Stefan Type Research Articles

The pressure drop in a porous material layer during combustion

During the combustion of a porous material layer, a manometer, which is attached to the cold end of the charge, records at the bottom of the layer a pressure reduction, which was discovered more than 20 years ago but which remains essentially unexplained up to the present. It is experimentally shown that this effect is similar to the pressure change in the cavities when a light gas (helium, hydrogen) diffuses from (or to) them under isothermal conditions and that it increases during the combustion mainly due to the accompanying Stefan type flow, and probably also as a result of the thermal diffusion. A pressure drop in the cavities is evidently made possible also by the pressure reduction in the flame which follows from the Hugoniot adiabatic theory.

Thermomechanics and the formulation of the Stefan problem for fully faceted interfaces

This paper develops a thermomechanics of two-phase heat conductors in which the interface between phases is fully faceted. The theory is based on balance of forces, balance of energy, and growth of entropy in conjunction with constitutive equations for the interface; and the chief result is a free-boundary problem of Stefan type in which the classical interface condition u = 0 u = 0 is replaced by a condition relating the integral of u u over each facet to the normal velocity of that facet.

Solution of Nonlinear Diffusion Problems by Linear Approximation Schemes

An approximation of degenerate parabolic problems (of porous medium and Stefan type) is studied. The enthalpy formulation and the variational technique are used. Nonstandard semidiscretization in time is used and Newton-like iterations are applied to solve the corresponding elliptic problems. Some numerical experiments are discussed and compared to other methods and to analytic solutions.

A globally convergent multi-grid algorithm for moving boundary problems of two-phase Stefan type

In this paper we deal with the numerical solution of moving boundary problems of two-phase Stefan type. Based on an implicit discretization in time and the use of continuous, piecewise linear finite elements in the space variables with respect to the weak formulation of the problem, a globally convergent multi-grid algorithm is developed. That algorithm strongly relies on the variational characterization of the fully discretized problem as the unconstrained minimization of a subdifferentiable convex objective functional. Numerical results indicate a significant improvement in efficiency compared with previous multi-grid approaches

Stefan-like problems

This paper presents an overview of the free boundary problems connected to the problem of Stefan type or, more generally, to problems describing a change of phase. After a sketch of the historical development of the research in the area, some specific questions are addressed, such as classical solvability of the problem in several space dimensions, regularization of supercooling, dynamical contact angle.

Induction period for the heating of metal particles by continuous laser radiation

8. A. A. Shishkov, S. D. Panin and B. V. Rumyantsev, Operating Processes in Solid Propellant Rocket Engines: A Reference Book [in Russian], Mashinostroenie, Moscow (1988). R. I. Nigmatulin, Fundamentals of the Mechanics of Heterogeneous Media [in Russian], Nauka, Moscow (1978). A. T. Nikitin and F. B. Yurevich, "A theoretical investigation of transient heating and the ablation of coking polymer materials," in: Heat and Mass Transport [in Russian], Vol. 2, Part 2, Izd. ITMO, Akad. Sci. Belor. SSR, Minsk (1972), p. 295. ii. A. M. Grishin, Mathematical Modeling of Some Transient Aero-Thermochemical Phenomena [in Russian], Izd. Tomsk Univ., Tomsk (1973). 12. O. F. Shlenskii, Thermal Properties of Glass-Reinforced Plastic [in Russian], Khimiya, Moscow (1973). 13. V. L. Strakhov and N. G. Chubakov, "Calculation of transient heating and mass removal on expansible coatings," Inzh.-Eiz. Zh., 55, No. 4, 571 (1988). 14. G. N. Kozhevnikov and A. G. Vodop'yanov, Elementary Oxides of Silicon and Aluminum in electrometallurgy [in Russian], Nauka, Moscow (1977). 15. B. M. Budak, N. L. Gol'dman, and A. B. Uspenskii, "Difference schemes with front straightening for the solution of Stefan type multifront problems," Dokl. Akad. Nauk SSSR, 167, No. 4, 735 (1966). 16. A. A. Samarskii, Theory of Difference Schemes [in Russian], Nauka, Moscow (1983). 17. V. A. Burakov and S. F. Sandu, "Numerical solution of problems of transient heat transfer and the removal of heat-shielding materials during condensed phase precipitation," Proc. All-Union Exhibition of Software for the Numerical Solution of Problems in Thermomechanics, Moscow St. Tech. Univ., Moscow (1990), p. i0. .

Boundary value problems to a certain class of non‐linear diffusion equations

AbstractThe first initial‐boundary value problem and the free boundary of Stefan's type for a certain class of non‐linear diffusion equations which arises in plasma physics are considered. For the first initial‐boundary value problem, the global existence of a classical and its large‐time behaviour are discussed. The free‐boundary problem is studied by the reduction to the first initial‐boundary value problem.

A fixed grid finite element method for nonlinear diffusion problems with moving boundaries

A comprehensive formulation for a class of diffusion problems with non-linear conductivities is derived by unifying and combining the freezing index and Kirchhoff transformation concepts. The transformed equations have appropriate continuity characteristics across the unknown moving boundary. The applicability of the fixed grid algorithm for the total solution domain is, accordingly, demonstrated. Associated finite element formulations and solution procedures for the transformed equations are detailed. In addition, selected numerical results for single and two phase Stefan type problems as well as fluid flow in a prescribed cavity are presented for solution verification and illustration.

Multi‐grid solution of two coupled Stefan equations arising in induction heating of large steel slabs

AbstractInduction heating of large steel slabs can be described by a coupled system of non‐linear evolution equations of Stefan type represnting the temporal and spatial distribution of the induced magnetic field and the generated temperature within the slab. Discretizing these equations implicitly in time and by finite differences in space, at each time step the solution of a system of difference inclusions is required. For the solution of that system two multi‐grid algorithms are given which, combined with a nested iteration type continuation strategy to proceed in time, result in computationally highly efficient schemes for the numerical simulation of the induction heating process.

Regularity results for a one-dimensional two-phase degenerate Stefan problem

(l.lh) 40) = 30, (l.li) where a,(u) > 0 and ai = 0 for i = 1,2. In the classical Stefan problem, the specific heat E(U) is required to be strictly positive, where a(u) = ml(u) if u is in Q,’ and ~~~(11) if u is in Q;. However, in more general free boundary problems various types of degeneracy in a can occur. In the electro-chemical machining process, the “specific heat” is zero in both phases resulting in elliptic differential equations, but with a Stefan type boundary condition. See [3] for a description of this model. In the study of saturated/unsaturated flow in

Existence and convergence theorems for doubly nonlinear partial differential equations of elliptic-parabolic type

Here T> 0, V,U = (&/ax,, au/ax,, . . . . au/ax,), and B, is a map from RN x (0, T) x R x RN into R for each i = 1, . . . . N, satisfying certain conditions to be specified at a later time. A precise formulation of (1.1) is given in Section 3. For given u0 and f we study the existence of weak solutions of (1.1). Although doubly nonlinear equations of the type (1.1) are of mathematical interest in their own right, they arise as a model for a variety of diffusion problems. In particular, they comprise in a unifying scheme, free-boundary problems of different natures. An example situation is nonstationary saturated-unsaturated flow of an incompressible fluid through a porous medium in the case of time-dependent water levels. A weak formulation of this problem leads to equations of the type (1.1); see [2]. However, a large held of application for our results is the diffusion problem involving a solid-liquid phase change of the Stefan type. An architypical example is heat transfer during solidification in a nonhomogeneous 205

Mathematical modeling of processes of pulsed melting and vaporization of a metal with clearly separated phase boundaries

The authors propose a method of numerical solution of problems of the Stefan type with two moving boundaries. As an example they solve the problem of pulsed melting and vaporization of an aluminum rod of finite length.

Numerical solution for film evaporation of a spherical liquid droplet on an isothermal and adiabatic surface

A numerical solution for the problem of film evaporation of a liquid droplet on a horizontal surface is presented. The droplets are small enough to be assumed spherical. Two principal cases are considered: (1) the horizontal surface is maintained at a constant temperature (case I), and (2) the surface is insulated while the ambience is hot (case II). The complete set of equations governing this problem were solved under the following assumptions : (1) evaporation is quasi-steady, (2) no internal liquid circulation, (3) constant properties, and (4) the droplet temperature is spatially uniform but temporally varying. The Lewis number is not assumed to be unity; gas phase viscous effects, Stefan type convection, and gas phase inertia are included in the analysis. The total droplet evaporation time was found to decrease with increasing plate (I) or ambient (II) temperature as expected, and the droplet progressively moves away from the plate as it evaporates. The numerical results agree with the analytical solution for film evaporation of a droplet above an adiabatic surface in a hot ambience in the limit of large effective Reynolds number (i.e. potential flow).

Generalized integral transform method for the solution of the heat-conduction equation in a region with moving boundaries

We develop a method of integral transforms for the solution of the heat-conduction equation in a region with moving boundaries. The method leads to analytical solutions of thermal problems of both classical and Stefan type in new functional forms.

A hyperbolic Stefan problem

A free-boundary problem of Stefan type is presented under constitutive assumptions on flux and energy which contain an effective time delay. This contains the hyperbolic telegraphers equation and, hence, has the feature that propagation speed of disturbances is bounded. With the appropriate physically consistent condition on the interface this is shown to lead to a well-posed weak formulation of the problem.

The Structure and Stability of Propagating Redox Fronts

This paper analyzes certain models for the deposition of minerals. The models describe the transport, by flow and diffusion, of an aqueous oxidant in an aquifer where it reacts with an immobile reductant to produce certain deposits. We prove the existence and the uniqueness of travelling wave solutions of the equations. In certain limits, the system can be modelled by a problem of Stefan type with planar fronts. We show these solutions are marginally stable and are linearly stable to transverse perturbations.

An evolutionary continuous casting problem of Stefan type

A simplified model for the solidification of an ingot being cast continuously by withdrawing it from a mould at constant rate is studied via the variational inequalities approach. Existence, uniqueness and regularity results are given for this one-phase Stefan type problem, together with a detailed analysis for the free boundary (the solid-liquid interface) particularly with respect to the asymptotic behavior as time t ↑ ∞ t \uparrow \infty where the steady state is attained.

A model for intumescent paints

A model is constructed for intumescent paints in which it is assumed that the transition to the swollen state occurs at a very thin zone or front. Across this front there is a discontinuity in the density, the velocity, and the temperature gradient. The temperature is continuous and fixed at the front at a value that is a property of the paint. All of the mass loss through outgassing occurs at the front, and this gives rise to a large increase in the volume of the paint particles as they are traversed by the front. On each side of the front, the physical processes consist exclusively of heat conduction. In the context of this model, a finite coating subject to a heat flux contains two moving boundaries: the front, and the outer surface of the coating at which the flux is applied. Through an elementary mathematical transformation, this problem is reduced to one in which the outer surface is fixed and only the inner front moves. This simplified problem is of Stefan type, easily amenable to numerical analysis. The time-dependent substrate temperature, when calculated in this way, tends to level off at a constant value until the front has completely traversed the coating; then it sharply increases again. This is a characteristic of much of the available experimental data and, combined with visual evidence, lends credence to our model.

Thermal model for lithospheric thinning and associated uplift in the neotectonic phase of intraplate orogenic activity and continental rifts

Many regions of recent tectonic activity are characterized by the existence of mantle inhomogeneties which are revealed by various geophysical methods. It is found that the lithosphere beneath the continental rifts and many recent uplifts (high plateaus) is abnormally thin. This thinning of the lithosphere is connected with the process of intensive convective heating. A study of the thermal evolution of the lithosphere, when the convective heat flow is supplied to its base, is performed in the framework of a Stefan's type problem. Solutions are derived giving the relationship between the evolution of the thickness of the lithosphere and the value of the heat flow at the base of the lithosphere at the neotectonic reactivation stage. The results of computations show that the thickness of the lithosphere can be reduced to one half of its initial value in a time interval of an order of several million years when the value of convective heat flow from the anomalous mantle is of an order of 40–100 mW/m 2. Important data on the evolution of the lithosphere can be obtained from the comparative analysis of the composition of volcanic rocks at different stages of continental rifting. It has been found that the source of magmas has a general tendency to migrate to a shallower depth with time. This process suggests successive thinning of the lithosphere. Based on this deduction, we can use these data to evaluate the time dependence of heat flow from the anomalous mantle. The study of this inverse problem leads to the conclusion that the beginning of rifting should be accompanied by a sudden (like a step-function) increase in heat flow supplied by a mantle plume to the base of the lithosphere. The required values of additional heat flow are in the range 55–102 mW/m 2. Vertical crustal movements are caused by a number of physical processes (thermal expansion of the lithosphere, isostatic forces associated with a density inversion at the base of the lithosphere, tension produced by the flow of heated material and finally the regional tectonic stress field). The role of these factors and their interactions in tectonic processes are discussed.

Continuation of the solution of a free boundary problem in cylindrical symmetry

We perform an a priori analysis of the behavior of the solution to a Stefan type free boundary problem in cylindrical symmetry, in arbitrarily large time intervals; a nonlinear flux condition is prescribed on the fixed boundary, for t>0.