STEFAN PROBLEMS - Maths.anu.edu.au

112STEFAN PROBLEMSJohn van der Hoek§O.IntroductionIn 1899 STEFAN [1] posed the following problem:material occupies thephase occupiescooccupies 0 x 00 spaceco x 00A heat conductingInitially (t 0), the liquid x 0 at temperature TI 0at temperature T2 O.and the solid phaseIt is required to determinetemperatures u1ex,t) and u 2 Cx,t) at position x and time t of theand solid phases, respectively, and the position x s (t) of the freeor moving boundary between the phases as a function of time (t).Stefanshowed that the following thermal balance operates between the twophaseswhere A latent heat, p density of the material in its originalphase state, Kj and K2 are coefficients of conductivity correspondingto the liquid and solid phases, respectively.Condition (1) will bereferred to as the Stefan free boundary condition.This relationshipholds, for example, at the interface between water and ice in theprocess of melting ice (see RUBINSTEIN [2]).The full formulation ofStefan's problem would also include in addition to (1),ddU I(2)ax(Kl ax ), set) x (3)ax(K2 dX ),ddU 2-coco,t 0 x set), t 0

113Tl 0,(4)(5)u 2 (x,0) (6)u1(s(t),t)T2 0,00 x 00 x 00 u 2 (s(t),t) 0,t 0where Cl and C2 are specific heats of the respective media which willin general depend on the spacial variable x. Stefan problems are freeboundary problems where the free boundary is characterised (in part)by the Stefan free boundary condi tion. Such free boundary problemsare also referred to as moving boundary problems since the positionof the free boundary is a function of time. For other types of movingboundary problems, the reader is referred to WILSON et al. [3], WILSONet al. [4], RUBINSTEIN [2] and FURZE LAND [5]. The Stefan free boundarycondition can also be expressed in space dimension n 2. (seeFRIEDMAN [6]).§l.One Phase Stefan ProblemWhen the temperature in the solid phase is maintained at the meltingtemperature, the two phase problem of §O becomes a one phase Stefanproblem. Under some simplifying assumptions and under scaling ofindependent variables, the problem in one space dimension becomes oneof finding functions u u(x,t) and s set) satisfying,(7)auat(8)u(x,O)where, ba 2u ax2'0 hex), s(O),h(b)(9)u(O,t) f(t) ,(10)u(s(t),t)(11)au- ax (s(t)-,t) 0,x set),0 t T0 x b 0,hex) 0 if x*b0 t T, f 00 t Tds dt(t), 0 t T .A recent survey of results concerning problem (7)-(11) is given inFRIEDMAN [7] for one and more space dimensional problems. The onephase Stefan problem yields several formulations, for example the

114weak formulation due to FRIEDMAN [6] and the formulation as avariational inequality due to DUVAUTFor the one dimensional onephase Stefan problem" many results have been obtained on existence andregulari ty of the solution u and on the natuTe al1d regulari ty of thefree bOlmdary x " s(t),For example, FRIEDMAN [9] has shown that iff(t) in (9) is real analytic in t fOT 0 t T then so is set) as afunction of ,For space dimension n ;;;. 2 very Ii ttle is known,Themost recent resul t is due to CAFFARELLI and FRIEDMAN [10] who provethat the temperature in the one phase n-dimensional Stefan problem iscontinuous.§3.Two and Mul tiphase Stefan ProblemsA recent survey of results for the general multiphase Stefanis CANNON [ll],In 1968, FRIEDMAl\l [6], [12] developed a theory ofweak solutions of these Stefan problems,As for one phase Stefanproblems, many results are known in the case of one dimensional spaceva.riable but little known for space dimension n ;;;. 2.For the onedimensional problem we refer the reader to CANNON, HENRY and KOTLOW[13] for existence of classical solutions and to SCHAEFFER [14] a.ndFRIEDMAN [9] for regularity of the free boundary.No results areknown for the regularity of temperatures or for the free boundary forspace dimension n ;;;. 20In 1975, DUVAUTgave a formulation of thetwo phase Stefan problem as an evolution variational inequality.Ithas been used by some authors (see for example KIKUCHI and ISHIKAWA[16]) to provide numerical results,§4.Some Related CommentsOne of the major differences between Stefan problems in one spacedimension and for space dimension n ;;;. 2 is the fact that the formerallows analysis via integral equation methods (see CANNON et aL [13])whereas the latter does not.As FRIEDMAN [6] points out thecomplications in space dimension n ;;;. 2 are of a physical nature."Thus,even if the data are very smooth the solution need not be smooth ingeneral.For example, when a body of ice having the shape

usACDkeeps growing, the interfaces AB and CD may eventually coincide.in the next movement the whole joint boundary will disappear.free boundary variesThus thein a discontinuous manner."The Stefan problems have been studied withconditions.Then,various types of boundaryThe most usual formulations have used temperature andheat flux specification on the boundary of interest.FASANO andPRIMICERIO [17] have studied in one dimension, boundary conditions whichinvolve a non linear relationship between the heat flux and thetemperature on the boundary.CANNON and VAN DER HOEK [18]-[20] havestudied the one and two phase Stefan problem in one space dimensionin the case where the boundary condition (for example (9)) is replacedby a specification of energy or heat content in one of the phases.Inthe one phase problem this amounted to specifying the quantitys (t)fu(x, t) dxoas a function of t for 0 t T.§4.[1]ReferencesSTEFAN, J. (1899), "tiber einige Prob1eme der Theorie derWarmeleitung", S.-B. Wien. Akad. Mat. Natur.[2]2!,173-484.RUBINSTEIN, L.I. (1971), "The Stefan Problem", Translations ofMath. Monog. Vol. 27, American Math. Soc., Providence R.I., U.S.A.

116[3]WILSON, D.G. and SOLOMON, A.D. (eds.) (1978), "Moving BoundaryProblems", Academic Press, Inc. New York.[4]WILSON, D.C., SOLOMON, A.D. and TRENT, J .S. (1979), "ABibliography of Moving Boundary Problems with Key Word Index",National Technical Information Service, U.S. Dept. of Commerce,5285 Port Royal Road, Springfield, VA 22161, U.S.A.[5]FURZE LAND , R.M. (1977), "A Survey of the Formulation andSolution of Free and Moving Boundary (Stefan) Problems",Tech. Rep. No. 76, Dept. of Mathematics, Brunel University,Uxbridge, Middlesex, U.K.[6]FRIEDMAN, A. (1968), "The Stefan Problem in Several SpaceVariables", Trans. American Math. Soc. 133, 51-87.[7]FRIEDMAN, A. (1978), "One Phase Moving Boundary Problems", in[3], 25-40.[8]DUVAUT, G. (1973), ''R &solution d'un probleme de Stefan (Fusiond'un bloc de glaceazero degre)" , C.R. Acad. Sc. Paris, 376,1461-1463.[9]FRIEDMAN, A.(1976), "Analyticity of the fl'ee boundary for theStefan problem" , Archive Rat. Mech. Anal., 61, 97-125.[10]CAFFARELLI, L.A. ane! FRIEDMAN, A. (1979), "Continuity of theTemperature in the Stefall P:coblem", Indiana Univ. Math. J. ,53-70.[11]CANNON,.I .R. (1978), "Multiphase Parabolic Free Boundary ValueProblems", in [3], 3-24.FRIEDMAN, A. (1968), "One dimensional Stefan problems with nonmonotone free boundary", Trans. American Math. Soc. 133, 89-114.CANNON, J.R., HENRY, D.B. and KOTLOW, D.B. (1976), "Classicalsolutions of the one-dimensional two-phase Stefan problem",Annali di Math. pura ed appl. (lV) eVIl, 311-341.[14]SCHAEFFER, D.G. (1976), "A ne\'l proof of the infinitedifferentiability of the free boundary in the Stefan problem",J. Diff. Eq. 20., 266-269.[15]DUVAUT, G. (1975), "The Solution of a two-phase Stefan Problemby a variational inequality", in J. R. Ockenden and W. R. Hodgkins(edi tors)., "Moving Boundary Value Problems in Heat Flow andDiffusion", Clarendon Press, Oxford, U.K., 173-181.

117[16]KIKUCHI, N. and ISHIKAWA, Y, (1979), "Numerical methods for atwo-phase Stefan problem by variational inequalities", Int. J.Num. Meth. in Engineering[17]li,1221-1239.FASANO, A. and PRIMICERIO, M. (1972), "Su un problemaunidimensionale di diffusione in un mezzo a contorno mobile cancondizioni ai limi ti non lineari", Annali di Math. pura ed appl.CIV) XCIII, 333-357.[18]CANNON,.J .R. and VAN DER HOEK, John (1980), "The one-phaseStefan problem subject to the Specifica'cion of Energy", toappear in J. Math. AnnaL and App!.[19]CANNON, J.R. and VAN DER HOEK, .John (1980), "The Existence ofilnd a Continuous Dependence Resul t for the Solution of the HeatEquation Subject to the Specification of Energy", to appear inSupplemento Boll. U.M. I.[20]CANNON, J .R. and VAN DER HOEK, John (1980), "The ClassicalSolution of the One-dimensional Two-Phase Stefan Problem withEnergy Specification", to appear in Annali di Mat. pura ed appLDepartment of Mathematics,The University of Adelaide,Adelaide,S.A. 5001AUSTRALIA

112 STEFAN PROBLEMS John van der Hoek §O. Introduction In 1899 STEFAN [1] posed the following problem: A heat conducting material occupies the space _co x 00 Initially (t 0), the liquid phase occupies _co x 0 at temperature TI 0 and the solid phase occupies 0 x 00 at temperature T2 O. It is required to determine