Stefan Problem
   HOME

TheInfoList



OR:

In mathematics and its applications, particularly to
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...
s in matter, a Stefan problem is a particular kind of
boundary value problem In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to ...
for a system of partial differential equations (PDE), in which the boundary between the phases can move with time. The classical Stefan problem aims to describe the evolution of the boundary between two phases of a material undergoing a phase change, for example the melting of a solid, such as
ice Ice is water frozen into a solid state, typically forming at or below temperatures of 0 degrees Celsius or Depending on the presence of impurities such as particles of soil or bubbles of air, it can appear transparent or a more or less opaq ...
to
water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
. This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At the interface between the phases (in the classical problem) the temperature is set to the phase change temperature. To close the mathematical system a further equation, the Stefan condition, is required. This is an energy balance which defines the position of the moving interface. Note that this evolving boundary is an unknown (hyper-)surface; hence, Stefan problems are examples of free boundary problems. Analogous problems occur, for example, in the study of porous media flow, mathematical finance and crystal growth from monomer solutions.


Historical note

The problem is named after Josef Stefan (Jožef Stefan), the Slovenian
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate ca ...
who introduced the general class of such problems around 1890 in a series of four papers concerning the freezing of the ground and the formation of sea
ice Ice is water frozen into a solid state, typically forming at or below temperatures of 0 degrees Celsius or Depending on the presence of impurities such as particles of soil or bubbles of air, it can appear transparent or a more or less opaq ...
. However, some 60 years earlier, in 1831, an equivalent problem, concerning the formation of the Earth's crust, had been studied by Lamé and Clapeyron. Stefan's problem admits a similarity solution, this is often termed the
Neumann Neumann is German language, German and Yiddish language, Yiddish for "new man", and one of the List of the most common surnames in Europe#Germany, 20 most common German surnames. People * Von Neumann family, a Jewish Hungarian noble family A ...
solution, which was allegedly presented in a series of lectures in the early 1860s. A comprehensive description of the history of Stefan problems may be found in Rubinstein.


Premises to the mathematical description

From a mathematical point of view, the phases are merely regions in which the solutions of the underlying PDE are continuous and differentiable up to the order of the PDE. In physical problems such solutions represent properties of the medium for each phase. The moving boundaries (or
interface Interface or interfacing may refer to: Academic journals * ''Interface'' (journal), by the Electrochemical Society * '' Interface, Journal of Applied Linguistics'', now merged with ''ITL International Journal of Applied Linguistics'' * '' Int ...
s) are infinitesimally thin
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
s that separate adjacent phases; therefore, the solutions of the underlying PDE and its derivatives may suffer discontinuities across interfaces. The underlying PDEs are not valid at the phase change interfaces; therefore, an additional condition—the Stefan condition—is needed to obtain closure. The Stefan condition expresses the local
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
of a moving boundary, as a function of quantities evaluated at either side of the phase boundary, and is usually derived from a physical constraint. In problems of
heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction ...
with phase change, for instance,
conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means tha ...
dictates that the discontinuity of
heat flux Heat flux or thermal flux, sometimes also referred to as ''heat flux density'', heat-flow density or ''heat flow rate intensity'' is a flow of energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity ...
at the boundary must be accounted for by the rate of
latent heat Latent heat (also known as latent energy or heat of transformation) is energy released or absorbed, by a body or a thermodynamic system, during a constant-temperature process — usually a first-order phase transition. Latent heat can be underst ...
release (which is proportional to the local velocity of the interface). The regularity of the equation has been studied mainly by Luis Caffarelli and further refined by work of Alessio Figalli, Xavier Ros-Oton and Joaquim Serra


Mathematical formulation


The one-dimensional one-phase Stefan problem

The one-phase Stefan problem is based on an assumption that one of the material phases may be neglected. Typically this is achieved by assuming that a phase is at the phase change temperature and hence any variation from this leads to a change of phase. This is a mathematically convenient approximation, which simplifies analysis whilst still demonstrating the essential ideas behind the process. A further standard simplification is to work in non-dimensional format, such that the temperature at the interface may be set to zero and far-field values to +1 or -1. Consider a semi-infinite one-dimensional block of ice initially at melting temperature u=0 for x \in ;+\infty). The most well-known form of Stefan problem involves melting via an imposed constant temperature at the left hand boundary, leaving a region [0;s(t)/math> occupied by water. The melted depth, denoted by s(t), is an unknown function of time. The Stefan problem is defined by :* The heat equation: \frac = \frac, \quad \forall (x,t) \in [0;s(t)] \times [0;+\infty] :* A fixed temperature, above the melt temperature, on the left boundary: u(0,t) = 1, \quad \forall t > 0 :* The interface at the melting temperature is set to u \left(s(t),t \right) = 0 :* The Stefan condition: \beta \frac s(t) = -\frac u \left(s(t), t \right) where \beta is the Stefan number, the ratio of latent to ''specific''
sensible heat Sensible heat is heat exchanged by a body or thermodynamic system in which the exchange of heat changes the temperature of the body or system, and some macroscopic variables of the body or system, but leaves unchanged certain other macroscopic vari ...
(where specific indicates it is divided by the mass). Note this definition follows naturally from the nondimensionalisation and is used in many texts however it may also be defined as the inverse of
this This may refer to: * ''This'', the singular proximal demonstrative pronoun Places * This, or ''Thinis'', an ancient city in Upper Egypt * This, Ardennes, a commune in France People with the surname * Hervé This, French culinary chemist Art ...
. :* The initial temperature distribution: u(x,0) = 0, \; \forall x \geq 0 :* The initial depth of the melted ice block: s(0) = 0 :The Neumann solution, obtained by using self-similar variables, indicates that the position of the boundary is given by s(t) = 2 \lambda \sqrt where \lambda satisfies the transcendental equation \beta \lambda = \frac\frac. The temperature in the liquid is then given by T=1-\frac.


Applications

Apart from modelling melting of solids, Stefan problem is also used as a model for the asymptotic behaviour (in time) of more complex problems. For example, PegoR. L. Pego. (1989). Front Migration in the Nonlinear Cahn-Hilliard Equation. ''Proc. R. Soc. Lond. A.'',422:261–278. uses matched asymptotic expansions to prove that Cahn-Hilliard solutions for phase separation problems behave as solutions to a non-linear Stefan problem at an intermediate time scale. Additionally, the solution of the
Cahn–Hilliard equation The Cahn–Hilliard equation (after John W. Cahn and John E. Hilliard) is an equation of mathematical physics which describes the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains p ...
for a binary mixture is reasonably comparable with the solution of a Stefan problem. In this comparison, the Stefan problem was solved using a front-tracking, moving-mesh method with homogeneous
Neumann boundary condition In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann. When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative appli ...
s at the outer boundary. Also, Stefan problems can be applied to describe phase transformations other than solid-fluid or fluid-fluid. The Stefan problem also has a rich inverse theory; in such problems, the meting depth (or
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
or hyper-surface) is the known datum and the problem is to find or .


Advanced forms of Stefan problem

The classical Stefan problem deals with stationary materials with constant thermophysical properties (usually irrespective of phase), a constant phase change temperature and, in the example above, an instantaneous switch from the initial temperature to a distinct value at the boundary. In practice thermal properties may vary and specifically always do when the phase changes. The jump in density at phase change induces a fluid motion: the resultant kinetic energy does not figure in the standard energy balance. With an instantaneous temperature switch the initial fluid velocity is infinite, resulting in an initial infinite kinetic energy. In fact the liquid layer is often in motion, thus requiring
advection In the field of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is a ...
or
convection Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the c ...
terms in the heat equation. The melt temperature may vary with size, curvature or speed of the interface. It is impossible to instantaneously switch temperatures and then difficult to maintain an exact fixed boundary temperature. Further, at the nanoscale the temperature may not even follow Fourier's law. A number of these issues have been tackled in recent years for a variety of physical applications. In the solidification of supercooled melts an analysis where the phase change temperature depends on the interface velocity may be found in Font ''et al''. Nanoscale solidification, with variable phase change temperature and energy/density effects are modelled in. Solidification with flow in a channel has been studied, in the context of lava and microchannels, or with a free surface in the context of water freezing over an ice layer. A general model including different properties in each phase, variable phase change temperature and heat equations based on either Fourier's law or the Guyer-Krumhansl equation is analysed in.


See also

* Free boundary problem *
Olga Oleinik Olga Arsenievna Oleinik (also as ''Oleĭnik'') HFRSE (russian: link=no, О́льга Арсе́ньевна Оле́йник) (2 July 1925 – 13 October 2001) was a Soviet mathematician who conducted pioneering work on the theory of partial di ...
* Shoshana Kamin *
Stefan's equation In glaciology and civil engineering, Stefan's equation (or Stefan's formula) describes the dependence of ice-cover thickness on the temperature history. It says in particular that the expected ice accretion is proportional to the square root of th ...


Notes


References


Historical references

*. An interesting historical paper on the early days of the theory; a
preprint In academic publishing, a preprint is a version of a scholarly or scientific paper that precedes formal peer review and publication in a peer-reviewed scholarly or scientific journal. The preprint may be available, often as a non-typeset version ...
version (in
PDF Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ...
format) is available her


Scientific and general references

*. Contains an extensive bibliography, 460 items of which deal with the Stefan and other free boundary problems, updated to 1982. * *. An important monograph from one of the leading contributors to the field, describing his proof of the existence of a
classical solution Classical may refer to: European antiquity *Classical antiquity, a period of history from roughly the 7th or 8th century B.C.E. to the 5th century C.E. centered on the Mediterranean Sea *Classical architecture, architecture derived from Greek and ...
to the multidimensional Stefan problem and surveying its historical development. *. The paper containing Olga Oleinik's proof of the existence and uniqueness of a generalized solution for the
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
Stefan problem, based on previous researches of her pupil S.L. Kamenomostskaya. *. The earlier account of the research of the author on the Stefan problem. *. In this paper the author proves the existence and uniqueness of a generalized solution for the
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
Stefan problem, later improved by her master Olga Oleinik. * *. A comprehensive reference, written by one of the leading contributors to the theory, updated up to 1962–1963 and containing a bibliography of 201 items. *. The impressive personal bibliography of the author on moving and free boundary problems (M–FBP) for the heat-diffusion equation (H–DE), containing about 5900 references to works appeared on approximately 884 different kinds of publications. Its declared objective is trying to give a comprehensive account of the existing western mathematical–physical–engineering literature on this research field. Almost all the material on the subject, published after the historical and first paper of Lamé–Clapeyron (1831), has been collected. Sources include scientific journals, symposium or conference proceedings, technical reports and books.


External links

* * *{{springer , title= Stefan problem, inverse , id= S/s087610 , last=Vasil'ev , first= F. P. , author-link= Partial differential equations Boundary value problems