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 exploiting 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, exploiting a concept of slightly (so-called “semi”) compressible materials. The $L^1$-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 week solutions is provided for an incomplete melting, exploiting a time-discretisation approximation.

Modelling InSb Czochralski Growth

A model of Czrochralski crystal growth is presented that reduces the issue of finding the temperature distribution as a Stefan problem. From the temperature distribution the corresponding distribution of thermal stress inside the growing crystal can be computed. This work will allow the rapid simulation of a variety of crystal growing strategies which would be prohibitively expensive in an experimental setting.

Verification of a mathematical model for the solution of the Stefan problem using the mushy layer method

The use of solar energy has limitations due to its periodic availability: solar plants do not operate at night and are ineffective in dull weather. The solution of this problem involves the introduction of energy storage and duplication systems into the conversion loop. Among the energy storage systems, solid–liquid phase transition modules have significant energy, ecologic, and cost advantages. Physical processes in modules of this type are described by a system of non-stationary nonlinear partial differential equations with specific boundary conditions at the phase interface. The verification of a method for solving the Stefan problem for a heat-storage material is presented in this paper. The use of the mushy layer method made it possible to simplify the classical mathematical model of the Stefan problem by reducing it to a nonstationary heat conduction problem with an implicit heat source that takes into account the latent heat of transition. The phase transition is considered to occur in an intermediate zone determined by the solidus and liquidus temperatures rather than in in infinite region. To develop a Python code, use was made of an implicit computational scheme in which the solidus and liquidus temperatures remain constant and are determined in the course of numerical experiments. The physical model chosen for computer simulation and algorithm verification is the process of ice layer formation on a water surface at a constant ambient temperature. The numerical results obtained allow one to determine the temperature fields in the solid and the liquid phase and the position of the phase interface and calculate its advance speed. The algorithm developed was verified by analyzing the classical analytical solution of the Stefan problem for the one-dimensional case at a constant advance speed of the phase interface. The value of the verification coefficient was determined from a numerical solution of a nonlinear equation with the use of special built-in Python functions. Substituting the data for the physical model under consideration into the analytical solution and comparing them with the numerical simulation data obtained with the use of the mushy layer method shows that the results are in close agreement, thus demonstrating the correctness of the computer algorithm developed. These studies will allow one to adapt the Python code developed on the basis of the mushy layer method to the calculation of heat storage systems with a solid-liquid phase transition with account for the features of their geometry, the temperature level, and actual boundary conditions.

Application of the vertical model of the seasonally thawed layer for the Tulik Lake area (Alaska)

Практический интерес в районах вечной мерзлоты представляет глубина сезонного оттаивания. Построена одномерная (в вертикальном направлении) упрощенная полуэмпирическая модель динамики вечной мерзлоты в “приближении медленных движений границ фазового перехода”, основанная на задаче Стефана и эмпирических соотношениях. Калибровочные параметры модели выбираются для исследуемого района с использованием натурных измерений глубины оттаивания и температуры воздуха. Проверка работоспособности численной модели проведена для района оз. Тулик (Аляска). Получено согласие рассчитанных значений глубины талого слоя и температуры поверхности почвы с результатами измерений Due to the change in global air temperature, the assessment of permafrost reactions to climate change is of interest. As the climate warms, both the thickness of the thawed soil layer and the period for existence of the talik are increased. The present paper proposes a small-size numerical model of vertical temperature distributions in the thawed and frozen layers when a frozen layer on the soil surface is absent. In the vertical direction, thawed and frozen soils are separated. The theoretical description of the temperature field in soils when they freeze or melt is carried out using the solution of the Stefan problem. The mathematical model is based on thermal conductivity equations for the frozen and melted zones. At the interfacial boundary, the Dirichlet condition for temperature and the Stefan condition are set. The numerical methods for solving of Stefan problems are divided into two classes, namely, methods with explicit division of fronts and methods of end-to-end counting. In the present work, the method with the selection of fronts is implemented. In the one-dimensional Stefan problem, when transformed to new variables, the computational domain in the spatial variable is mapped onto the interval [0 , 1]. In the presented equations, the convective terms characterize the rate of temperature transfer (model 1). A simplified version of the Stefan problem solution is considered without taking into account this rate (“approximation of slow movements of the boundaries of the phase transition”, model 2). The model is tuned to a specific object of research. Model parameter values can vary significantly in different geographic regions. This paper simulates the dynamics of permafrost in the area of Lake Tulik (Alaska) in summer. Test calculations based on the proposed simplified model show its adequacy and consistency with field measurements. The developed model can be used for qualitative studies of the long-term dynamics of permafrost using data of the air temperature, relative air humidity and precipitation

A SOLUTION TO STEFAN PROBLEM USING EULERIAN TWO FLUID VOF MODEL

A novel approach for the solution of Stefan problem within the framework of the multi fluid model supplemented with Volume of Fluid (VOF) method, i.e. two-fluid VOF, is presented in this paper. The governing equation set is comprised of mass, momentum and energy conservation equations, written on a per phase basis and supplemented with closure models via the source terms. In our method, the heat and mass transfer is calculated from the heat transfer coefficient, which has a fictitious function and depends on the local cell size and the thermal conductivity, and the implementation is straightforward because of the usage of the local value instead of a global parameter. The interface sharpness is ensured by the application of the geometrical reconstruction scheme implemented in VOF. The model is verified for three types of computational meshes including triangular cells, and good agreement was obtained for the interface position and the temperature field. Although the developed method was validated only for Stefan problem, the application of the method to engineering problems is considered to be straightforward since it is implemented to a commercial CFD code only using a local value; especially in the field of naval hydrodynamics wherein the reduction of ship resistance using boiling flow can be computed efficiently since the method handles phase change processes using low resolution meshes.

On the Stefan Problem With Nonlinear Thermal Conductivity

The solution is obtained and validated by an existence and uniqueness theorem for the following nonlinear boundary value problem \[ \frac{d}{dx}(1+\delta y+\gamma y^{2})^{n}\frac{dy}{dx}]+2x\frac{dy}{dx}=0,\,\,\,x>0,\,\,y(0)=0,\,\,\,y(\infty)=1, \] which was proposed in 1974 by [1] to represent a Stefan problem with a nonlinear temperature-dependent thermal conductivity on the semi-infinite line (0;1). The modified error function of two parameters $\varphi_{\delta,\gamma}$ is introduced to represent the solution of the problem above, and some properties of the function are established. This generalizes the results obtained in [3, 4].