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
or
.
Consider a semi-infinite one-dimensional block of ice initially at melting temperature
for