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| 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 (matter)|phase]] boundary can move with time. The '''classical Stefan problem''' aims to describe the temperature distribution in a [[Homogeneity (physics)|homogeneous medium]] undergoing a [[phase transition|phase change]], for example [[ice]] passing to [[water]]: this is accomplished by solving the [[heat equation]] imposing the [[initial value problem|initial temperature distribution]] on the whole medium, and a particular [[boundary value problem|boundary condition]], the [[Stefan condition]], on the evolving boundary between its two phases. Note that this evolving boundary is an unknown [[hypersurface|(hyper-)surface]]: hence, Stefan problems are examples of [[free boundary problem]]s.
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| == Historical note ==
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| The problem is named after [[Joseph Stefan|Jožef Stefan]], the [[Slovenes|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 [[Gabriel Lamé|Lamé]] and [[Benoît Paul Émile Clapeyron|Clapeyron]].
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| == Premises to the mathematical description ==
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| 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 (chemistry)|interface]]s) are infinitesimally thin [[surface]]s that separate adjacent phases; therefore, the solutions of the underlying PDE and its derivatives may suffer discontinuities across interfaces.
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| The underlying PDE is not valid at phase change interfaces; therefore, an additional condition—the '''Stefan condition'''—is needed to obtain [[Well-posed problem|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. | |
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| == Mathematical formulation ==
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| === The one-dimensional one-phase Stefan problem ===
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| Consider an semi-infinite one-dimensional block of ice initially at melting temperature <math>u</math>≡<math>0</math> for <math>x</math> ∈ [0,+∞). Heat flux of <math>f(t)</math> is introduced at the left boundary of the domain causing the block to melt down leaving an interval <math>[0,s(t)]</math> occupied by water. The melted depth of the ice block, denoted by <math>s(t)</math>, is an unknown function of time; the solution of the Stefan problem consists of finding <math>u</math> and <math>s</math> such that
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| : <math>\begin{align}
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| \frac{\partial u}{\partial t} &= \frac{\partial^2 u}{\partial x^2} &&\text{in } \{(x,t): 0 < x < s(t), t>0\}, && \text{the heat equation},\\
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| -\frac{\partial u}{\partial x}(0, t) &= f(t), && t>0, &&\text{the Neumann condition at the left end of the domain describing the inlet heat flux}, &&\\
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| u\big(s(t),t\big) &= 0, && t>0, &&\text{the Dirichlet condition at the water-ice interface: setting melting/freezing temperature},\\
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| \frac{\mathrm{d}s}{\mathrm{d}t} &= -\frac{\partial u}{\partial x}\big(s(t), t\big), && t>0, &&\text{Stefan condition},\\
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| u(x,0) &= 0, && x\geq 0, &&\text{initial temperature distribution},\\
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| s(0) &= 0, && &&\text{initial depth of the melted ice block}.
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| \end{align}
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| </math>
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| The Stefan problem also has a rich inverse theory, where one is given the curve <math>s</math> and the problem is to find <math>u</math> or <math>f</math>.
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| ==Applications==
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| Stefan problems are also used as models for the asymptotic behavior with respect to time of more complex problems: for example, Pego<ref name="Pego">R. L. Pego. (1989). Front Migration in the Nonlinear Cahn-Hilliard Equation. ''Proc. R. Soc. Lond. A.'','''422''':261–278.</ref> uses matched asymptotic expansions to prove that Cahn-Hilliard solutions for phase separation problems behave as solutions to a nonlinear Stefan problem at an intermediate time scale. Also the solution of the [[Cahn–Hilliard equation]] for a binary mixture is reasonably comparable with the solution of a Stefan problem.<ref>F. J. Vermolen, M.G. Gharasoo, P. L. J. Zitha, J. Bruining. (2009). Numerical Solutions of Some Diffuse Interface Problems: The Cahn-Hilliard Equation and the Model of Thomas and Windle.'' Int. J. Mult. Comp. Eng.'','''7(6)''':523–543.</ref> In this comparison, the Stefan problem was solved using a front-tracking, moving-mesh method with homogeneous [[Neumann boundary condition]]s at the outer boundary.
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| == See also ==
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| *[[Free boundary problem]]
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| *[[Moving boundary problem]]
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| *[[Olga Arsenievna Oleinik]]
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| *[[Shoshana Kamin]]
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| *[[Stefan's equation]]
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| == Historical references ==
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| *{{Citation
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| | last = Vuik
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| | first = C.
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| | author-link = Kees Vuik
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| | title = Some historical notes about the Stefan problem
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| | journal = [[Nieuw Archief voor Wiskunde]]
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| | series = 4e serie
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| | volume = 11
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| | issue = 2
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| | pages = 157–167
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| | year = 1993
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| | url =
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| | mr = 1239620
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| | zbl = 0801.35002
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| }}. An interesting historical paper on the early days of the theory: a [[preprint]] version (in [[PDF]] format) is available here [http://ta.twi.tudelft.nl/nw/users/vuik/wi1605/opgave1/stefan.pdf].
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| ==References== | |
| {{reflist}}
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| *{{Citation
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| | last = Cannon
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| | first = John Rozier
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| | author-link = John Rozier Cannon
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| | title = The One-Dimensional Heat Equation
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| | place = [[Reading, Massachusetts|Reading]]–[[Menlo Park, California|Menlo Park]]–[[London]]–[[Don Mills]]–[[Sydney]]–[[Tokyo]]/ [[Cambridge]]–[[New York]]–[[New Rochelle]]–[[Melbourne]]–[[Sydney]]
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| | publisher = [[Addison–Wesley|Addison-Wesley Publishing Company]]/[[Cambridge University Press]]
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| | year = 1984
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| | series = Encyclopedia of Mathematics and Its Applications
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| | volume = 23
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| | edition = 1st
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| | pages = XXV+483
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| | url = http://books.google.com/?id=XWSnBZxbz2oC&printsec=frontcover#v=onepage&q&f=true
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| | id =
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| | mr = 0747979
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| | zbl = 0567.35001
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| | isbn =978-0-521-30243-2 }}. Contains an extensive bibliography, 460 items of which deal with the Stefan and other [[free boundary problem]]s, updated to 1982.
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| *{{Citation
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| | last = Kirsch
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| | first = Andreas
| |
| | author-link =
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| | title = Introduction to the Mathematical Theory of Inverse Problems
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| | place = Berlin–Heidelberg–New York
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| | publisher = [[Springer Verlag]]
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| | year = 1996
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| | series= Applied Mathematical Sciences series
| |
| | volume = 120
| |
| | pages = x+282
| |
| | url = http://books.google.com/books?id=llNUaSKHj3gC&printsec=frontcover#v=onepage&q&f=true
| |
| | doi =
| |
| | id =
| |
| | mr = 1479408
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| | zbl = 0865.35004
| |
| | isbn = 0-387-94530-X}}
| |
| *{{Citation
| |
| | last = Meirmanov
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| | first = Anvarbek M.
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| | author-link = Anvarbek Meirmanov
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| | title = The Stefan Problem
| |
| | place = Berlin – New York
| |
| | publisher = [[Walter de Gruyter]]
| |
| | year = 1992
| |
| | series = De Gruyter Expositions in Mathematics
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| | volume = 3
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| | pages = x+245
| |
| | url = http://books.google.com/?id=ae1VlQjOtJQC&printsec=frontcover#v=onepage&q=
| |
| | id =
| |
| | mr = 1154310
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| | zbl = 0751.35052
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| | isbn = 3-11-011479-8}}.
| |
| *{{Citation
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| | last = Oleinik
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| | first = O. A.
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| | author-link = Olga Arsenievna Oleinik
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| | title = A method of solution of the general Stefan problem
| |
| | journal = [[Proceedings of the USSR Academy of Sciences|Doklady Akademii Nauk SSSR]]
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| | language = Russian
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| | volume = 135
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| | pages = 1050–1057
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| | year = 1960
| |
| | id =
| |
| | mr = 0125341
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| | zbl = 0131.09202
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| }}. The paper containing [[Olga Oleinik]]'s proof of the existence and uniqueness of a [[generalized solution]] for the [[Dimension|three-dimensional]] Stefan problem, based on previous researches of her pupil [[S.L. Kamenomostskaya]].
| |
| *{{Citation
| |
| | last = Kamenomostskaya
| |
| | first = S. L.
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| | author-link = Shoshana Kamin
| |
| | title = On Stefan Problem
| |
| | journal = Nauchnye Doklady Vysshey Shkoly, Fiziko-Matematicheskie Nauki
| |
| | volume = 1
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| | issue = 1
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| | pages = 60–62
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| | year = 1958
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| | language = Russian
| |
| | url =
| |
| | doi =
| |
| | id =
| |
| | zbl = 0143.13901
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| }}. The earlier account of the research of Shoshana Kamin on the Stefan problem.
| |
| *{{Citation
| |
| | last = Kamenomostskaya
| |
| | first = S. L.
| |
| | author-link = Shoshana Kamin
| |
| | title = On Stefan's problem
| |
| | journal = [[Matematicheskii Sbornik]]
| |
| | language = Russian
| |
| | volume = 53(95)
| |
| | issue = 4
| |
| | pages = 489–514
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| | year = 1961
| |
| | url = http://mi.mathnet.ru/eng/msb/v95/i4/p489
| |
| | id =
| |
| | mr = 0141895
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| | zbl = 0102.09301
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| }}. In this paper the author proves the existence and uniqueness of a [[generalized solution]] for the [[Dimension|three-dimensional]] Stefan problem, later improved by her master [[Olga Oleinik]].
| |
| *{{Citation
| |
| | last = Rubinstein
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| | first = L. I.
| |
| | author-link =
| |
| | title = The Stefan Problem
| |
| | place = [[Providence, R.I.]]
| |
| | publisher = [[American Mathematical Society]]
| |
| | year = 1971
| |
| | series = Translations of Mathematical Monographs
| |
| | volume = 27
| |
| | pages = viii+419
| |
| | url = http://books.google.com/?id=lDnLwUyiGAwC&printsec=frontcover#v=onepage&q=
| |
| | id =
| |
| | mr = 0351348
| |
| | zbl = 0219.35043
| |
| | isbn = 0-8218-1577-6}}. A comprehensive reference updated up to 1962–1963, with a bibliography of 201 items.
| |
| *{{Citation
| |
| | last = Tarzia
| |
| | first = Domingo Alberto
| |
| | author-link =
| |
| | title = A Bibliography on Moving-Free Boundary Problems for the Heat-Diffusion Equation. The Stefan and Related Problems
| |
| | journal = MAT
| |
| | series = Series A: Conferencias, seminarios y trabajos de matemática.
| |
| | volume = 2
| |
| | pages = 1–297
| |
| |date=Julio 2000
| |
| | url = http://web.austral.edu.ar/cienciasEmpresariales-investigacion-mat-A-02.asp
| |
| | issn = 1515-4904
| |
| | mr = 1802028
| |
| | zbl = 0963.35207
| |
| }}. 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 western existing 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 condition
| |
| | id= S/s087590
| |
| | last=Vasil'ev
| |
| | first= F. P.
| |
| | author-link=
| |
| }}
| |
| *{{springer
| |
| | title= Stefan problem
| |
| | id= S/s087600
| |
| | last=Vasil'ev
| |
| | first= F. P.
| |
| | author-link=
| |
| }}
| |
| *{{springer
| |
| | title= Stefan problem, inverse
| |
| | id= S/s087610
| |
| | last=Vasil'ev
| |
| | first= F. P.
| |
| | author-link=
| |
| }}
| |
| | |
| [[Category:Partial differential equations]]
| |
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