Stefan problem
Encyclopedia
In mathematics
and its applications, particularly to phase transition
s in matter, a Stefan problem (also Stefan task) is a particular kind of boundary value problem
for a partial differential equation
(PDE), adapted to the case in which a phase
boundary can move with time. The classical Stefan problem aims to describe the temperature distribution in a homogeneous medium
undergoing a phase change
, for example ice
passing to water
: this is accomplished by solving the heat equation
imposing the initial temperature distribution
on the whole medium, and a particular boundary condition
, the Stefan condition, on the evolving boundary between its two phases. Note that this evolving boundary is an unknown (hyper-)surface
: hence, Stefan problems are examples of free boundary problem
s.
, the Slovene physicist
who introduced the general class of such problems around 1890, in relation to problems of ice
formation. This question had been considered earlier, in 1831, by Lamé
and Clapeyron
.
s) are infinitesimally thin surface
s that separate adjacent phases; therefore, the coefficients of the underlying PDE and its derivatives may suffer discontinuities across interfaces.
The underlying PDE is not valid at phase change interfaces; therefore, an additional condition—the Stefan condition—is needed to obtain closure
. The Stefan condition expresses the local velocity
of a moving boundary, as a function of quantities evaluated at both sides of the phase boundary, and is usually derived from a physical constraint. In problems of heat transfer
with phase change, for instance, the physical constraint is that of conservation of energy
, and the local velocity of the interface depends on the heat flux
discontinuity at the interface.
≡
for 
∈ [0,+∞[. The ice is heated from the left with heat flux 
. The flux causes the block to melt down leaving an interval 
occupied by water. The melt depth of the ice block, denoted by 
, is an unknown function of time; the solution of the Stefan problem consists of finding 
and 
such that
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and its applications, particularly to phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
s in matter, a Stefan problem (also Stefan task) is a particular kind of boundary value problem
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 restraints, called the boundary conditions...
for a partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
(PDE), adapted to the case in which a phase
Phase (matter)
In the physical sciences, a phase is a region of space , throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, and chemical composition...
boundary can move with time. The classical Stefan problem aims to describe the temperature distribution in a homogeneous medium
Homogeneity (physics)
In general, homogeneity is defined as the quality or state of being homogeneous . For instance, a uniform electric field would be compatible with homogeneity...
undergoing a phase change
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
, for example ice
Ice
Ice is water frozen into the solid state. Usually ice is the phase known as ice Ih, which is the most abundant of the varying solid phases on the Earth's surface. It can appear transparent or opaque bluish-white color, depending on the presence of impurities or air inclusions...
passing to water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
: this is accomplished by solving the heat equation
Heat equation
The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...
imposing the initial temperature distribution
Initial value problem
In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...
on the whole medium, and a particular boundary condition
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 restraints, called the boundary conditions...
, the Stefan condition, on the evolving boundary between its two phases. Note that this evolving boundary is an unknown (hyper-)surface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
: hence, Stefan problems are examples of free boundary problem
Free boundary problem
In mathematics, a free boundary problem is a partial differential equation to be solved for both an unknown function u and an unknown domain Ω. The segment Γ of the boundary of Ω which is not known at the outset of the problem is the free boundary....
s.
Historical note
The problem is named after Jožef StefanJoseph Stefan
Joseph Stefan was a physicist, mathematician, and poet of Slovene mother tongue and Austrian citizenship.- Life and work :...
, the Slovene physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
who introduced the general class of such problems around 1890, in relation to problems of ice
Ice
Ice is water frozen into the solid state. Usually ice is the phase known as ice Ih, which is the most abundant of the varying solid phases on the Earth's surface. It can appear transparent or opaque bluish-white color, depending on the presence of impurities or air inclusions...
formation. This question had been considered earlier, in 1831, by Lamé
Gabriel Lamé
Gabriel Léon Jean Baptiste Lamé was a French mathematician.-Biography:Lamé was born in Tours, in today's département of Indre-et-Loire....
and Clapeyron
Benoit Paul Émile Clapeyron
Benoît Paul Émile Clapeyron was a French engineer and physicist, one of the founders of thermodynamics.-Life:...
.
Premises to the mathematical description
From a mathematical point of view, the phases are merely regions in which the coefficients of the underlying PDE are continuous and differentiable up to the order of the PDE. In physical problems such coefficients represent properties of the medium for each phase. The moving boundaries (or interfaceInterface (chemistry)
An interface is a surface forming a common boundary among two different phases, such as an insoluble solid and a liquid, two immiscible liquids or a liquid and an insoluble gas. The importance of the interface depends on which type of system is being treated: the bigger the quotient area/volume,...
s) are infinitesimally thin surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
s that separate adjacent phases; therefore, the coefficients of the underlying PDE and its derivatives may suffer discontinuities across interfaces.
The underlying PDE is not valid at phase change interfaces; therefore, an additional condition—the Stefan condition—is needed to obtain closure
Well-posed problem
The mathematical term well-posed problem stems from a definition given by Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that# A solution exists# The solution is unique...
. The Stefan condition expresses the local velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
of a moving boundary, as a function of quantities evaluated at both sides of the phase boundary, and is usually derived from a physical constraint. In problems of heat transfer
Heat transfer
Heat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...
with phase change, for instance, the physical constraint is that of conservation of energy
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
, and the local velocity of the interface depends on the heat flux
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
discontinuity at the interface.
The one-dimensional one-phase Stefan problem
Consider an semi-infinite one-dimensional block of ice initially at melting temperature≡ for ∈ [0,+∞[. The ice is heated from the left with heat flux . The flux causes the block to melt down leaving an interval occupied by water. The melt depth of the ice block, denoted by , is an unknown function of time; the solution of the Stefan problem consists of finding and such that-
The Stefan problem also has a rich inverse theory, where one is given the curve
and the problem is to find 
or 
.
Applications
The solution of the Cahn–Hilliard equationCahn–Hilliard equationThe Cahn–Hilliard equation 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 pure in each component...
for a binary mixture demonstrated to coincide well with the solution of a Stefan problem.
See also
- Free boundary problemFree boundary problemIn mathematics, a free boundary problem is a partial differential equation to be solved for both an unknown function u and an unknown domain Ω. The segment Γ of the boundary of Ω which is not known at the outset of the problem is the free boundary....
- Moving boundary problem
- Olga Arsenievna OleinikOlga Arsenievna OleinikOlga Arsenievna Oleinik was a Soviet mathematician who conducted pioneering work on the theory of partial differential equations, the theory of strongly inhomogeneous elastic media, and the mathematical theory of boundary layers. She was a student of Ivan Petrovsky...
- Shoshana KaminShoshana KaminShoshana Kamin , also referred as S.L. Kamenomostskaya , is a Soviet Union-born Israeli mathematician, working on the theory of parabolic partial differential equations and related mathematical physics problems: she gave, at the same time and independently of Olga Oleinik, the first proof of the...
- Stefan's equationStefan's equationIn glaciology and civil engineering, Stefan's equation 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 the number of degree days below freezing...
- Free boundary problem