Hermite polynomials. The adaptive mesh refinement technique with quadtree grid data structure is implemented to capture the liquid-vapor interface in simulations, in which the mesh can be locally refined by subdividing blocks identified by user-defined criterion. The Lax-Wendroff finite-difference scheme is adopted to calculate the whole computational domain continuously under nonuniform grids. The present method is validated by the classical Stefan problem and then performed to simulate the film boiling heat transfer. The numerical experiments verify the ability of the proposed method to capture the correct evolution of the interface, guarantee the numerical accuracy and remarkably save the computational cost.

An analytical solution for a phase change nano-capsule cooled by forced convection

Great efforts have been made on the solidification of phase change nano-capsules (PCNCs). However, the analytical solutions received less attention when its initial temperature is not at the fusion temperature. Therefore, this work aims to analytically study the thermal behavior of PCNC with the same solid-liquid phase density during forced convective cooling when its initial temperature is greater than the fusion temperature. Asymptotic solutions are developed for long- and small-time scale. Asymptic solutions in the limit of large Stefan number to the solid phase temperature and the position of the free boundary were derived using the perturbation method with boundary-fixing transformations. The results predicted by the derived solutions are in good agreement with the data reported in the reference. It is shown that the surface temperature decreases negatively and exponentially under the Robin boundary conditions during the pre-cooling process. Increasing the values of the interfacial tension or the large Stefan number has the effect of slowing down the movement of the free boundary. The rate of change of the solidification time with respect to the Stefan number is maintained constant for a stationary free boundary. The analytical solutions can also be used to solve the classical Stefan problem (large Stefan number) by ignoring the surface tension. The analytic estimate provides a more convenient option for engineering applications than the complex numerical solution.

The Stefan problem in a thermomechanical context with fracture and fluid flow

The classical Stefan problem, concerning mere heat‐transfer during solid‐liquid phase transition, is here enhanced towards mechanical effects. The Eulerian description at large displacements is used with convective and Zaremba‐Jaumann corotational time derivatives, linearized by using the additive Green‐Naghdi's decomposition in (objective) rates. In particular, the liquid phase is a viscoelastic fluid while creep and rupture of the solid phase is considered in the Jeffreys viscoelastic rheology exploiting the phase‐field model and a concept of slightly (so‐called “semi”) compressible materials. The ‐theory for the heat equation is adopted for the Stefan problem relaxed by allowing for kinetic superheating/supercooling effects during the solid‐liquid phase transition. A rigorous proof of existence of weak solutions is provided for an incomplete melting, employing a time discretization approximation.

A general non-Fourier Stefan problem formulation that accounts for memory effects

The Stefan problem is the classical model of a melting phase change. In heterogeneous systems, such phase changes can exhibit non-Fourier (anomalous) behaviors, where the advance of the melt interface does not follow the expected time scaling. These situations can be modeled by replacing the derivatives, in the governing partial differential equations, with fractional order derivatives. In particular, replacing the time derivatives leads to non-Fourier models that account for memory effects in the system. In this work, by using appropriate time convolution integrals, a general thermodynamic balance statement for melting phase problems, explicitly accounting for memory effects, is developed. From this balance, a general model formulation applicable to problems involving melting over a temperature range (i.e., a mushy region) is derived. A key component in this model is the representation of memory effects through the use of fractional derivative based constitutive models of the enthalpy and heat flux. On shrinking the mushy region to a single isotherm, a general sharp interface melting model is obtained. Here, in contrast to the classic Stefan problem, the fractional derivatives induce a natural regularization, such that the constitutive models for enthalpy and heat flux are continuous at the melt interface; a result confirmed through numerical simulation. To further support the theoretical findings, a physical example of a non-Fourier Stefan problem is presented. Overall the development and results in this paper underscore the importance of explicitly relating the development of fractional calculus models to the appropriate thermodynamic balance statements.

Pattern formation and front stability for a moving-boundary model of biological invasion and recession

We investigate pattern formation in a two-dimensional (2D) Fisher–Stefan model, which involves solving the Fisher–KPP equation on a compactly-supported region with a moving boundary. The boundary evolves analogously to the classical one-phase Stefan problem, such that boundary speed is proportional to the local population density gradient. By combining the Fisher–KPP and classical Stefan theory, the Fisher–Stefan model alleviates two limitations of the Fisher–KPP equation for biological populations. Unlike the Fisher–KPP equation, solutions to the Fisher–Stefan model have compact support, explicitly defining the region occupied by the population. Furthermore, the Fisher–Stefan model admits travelling wave solutions with non-negative density for all wave speeds c∈(−∞,∞), and can thus model both population invasion and population recession. In this work, we investigate whether the 2D Fisher–Stefan model predicts pattern formation, by analysing the linear stability of planar travelling wave solutions to sinusoidal transverse perturbations. Planar fronts of the Fisher–KPP equation are linearly stable. Similarly, we demonstrate that invading planar fronts (c>0) of the Fisher–Stefan model are linearly stable to perturbations of all wave numbers. However, our analysis demonstrates that receding planar fronts (c<0) of the Fisher–Stefan model are linearly unstable for all wave numbers. This is analogous to unstable solutions for planar solidification in the classical Stefan problem. Introducing a surface tension regularisation stabilises receding fronts for short-wavelength perturbations, giving rise to a range of unstable modes and a most unstable wave number. We supplement linear stability analysis with level-set numerical solutions that corroborate theoretical results. Overall, front instability in the Fisher–Stefan model suggests a new mechanism for pattern formation in receding biological populations.

Effect of dopants on the optical properties of benzil crystals

Single-doped [Ag, Na, iodine, naphthalene (Naph), meta-dintrobenzene (m-DNB)] and double-doped, organic–inorganic (Naph + iodine; m-DNB + iodine) benzil (Bz) crystals have been grown from melt in a Bridgman-Stockbarger configuration. All crystals have been grown in the same experimental conditions, temperature variation at the growth interface = 30 K and moving speed of the growth ampoule = 1 mm/h. These conditions have been estimated from the theoretical analysis involving the solution of the classical (conventional) Stefan problem for flat solid–liquid interface in the boundary conditions imposed by our growth system. The effect of doping on the disorder degree, photoluminescence and dielectric constant of the Bz matrix has been investigated. The disorder in the crystals increased with the increase of dopant concentration. The highest disorder was induced by the organic dopant m-DNB, whose molecules does not show an adequate geometrical similarity with Bz molecules. The position of the photoluminescence peak, both at excitation with 335 nm and 435 nm, has not been affected by the presence of dopant(s). All doped samples show Second Harmonic Generation, even those with a significant degree of disorder. The effect of the single and double-doping on the second-order ONL coefficient was analyzed by comparison with the second-order ONL coefficient of pure Bz crystals. In the case of the double-doping, the highest second-order optical nonlinear coefficient has been evidenced in the Bz crystals simultaneously doped with m-DNB and iodine in the same concentration (1%).

A Fisher-KPP Model with a Nonlocal Weighted Free Boundary: Analysis of How Habitat Boundaries Expand, Balance or Shrink.

In this paper, we propose a novel free boundary problem to model the movement of single species with a range boundary. The spatial movement and birth/death processes of the species found within the range boundary are assumed to be governed by the classic Fisher-KPP reaction-diffusion equation, while the movement of a free boundary describing the range limit is assumed to be influenced by the weighted total population inside the range boundary and is described by an integro-differential equation. Our free boundary equation is a generalization of the classical Stefan problem that allows for nonlocal influences on the boundary movement so that range expansion and shrinkage are both possible. In this paper, we prove that the new model is well-posed and possesses steady state. We show that the spreading speed of the range boundary is smaller than that for the equivalent problem with a Stefan condition. This implies that the nonlocal effect of the weighted total population on the boundary movement slows down the spreading speed of the population. While the classical Stefan condition categorizes asymptotic behavior via a spreading-vanishing dichotomy, the new model extends this dichotomy to a spreading-balancing-vanishing trichotomy. We specifically analyze how habitat boundaries expand, balance or shrink. When the model is extended to have two free boundaries, we observe the steady state scenario, asymmetric shifts, or even boundaries moving synchronously in the same direction. These are newly discovered phenomena in the free boundary problems for animal movement.

On the Reformulation of the Classical Stefan Problem as a Shape Optimization Problem

This article is concerned with the multi-dimensional one-phase Stefan problem, which belongs to the class of moving boundary problems. We suggest to reformulate the classical Stefan problem as a shape optimization problem, consisting of an objective functional for the moving boundary and a partial differential equation corresponding to a heat type equation. Minimizing the objective functional subject to the differential equation under consideration is equivalent to solving the Stefan problem. In order to apply gradient-based optimization algorithms, we analytically compute the shape gradient of the objective functional. A numerical example justifies our approach.

Solidification and dynamic wetting: A unified modeling framework

A conceptual and mathematical framework for the singularity-free modeling of non-equilibrium solidification/melting and non-isothermal dynamic wetting is developed where both processes are embedded into a broader class of physical phenomena as particular cases. This allows one to consider problems describing fluid flows with phase transitions and dynamic wetting occurring independently or interactively in a regular conceptually consistent way without ad hoc assumptions. The simplest model formulated on the basis of this approach explains, at this stage qualitatively, the arrest of the moving contact line observed experimentally in the impact and spreading of a molten drop on a cold substrate. The classical Stefan problem and the model of isothermal dynamic wetting as an interface formation process are recovered as limiting cases.

Development and validation of an asymptotic solution for a two-phase Stefan problem in a droplet subjected to convective boundary condition

Droplet solidification is governed by classical Stefan problems which have been commonly treated as a single-phase problem by the majority of the studies in the literature. This approach, however, is unable to capture the initial temperature and the start of freezing time correctly. The treatment of two-phase Stefan problem in spherical coordinates is limited. No known exact solution exists, albeit numerical solutions and asymptotics have proven to be useful. We present a singular perturbation solution in the limit of low Stefan number and arbitrary Biot number for the two-phase Stefan problem in a finite spherical domain. An asymptotic solution is developed for a droplet at a non-freezing initial temperature subjected to a convective boundary condition at the surface. The solution is developed for both long-time and short-time scales. The results from asymptotic expansion method are validated with the experimental results in the literature and are further verified by a numerical model of a freezing droplet using enthalpy–porosity method. The sensitivity of the asymptotic solution to the droplet initial temperature, Biot number, and Stefan number has also been studied. The results indicate that the solution from perturbation series and enthalpy–porosity method agrees to within 1%–10% for temperature profile and overall freezing times over a wide range of practical values for initial temperature, Stefan and Biot numbers for the application of spray freezing. Our perturbation series solution is also able to capture the effect of initial temperature on the overall freezing time of the droplet.

A stable and accurate scheme for solving the Stefan problem coupled with natural convection using the Immersed Boundary Smooth Extension method

The dissolution of solids has created spectacular geomorphologies ranging from centimeter-scale cave scallops to the kilometer-scale “stone forests” of China and Madagascar. Mathematically, dissolution processes are modeled by a Stefan problem, which describes how the motion of a phase-separating interface depends on local concentration gradients, coupled to a fluid flow. Simulating these problems is challenging, requiring the evolution of a free interface whose motion depends on the normal derivatives of an external field in an ever-changing domain. Moreover, density differences created in the fluid domain induce self-generated convecting flows that further complicate the numerical study of dissolution processes. In this contribution, we present a numerical method for the simulation of the Stefan problem coupled to a fluid flow. The scheme uses the Immersed Boundary Smooth Extension method to solve the bulk advection-diffusion and fluid equations in the complex, evolving geometry, coupled to a θ-L scheme that provides stable evolution of the boundary. We demonstrate 3rd-order temporal and pointwise spatial convergence of the scheme for the classical Stefan problem, and 2nd-order temporal and pointwise spatial convergence when coupled to flow. Examples of dissolution of solids that result in high-Rayleigh number convection are numerically studied, and qualitatively reproduce the complex morphologies observed in recent experiments.

Modeling polymer crystallisation induced by a moving heat sink.

Recent experimental work has shown that polymer crystallisation can be used to "move" and organize nanoparticles (NP). As a first effort at modeling this situation, we consider the classical Stefan problem but with the modification that polymer crystallisation does not occur at a single temperature. Rather, the rate of crystallisation is proportional to its subcooling, and here we employ a form inspired by the classical Avrami model to describe this functional form. Our results for the movement of the polymer crystallisation front, as defined as the point where the crystallinity is 50%, closely track the results of the classical Stefan problem. Thus, at this level of approximation, the crystallisation kinetics of the polymer do not cause qualitative changes to the physics of this situation. Inspired by this fact we study the more interesting situation where the directional recrystallisation of a polymer melt is considered, e.g., through the application of a moving heat sink over an initially molten polymer, reminiscent of a processing technique termed zone annealing. The polymer crystallisation shows that a steady state exists for a range of sink velocities. The solid-melt interface moves slightly ahead of the sink but at the same velocity. The steady-state distance between the sink and the interface decreases with increasing sink velocity - this is a consequence of the excess cooling provided by the sink over what is required to crystallise the melt. The most interesting new result is that the temperature of the crystal-melt interface decreases with increasing sink velocity. This is in line with the ansatz of Lovinger and Gryte who suggested that larger zone annealing velocities correspond to progressively larger effective undercoolings at which polymer crystallisation occurs.

Approximate solutions to one-phase Stefan-like problems with space-dependent latent heat

The work in this paper concerns the study of different approximations for one-dimensional one-phase Stefan-like problems with a space-dependent latent heat. It is considered two different problems, which differ from each other in their boundary condition imposed at the fixed face: Dirichlet and Robin conditions. The approximate solutions are obtained by applying the heat balance integral method (HBIM), the modified HBIM and the refined integral method (RIM). Taking advantage of the exact analytical solutions, we compare and test the accuracy of the approximate solutions. The analysis is carried out using the dimensionless generalised Stefan number (Ste) and Biot number (Bi). It is also studied the case when Bi goes to infinity in the problem with a convective condition, recovering the approximate solutions when a temperature condition is imposed at the fixed face. Some numerical simulations are provided in order to assert which of the approximate integral methods turns out to be optimal. Moreover, we pose an approximate technique based on minimising the least-squares error, obtaining also approximate solutions for the classical Stefan problem.

A Novel 2D Model for Freezing Phase Change Simulation during Cryogenic Fracturing Considering Nucleation Characteristics

A 2D computational fluid dynamics (CFD) model in consideration of nucleation characteristics (homogeneous/heterogeneous nucleation) using the volume of fluid (VOF) method and Lee model was proposed. The model was used to predict the process of a multiphase flow accompanied by freezing phase change during cryogenic fracturing. In this model, nucleation characteristic (homogeneous and heterogeneous nucleation) during the freezing process and the influence of the formed ice phase on the flowing behavior was considered. Validation of the model was done by comparing its simulation results to Neumann solutions for classical Stefan problem. The comparison results show that the numerical results are well consistent with the theoretical solution. The maximum relative differences are less than 7%. The process of multiphase flow accompanied by the freezing of water was then simulated with the proposed model. Furthermore, the transient formation and growth of ice as well as the evolution of temperature distribution in the computational domain was studied. Results show that the proposed method can better consider the difference between homogeneous nucleation in the fluid domain and heterogeneous nucleation on the wall boundary. Finally, the main influence factors such as the flow velocity and initial distribution of ice phase on the fracturing process were discussed. It indicates that the method enable to simulate the growth of ice on the wall and its effect on the flow of multiphase fluid.

An implementation of the phase-field model based on coupled thermomechanical finite element solvers for large-strain twinning, explicit dynamic fracture and the classical Stefan problem

The implementation of a phase-field model in finite elements usually requires significant expertise and involves the development of a user element with additional degrees of freedom. An alternative implementation of the phase-field model within a thermo-mechanical finite element simulation package was presented in (Cho et al 2012 Int. J. Solids Struct. 49 1973–1992), where the phase-field variable is treated as the temperature degree of freedom. However, this approach has only been used for small strain phase-field modelling of martensitic transformations and quasistatic phase-field modelling of fracture. In this work, we present a phase-field finite element implementation via the temperature degree of freedom for several additional cases from the literature: (i) the large-strain phase-field description of deformation twinning presented in (Clayton and Knap 2011 Physica D 240 841–858), (ii) phase-field description of brittle fracture with inertial effects based on the theory from (Molnár and Gravouil 2017, Finite Elem. Anal. Des. 130 27–38) and (Miehe et al 2010 Int. J. Numer. Methods Eng. 83 1273–1311) and (iii) the classical Stefan problem of solidification presented in (Mackenzie and Robertson 2002 J. Comput. Phys. 181 526–544). The last problem involves the temperature and phase-field variables as unknowns.

On the Two-phase Fractional Stefan Problem

Abstract The classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing the main properties of the classical problem that are of interest to us. Then we introduce the fractional Stefan problem and develop the basic theory. After that we center our attention on selfsimilar solutions, their properties and consequences. We first discuss the results of the one-phase fractional Stefan problem, which have recently been studied by the authors. Finally, we address the theory of the two-phase fractional Stefan problem, which contains the main original contributions of this paper. Rigorous numerical studies support our results and claims.