10.1090/mmono/027
TRANSLATIONS OF MATHEMATICAL MONOGRAPHS
Volume 27
THE STEFAN PROBLEM
by L I. RUBENSTEIN
AMERICAN MATHEMATICAL SOCIETY
Providence, Rhode Island 02904
1971 nPOBJIEMA CTE$AH A
JI. M . PYBWHU1TEMH
JIaTBMMCKMM rocyAapcTBeHHH M yHMBepcMTe T MMeHw nETPA CTY^IK M BbHMCJIMTEJlbHblM UEHT P
M3AaTejibCTB0 „3Baiir3He " Pural967
Translated fro m the Russian by A. D. Solomon
Library o f Congress Card Number 75-168253 International Standard Book Number 0-8218-1577- 6
Copyright© 197 1 by the American Mathematical Society
Printed in the United States of America All Rights Reserved May not be reproduced in any form without permission of the publishers PREFACE In recen t year s w e hav e witnesse d a n intensiv e developmen t o f th e Stefan problem , wit h whic h th e autho r ha s lon g bee n concerned . This monograp h wa s originall y intende d t o examin e th e importan t results concerned wit h th e classica l Stefa n proble m an d it s generaliza - tions, up to the present time, but thi s progra m coul d no t b e carrie d ou t completely, first of all because the literature concerned with this problem is continuousl y increasing , an d ver y importan t work , i n whic h th e Stefan problem i s treated a s a problem o f numerica l analysis , ha s bee n only partiall y publishe d an d partiall y announce d i n publication s an d at scientifi c conferences; 1* a t th e sam e tim e th e wor k planne d fo r ou r monograph was to a large extent complete . I n thi s regar d th e numerica l solution o f problem s o f Stefa n typ e i s usuall y discusse d i n monograph s in an extremely concis e manner. We will limit ourselve s t o brie f descrip - tions o f finite differenc e algorithm s whic h hav e bee n propose d fo r th e solution o f th e Stefa n proble m (Par t 2 , Chapte r VIII) ; a s a rul e w e will not give a justification o f th e algorith m bu t refe r th e reade r t o th e original literature. The autho r is , o n th e on e hand , mainl y intereste d i n the physical formulatio n o f a problem which reduces to some form of the Stefan problem (Par t 1 , Chapters I—IV) and , on the other hand , i n th e examination of general theoretical problems, i.e. problems of a qualitative character (Par t 2 , Chapter s I—VII) . I n Supplemen t 3 w e presen t illustrative examples o f th e numerica l solutio n t o problem s examine d i n the first part . I n th e first tw o supplement s w e presen t certai n know n results whic h ar e use d i n th e mai n part s o f th e book . At th e en d o f th e boo k w e includ e a bibliograph y o f th e problem . Unfortunately, man y source s wer e no t availabl e t o th e author . I t i s probable tha t man y work s exis t whic h hav e excape d hi s attention . Therefore w e mak e n o clai m o f completenes s fo r th e bibliography . The autho r take s thi s opportunit y t o expres s hi s appreciatio n t o his colleague s N . Avdoijins , M . Antimirov , A . Kuikis , H . German ,
We have in mind the joint work of B. M . Budak, F. P. VasiTev and A. B. Uspenskil, as presented in [15 ] and announced in their articles in a number of reports of conference s on numerica l analysi s (Mosco w Stat e Univ. , 1965) .
in iv PREFACE
E. Enikeeva, M. Zavorina, B. Martuzans, E. Pi^enkova and A. Skroman, who checked the accuracy o f th e numerica l solution s o f th e example s o f Supplement 3 and aide d th e autho r i n eliminatin g man y errors , a s wel l as t o E . Riekstiij s an d V . Abolinja , wh o rea d th e manuscrip t o f th e monograph. Th e autho r expresse s specia l thank s t o th e Recto r o f th e Stucki Institut e o f th e Latvia n Stat e University , Professo r V . A . Steinberg, an d t o th e directo r o f th e Universit y computin g center , Docent E . I . Arin , withou t whos e efficien t ai d th e monograp h coul d not hav e bee n brough t t o completion .
L. RUBINSTEI N TABLE O F CONTENT S
PREFACE ii i
INTRODUCTION 1 1. Historica l survey 1 2. Notatio n and terminolog y 1 5 PART ONE . Physica l Problems Reducing to Problems of the Stefan Typ e CHAPTER I . DIFFUSIO N O F HEA T B Y CONDUCTIO N I N A MEDIU M WITH A CHANGE O F PHASE STAT E 1 8 1. Th e Stefan condition s 1 8 2. Freezin g o f the ground 2 0 3. Crystallizatio n o f a melt when a plate is immersed i n it . . . 2 5 4. Formatio n o f a continuous ingot 3 2 5. Zona l noncrucible melting o f a cylindrical rod 3 7 CHAPTER II . THERMA L DIFFUSIO N PROCESSE S I N A MEDIU M WITH VARIABL E PHAS E STAT E 3 9 1. Dissolutio n o f a gas bubble in a liquid 3 9 2. Dynamic s o f one-dimensiona l nonisotherma l evaporatio n of an ideal liquid mixture 4 3 3. Crystallizatio n o f a binary allo y 5 2 CHAPTER III . PROBLEM S O F THE THEOR Y O F FILTRATION 6 1 1. Forcin g o f a hydrauli c solutio n int o th e groun d (proble m ofVerigin) 6 1 2. Advanc e o f a water-oil interface i n an elastic regime 6 3 3. Invers e problems 7 1 CHAPTER IV . SOM E PROBLEM S O F MECHANIC S O F CONTINUOU S MEDIA THA T REDUCE T O PROBLEMS OF STEFAN TYP E ... . 8 1 1. A proble m o f convectio n arisin g fro m crystallizatio n o f a supercooled melt 8 1 2. Hig h speed flow o f a soli d bod y i n a viscou s incompressibl e fluid 8 5 3. Nonstationar y flows o f a viscous plastic medium 8 9
v VI CONTENTS
PART TWO . Th e Classical Stefan Proble m and its Generalizations
CHAPTER I . TH E SINGLE-PHAS E STEFA N PROBLE M WIT H STRON G NONLINEARITY 9 4 1. Statemen t o f the problem. Formulation o f the basic results. 9 4 2. Reductio n to an integral equation. Theorem o f equivalence. 9 7 3. Existenc e o f the solution in the small 10 4 4. Uniquenes s of the solution 11 4 5. Stabilit y o f the solution 13 0 6. Furthe r observations 13 5 CHAPTER II . TH E CLASSICA L TWO-PHAS E STEFA N PROBLEM . TH E CASE O F A TWO-PHAS E INITIA L STAT E WIT H CONTINUOU S AGREEMENT O F BOUNDARY AN D INITIA L CONDITION S 14 1 1. Statemen t o f the problem 14 1 2. Th e first boundar y proble m o n a segmen t fo r a two-phas e initial stat e an d continuou s agreemen t o f boundar y an d initial conditions. Existence o f the solution in the large . . . 14 3
3. Asymptoti c behavior o f the solution to Problem B x 15 5 4. Degeneratio n o f one of the phases for t > t 0 16 1 5. Th e thir d boundar y problem . Existenc e o f th e solutio n in the large 16 4 6. Th e thir d boundar y problem . Asymptoti c behavio r o f the solution 17 0 7. Th e Cauchy-Stefan proble m 18 1 CHAPTER III . TH E INITIA L VELOCIT Y O F ADVANC E O F TH E INTER - PHASE BOUNDAR Y 190
1. Th e integral equation s o f Problem s B x an d A x withou t th e continuous agreement o f boundary and initial data 190 2. Initia l spee d o f th e interphas e boundary . Formulatio n o f the result 19 5 3. ProofofTheoremll.Thecase/xCO ) ^ 0 19 9 4. ProofofTheoremll.CaseACO ) = 0 20 5 5. Proo f o f Theorem 1 2 21 3
CHAPTER IV . SOLUTIO N O F PROBLEMS A X AND A 3 21 7 1. Th e first boundary problem (A x). Subsidiary function s . . . 21 7 2. Existenc e o f a solution to problem Ai 22 1
3. Th e third boundary problem (A 3) 23 1 CONTENTS vi i
CHAPTER V . TH E DOUBLE-LAYE R STEFA N PROBLE M WIT H DE - GENERATION O F ON E O F TH E PHASE S A T TH E INITIA L MOMENT AN D A DISCONTINUIT Y I N TH E BOUNDAR Y AN D INITIAL CONDITION S 23 8 1. Statemen t o f th e problem . Existenc e an d uniquenes s theorem 23 8 2. Reductio n to an integral equation. Proof o f Lemma 10 . . . . 24 2 3. Proo f o f Lemma 1 1 25 0 4. Existenc e o f the solution 25 4 5. Investigatio n o f the initial melting process 26 4
CHAPTER VI . TH E SPATIA L STEFA N PROBLEM . REDUCTIO N T O ON E DIMENSION. TH E MULTI-PHAS E STATIONAR Y PROBLEM . A FILTRATIO N ANALO G T O THE STEFA N PROBLE M 27 3 1. Th e Stefan proble m with spherical symmetry 27 3 2. Th e Stefan proble m with cylindrical symmetry 27 9 3. Th e multi-phase stationary Stefan problem 28 4 4. Filtratio n analog o f the Stefan proble m 28 7 CHAPTER VII . TH E SINGLE-PHAS E STEFA N PROBLE M FO R A GENERAL ONE-DIMENSIONA L EQUATIO N O F PARABOLI C TYPE. TH E GENERALIZE D SOLUTIO N O F TH E STEFA N PROBLEM 29 5 1. Th e single-phas e Stefa n problem . Kyner' s existenc e an d uniqueness theorem 29 5 2. Th e generalized solution o f the Stefan proble m 31 0 CHAPTER VIII . NUMERICA L METHOD S FO R TH E SOLUTIO N O F TH E STEFAN PROBLE M 32 4 1. Method s base d o n a reductio n t o a countabl e syste m o f ordinary differentia l equation s 32 4 2. Differenc e method s fo r th e solutio n o f a proble m o f Stefa n type 32 7 3. Additiona l remarks on the method o f integral equations. . . 34 2 CHAPTER IX . PROBLEM S 35 1
SUPPLEMENT 1 . TH E MAXIMU M PRINCIPL E AN D TH E VtBORNt - FRIEDMAN THEORE M 35 6 1. Notatio n 35 6 2. Th e weak maximum principl e 35 7 Vlll CONTENTS
3. Th e strong maximum principle o f Nirenberg 36 0 4. Theore m o f Vyborny-Friedman 36 5
SUPPLEMENT 2 . TH E LIMITIN G BOUNDAR Y VALUE S O F TH E DERIV - ATIVE O F A PARABOLI C FUNCTION . GREEN' S FUNCTIO N O F A DOUBLE-LAYE R BOUNDAR Y PROBLE M O N TH E LIN E AN D HALFLINE 36 8 1. Th e limitin g boundar y value s o f th e derivativ e o f a para - bolic function 36 8 2. Green' s functio n o f th e double-laye r boundar y proble m on the line and halfline 37 1 SUPPLEMENT 3 . NUMERICA L ILLUSTRATION S 38 0 1. Exampl e o f the numerica l solutio n o f the integra l equation s of the Stefan problem 38 0 2. Crystallizatio n o f a mel t unde r insertio n o f a plat e o r a sphere 38 7 3. Dynamic s o f growth o f a bubble o f ai r i n water 4. Dynamic s o f evaporatio n o f a solutio n o f norma l sextan e and heptane i n normal pentane 39 8 5. Zona l noncrucible melting o f a cylindrical rod 40 0
BIBLIOGRAPHY 41 0 SUPPLEMENT 1 THE MAXIMUM PRINCIPL E AND THE VYBORNY-FRIEDMAN THEORE M §1. Notatio n In the followin g w e wil l mak e us e o f th e followin g notation : Sfn i s n -dimensional Euclidea n space . Rn = $fnX (— o o < r < oo ) i s th e topologica l produc t o f S£ n wit h the line . (xu ••-,*„ ) ar e rectangula r Cartesia n coordinate s i n Sf n;
(xu •••,X II,T) ar e rectangula r Cartesia n coordinate s i n th e spac e R n; here the x-coordinate s ar e spac e variable s an d r i s the time . Th e poin t P = (x u • • -,xn,r) wil l b e writte n a s (p,r) o r (x, r). A functio n o f th e point P wil l be written a s u(P) o r u(p,r) o r u(x f T) . Let D be a domain of Rn, lyin g between the planes r = t 0 and T = t> t 0, D the closur e o f D i n R n an d D ro the intersectio n o f D wit h th e plan e r = r 0. Th e se t o f interio r point s o f D TQ relative t o S£ n i s denote d b y DTQ. Assume that D T is a domai n fro m Sf n fo r £ 0 < T ^ ^ W e se t
D/0= U D T; 2 T = 5 T\A; «o<'-s< (51.1.1) . . —_ — 2«'0= U 2 T; r| 0=sJ0UA0.
We shal l cal l D{ a paraboli c domain , r j it s paraboli c boundary , 2( 0 the latera l boundar y an d Dl 0 the base . Fo r th e examinatio n o f D\ Q we will alway s conside r t as a variable . I n th e cas e ^ , = 0we wil l write D\ l T an d 2 ' i n plac e o f A' 0, r*' 0 an d 2* 0, respectively . Let A be th e Laplac e operato r relativ e t o th e spac e coordinates , 2 2 a = cons t > 0 an d L a = a A — d/dt I f u(p, r) ha s th e derivative s in D* which ente r int o L a, w e wil l sa y tha t U i s a-superparabolic , a- parabolic o r a-subparaboli c i n D\ i f
(51.1.2) L au S 0 , L au = 0 o r L au ^0; P&D'. In th e cas e a 2 = 1 w e wil l refe r simpl y t o a superparabolic , paraboli c 356 §2. TH E WEAK MAXIMUM PRINCIPLE 35 7 or subparaboli c function . Alon g wit h th e operato r L a w e wil l conside r the operato r
(Sl.1.3) L*u = div[k(p, r) grad u] + £ Mp , r) ^- - e(p , r) ^, assuming c and 6 , (i = 1,2,3 , • • •,ra) ar e continuous an d & continuously t differentiate i n D wit h respec t t o the spatial coordinates , wher e c^c 0 >0 an d k^k 0>0. §2. Th e weak maximum principle THEOREM Sl.l. Let D'bea bounded parabolic domain, let u be continuous in D l and L*u ^ 0 (L*u ^ 0 ) within D\ Then the maximum {minimum) of u is attained on Tl. Let u s assum e th e contrary , i.e . tha t t L*u^0; (p fr)eD ; (51.2.1) ^ , . m = ma x u < M = ma x u . (p,r)er< (P,T)GB { Then th e value M i s assumed a t som e poin t (p 0, TO) OzD*. We examin e the functio n M m (51.2.2) V(p,r) = U( P,T) - 2~ ( r - r 0) in D To. We hav e M — m , , „ 0 ^ J» H ^ < M fo r (p , r) G rTo; (51.2.3) 2 *KPo, r0) = M. Consequently th e maximum o f v is attained a t som e poin t (q, r) £ D T°. At thi s poin t gradi ; = 0, Av ^ 0 an d cto/ctf^O ; henc e (51.2.4) L*[i;(g,r)]^0 , which is impossible, since by the definition o f v it follow s that everywher e in DTo (51.2.5) L*v = L*u + M~~m > o. 2 r0 The theore m i s proved . t REMARK 1. Let u be continuous in D an d within D* let it satisf y th e condition L*u + du ^ 0 (L* u + &u ^ 0 ) wit h 5^0 . The n th e negativ e 358 S. 1. TH E MAXIMUM PRINCIPLE minimum (positiv e maximum ) o f u i s attaine d o n r' . (Th e proo f i s obvious.) 2 REMARK 2 . I f k = a = cons t > 0, 6 , = 0 (i = 1, • • •,n) an d c = 1, then the operator L* coincide s wit h th e operator L a. Consequentl y th e maximum (minimum ) o f a continuou s a-subparaboli c (a-superparabolic ) function withi n D t is attaine d o n r* . l THEOREM Sl.2. Let D be a bounded parabolic domain such that 20 can be contained within a domain of arbitrarily small n-dimensional measure. Further, let u be a uniformly bounded function in D\ continuous in D*\2 0 and satisfying within D l the condition L*u^0 (L*u ^ 0) . Then the value u within D l does not exceed (is not less than) the least upper (greatest lower) bound of its value on r* . We limit ourselves to proving the theorem fo r subparabolic functions . The proo f i s given fo r a genera l linea r paraboli c operator , an d hence in particular fo r the operato r L* , introduce d fo r exampl e i n [62]. Thus fo r u continuou s i n D'\20 we hav e (51.2.6) su p u = m; su p | u | = M > 0; Lu ^ 0 ; (p , r) G D\ (p,T)er'\2o (p,r)GD * Without los s of generality, w e may assume that m = 0. Otherwise we examine v = u — m. We will sho w that i f (p 0, t) i s any point o f D t, the n u(pft) ^0 . In fact , fro m th e continuit y o f u i n D'\20 it follow s tha t for any c > 0 we can find som e rj Q < t such tha t fo r any fixed i\ €E (0, i?o) and domai n G nC&n suc h tha t G,C A w e wil l hav e
(51.2.7) u(p, T) < e; (p, T)GG,X(0^T^ 0).
We no w choos e a domai n G in (i — 1,2,3) i n S£ n determine d b y th e conditions
(51.2.8) G u C Gx, C G2tn CACAC G 3„;
n mes G^\Dn = ^<^ (2a\/*(t - v )) = Xe; (Sl.2.9) f mesD,\G 2,1, = M2 UP)^M; P eD,\G2„; (Si.2.10) *,(P)=< ; ped,, ; 0<*„(p) t<+,(p) (SI 2.11) n,(p , r) = J* ^ *,(g) tfv^C - ^) ) ""exp ( - ^/j^) n,(p,r)^0; (p,r)G2; ; (51.2.12) n,(p , T ) ^ « ; (p , T) G D,; L[n,(p,r)] = 0 ; (p.r)eA' . Comparing (Sl.2.12 ) an d (Sl.2.6) , w e find tha t i f (51.2.13) v(p,r) = u(p, T) - n,(p,r) , then (51.2.14) Li;£0 ; (P,T)G^ ; ^0 ; (p,i)£r, ' and moreover y(p,r) i s continuous i n D'„. Henc e fro m Theore m Sl.l i t follows that v(p,r) ^ 0 fo r (p,r) G-D,' , i-e . tha t (51.2.15) u(p,T) ^ n,(p,r); (p,T ) G #,'• But fro m (Sl.2.9)—(Sl.2.11 ) i t follow s tha t n,(p0, t) < M(2aV*(t - „) ) -"(Mi + H2 + us) (51.2.16) r , / j* \ + «^ (51.2.17) u(p 0,t) S2e. Since € > 0 i s arbitraril y smal l an d (p 0, t) i s a n arbitrar y poin t o f D t, we find tha t (51.2.18) u(p,r)^0; {p,r)(ED\ which complete s th e proof . 360 S. 1. TH E MAXIMUM PRINCIPL E REMARK. Theore m S1. 2 remain s vali d fo r a n unbounde d domai n D\ i f to the conditions o f the theorem we add the conditio n tha t U(p, t) tends uniformly t o 0 at oo . In fact , i n this case it suffice s t o examin e th e cylinder (51.2.19) Q^\ixfSRh 0 LEMMA Sl.l. Let L au ^ 0 in D* and let max Dtu = M be attained at the point (p 0, r0) of the boundary S of a sphere K belonging, together with S, to Dl; moreover, within K let u(p,r) < M. Then the x-coordinates of the point (p 0, r0) coincide with the x-coordinates of the center of the sphere K. The proof s o f Theorems Sl.l an d Sl.2 ar e due to A. N . Tihono v [177] . §3. TH E STRON G MAXIMU M PRINCIPL E 361 In fact , le t u s plac e th e origi n a t th e cente r o f K. Withou t los s o f generality w e may assum e that (p 0, r0) i s the unique point o f S a t whic h u admit s th e valu e M . For a n arbitrar y poin t (P,T) = (x u • • -,xn, r) w e wil l writ e 2 2 2 (51.3.1) P = £xh r = P*+T . Thus r i s th e distanc e o f (p,r) fro m th e cente r o f K. We wil l assume , i n contradictio n t o th e assertio n o f the lemma , tha t the x-coordinate s o f (Po , r0) d o no t coincid e wit h th e x-coordinate s o f the cente r o f K, i.e . tha t pi > 0. I n thi s cas e w e automaticall y hav e T0 < t, o r K C D*. This show s tha t ther e exist s a n r x > 0 s o smal l tha t the sphere K x wit h center a t (p 0, r0) an d radiu s r x < p0 belongs, togethe r l with it s boundar y S u t o D . The sphere K divide s S x int o tw o part s S[ an d Sf , o f whic h th e first belongs to K an d th e secon d t o th e complemen t o f K. Sinc e (p 0, r0) i s the unique point o f K a t which u = M, ther e exists some rj > 0 such tha t (51.3.2) U( P,T)SM-V; (p,r)GS{ . At th e sam e tim e (51.3.3) U(P,T)^M; (p,r)eS'{. Let us se t (51.3.4) h(p, r) = exp { - a(p 2 + r 2)} - exp { - arl), where r0 is the radius of K and a = cons t > 0 will be chosen later. We have h(p,r)>0; (p,r) By our assumptions r x < p0. Consequently p > 0 for the point (p , r) £ K ^ Therefore i t i s possible to tak e a > 0 s o large tha t (51.3.7) L ah>0; (p,r)GKi. We se t 362 S. 1. TH E MAXIMUM PRINCIPLE (51.3.8) v(p f T) = u(p, r) + eh(p, r); € = cons t > 0 . By virtu e o f (Sl.3.7 ) an d th e condition s o f th e lemm a (51.3.9) L av = L au + eL ah > 0. Consequently, b y virtu e o f th e wea k maximu m principle , v attain s it s maximum o n oj . Fro m (Sl.3.5 ) i t follow s tha t (51.3.10) v £ M + eh< M fo r (p , r) G Sf . Furthermore, o n S { (Sl.3.10*) v^M-rj + eh. Consequently i t i s possibl e t o tak e e > 0 s o smal l tha t (51.3.11) v < M fo r (p , r) G S{. For suc h a choic e o f € > 0 w e hav e v (51.3.12) u(p , r) = cons t = M(p, r) G DTQ. 7 In fact, let J^ be the set of all those points o f D TQ at whic h u(p, T) = M . 7 J^ is closed relative to D TQ. We suppose, in contradiction to the assertio n of the lemma, that D T0\S^ i s not empty; then i t contain s a poin t ((fc^o ) whose distance p 0 to th e se t Sf mus t b e les s tha n it s distanc e t o th e boundary 2 TQ of the domai n D TQ. Thus u(q 0, r0) < M, bu t b y continuit y there i s a neighborhoo d U o f th e poin t (<7o»*o ) containe d i n D* and such tha t (51.3.13) u(p , r) (51.3.15) u < M fo r (p , r) £ A x, and moreover , o n S x ther e i s a poin t (ft , rx) suc h tha t (51.3.16) u(q un) = M . We no w inscrib e i n A x a spher e X touchin g S x a t a poin t (ft , ri). W e then ca n appl y Lemm a Sl.l. Thu s th e jc-coordinate s o f th e cente r o f K coincid e with th e x-coordinate s o f th e poin t (ft,rx) , whic h i s possibl e only whe n (qun) i s th e verte x o f th e ellips e A x, coinciden t wit h on e of the endpoint s o f the lin e r . Thu s (51.3.17) u(ft, Tl) = M ; (ft,rjGr C U. But thi s contradict s (Sl.3.13) . Th e lemm a i s thus proved . PROOF O F THEORE M Si.2 . Le t (51.3.18) L au > 0 in D'; ma x u = u(p 0, t) = M an d (q Q, r0) G C(p0, t). (p,r)eD* ( We examine a n admissibl e curv e LCJ5 connectin g th e point s (q 0,r0) and (Po,t). W e assume, in contradiction to th e assertio n o f th e theorem , that (51.3.19) u(q 0, r0) < M; r 0< t. Then o n L ther e i s a poin t (ft , rx) neares t t o (q 0, r0) a t whic h u = M. Since r 0 < n < t an d D T1 is a domai n i n j^ n, i t i s possibl e t o construc t two n-dimensional cube s Q i and Q 2 with center s a t th e poin t (ft,^) , be - longing to the plane r = T U and on them, as bases, two n + 1 dimensional parallelograms R x an d R 2 suc h tha t (51.3.20) Q2CQ1 ; R2CRiCD*\ Tl with moreove r th e poin t (q 0,T0) GD \Rt. By virtu e o f Lemm a SI. 2 (51.3.21) U( P,T) 2 2 2 (81.3.22) h(p, T ) = r -j: (Xi - xf) - ( r - r*) . 364 S . 1. TH E MAXIMUM PRINCIPL E We hav e 2 (51.3.23) L ah = 2{T-T*- 2a n}. In Ri w e have r > r*. Thu s i t i s possible t o choos e r s o large tha t (51.3.24) L ah > 0 fo r (p , r) G R x. We se t (51.3.25) V(p f r) = U(p, r) + eh(p f r) , where € = cons t > 0 wil l b e chose n later . Let r b e a portio n o f th e boundar y o f R 2 belongin g t o K 9 an d C a part o f K belongin g t o R 2. Sinc e ever y poin t o f r i s a n interio r poin t of R t suc h that u v Finally, throughou t th e domai n G = KDR 2 (51.3.29) L av = L au + eL ah > 0. From (Sl.3.27)—(Sl.3.29 ) an d th e wea k maximu m principle , i t follow s that (51.3.30) v(p fr)^M; (q,r) (51.3.31) ± v(QlfTl)^o, or Therefore w e hav e (51.3.32) A u(q u r x) = ~ u(q u n ) - 2«(r x - r*) . or o r But b y constructio n r x — r* > 0. Thu s §4. TH E VYBORNY-FRIEDMAN THEORE M 36 5 (51.3.33) i - u(q ltn) ^ 2«(r , - r* ) > 0. or Furthermore, a t th e poin t (?I,TI) , a s at th e maximu m poir t o f U(P,T), we have (51.3.34) Au ^ 0 . From (Sl.3.33 ) an d (Sl.3.34 ) i t follow s tha t 2 (51.3.35) L au(Qu n) = a Au - ^ < 0, or which contradicts th e assumptions . Th e theore m i s proved . §4. Theore m of V#>ontf-Friedman [44] , [196] Below w e formulat e an d prov e a theore m du e t o A . Friedman . A n analogous theore m wa s prove d i n [62] . THEOREM S1.3 . Let u be continuous in the closure of the parabolic l domain D and a-subparabolic (a-superparabolic) within it. Let (q 0, r0) £ r ' be a point at which u attains its maximum (minimum) M. Assume that there exists a sphere K with the following properties: (qo,T0)EK\K = S; the radius drawn to the point (q 0y T0) is not parallel to the T axis; «4 u(p, T) Let v be a direction issuing from a point (q 0, r0) within K. Assume that du(q0,T0)/dv exists. Then (Sl.4.1) ^-u(q Q, T 0) <0 (^-u(q 0, T 0) >O) . OP \dv / REMARK. Th e theorem i s valid fo r a general linear parabolic operator . The proo f give n her e i s a reproductio n o f Friedman' s origina l proof , with appropriat e simplifications . PROOF. W e transfer th e origi n t o th e cente r o f K, an d construc t a cylinder C whos e axi s coincide s wit h th e tim e axis , an d whos e radiu s 5 > 0 i s s o smal l tha t 366 S . 1. TH E MAXIMUM PRINCIPL E (51.4.2) (q 0,T0) (51.4.3) S g = G\G; S* = K^K' be the boundarie s o f G and K l respectively. W e se t (si.4.4) si = sgns; s ; = sgnc- We examin e th e function s 2 2 2 2 (51.4.5) h = exp{ - a( P +r )} - exp( - aR ) » h* - exp( - aR ), where R is th e radiu s o f K and p 2 = ^?=ixf. W e hav e 2 2 (51.4.6) L aA = 2aa h* \2ap - n + \r ) . Since p ^ 6 > 0 fo r (P,T)£G , w e se e tha t fo r a > 0 sufficientl y larg e (51.4.7) L a/*>0; (p,r)GG . In additio n (51.4.8) A>0 ; (p,r)eK. We se t (51.4.9) V(p,r) = li(p,r) +eh(p,r). We the n hav e (51.4.10) L av = Lau + eL ah>0; (p,r)£G, since by assumption L au ^ 0. Furthermore , (51.4.11) v(p, r) = u(p, r) £ M fo r (p , r) G S;. Finally, by virtu e o f th e fact s tha t u < M fo r (p , r) G S/, an d tha t th e distance from Sg t o (q 0, r0) i s not les s than 8 > 0, we see that ther e exist s some rj > 0 suc h tha t (51.4.12) u(p , r) £ M - r, for (p , r) G S/. Consequently fo r c > 0 sufficientl y smal l (51.4.13) v(p, r) ^M-n + eh (51.4.14) v(p,r) (51.4.15) ^-v(q 0,r0)^0, ov since v is directed into G and v(q 0, r0) = u(q 0, r0) = M. Bu t fro m (Sl.4.15 ) and (SI.4.9 ) i t follow s tha t (51.4.16) — u(q0,rQ) ^ - t — h(q 0,T0). OV OV We hav e 2 (51.4.17) ^-h(q 0,T0) = - aRexpi- aR )cos(R,7), OV where R i s a radiu s vecto r t o th e poin t (q 0, r0) wit h initia l poin t a t th e center of K. Sinc e v is directed fro m (q 0, r0) int o K, w e find that cos(i? , v) < 0. This show s tha t (51.4.18) |-u(g 0,ro)<0, OV which complete s th e proof . REMARK. I f the point o f the boundar y maximu m (minimum ) belong s to Do , the n th e inequalit y (Sl.4.18 ) canno t b e valid , sinc e a t suc h a point conditio n a 3 automaticall y fail s t o hold . The essentia l importanc e of this fac t i s evident fro m th e followin g example . Le t u = — r 2. The n Lau = 2 T > 0 for r > 0 an d th e boundar y maximu m o f u, equa l t o zero , is attaine d o n D 0. A t th e sam e time , D'u < 0 an d du/dr\ DQ = 0. SUPPLEMENT I I THE LIMITING BOUNDAR Y VALUE S OF THE DERIVATIVE OF A PARABOLIC FUNCTION . GREEN'S FUNCTIO N OF A DOUBLE-LAYER BOUNDAR Y PROBLE M ON THE LINE AND H ALFLINE §1. Th e limiting boundary values of the derivative of a parabolic function In ou r examinatio n o f problem s A „ B , an d C i n Chapter s II-I V o f Part 2 , and i n ou r investigatio n o f the doubl e laye r single-phas e Stefa n problem i n Chapte r V o f Part 2 , th e value s o f du(x,t)/dx a t point s o f the boundary x = y(t) wer e alway s regarde d a s th e limitin g value s o n approach to the boundary fro m withi n the domain o f definition o f u(x, t). At th e sam e time , i n som e place s o f th e tex t i t wa s estimate d a s th e actual boundary value s without proo f o f the possibilit y o f suc h a n esti - mate (se e fo r exampl e th e proo f o f Lemm a 1 i n Par t 2 , Chapte r II) . The legitimac y o f thi s procedur e follow s fro m th e almos t obviou s Theorem S2.1 : THEOREM S2.1 . Let u(x,t) be continuous in the closure of the parabolic domain (52.1.1) iy-{X 1(T) (52.1.2) u|,.x,. w = fi(t) (i = 1,2) ; a| #_0 = *(x) ; X^O ) < x < X2(0), where X*, /, and uM u{Xiit) t) (52.1.3) li m - ' = Vi(t)^ li m * (£-1,2) , x-X (t)±0 X — Xi(t) *-X(0± O OX where the existence of the limit on the left is a consequence of the existence ofvt(t). The sig n - f i s used fo r i = 1 , an d — fo r i = 2 . 368 §1. LIMITIN G BOUNDAR Y VALUE S 36 9 In fact , fo r Xx(0 rx2(o) "(*>*)= L,'X!(0m) 4>(Z)E(x-Z,t)di; J Ax(0 ) (S2.1.4) + Z (- D * J['[ vM E(x - XMJ - r) - fiir) ( y - X,(r ) ) E(X- X,-(r), t - r ) ] dr Here E(jt, 0 i s the fundamenta l solutio n o f the hea t equation , define d in agreemen t wit h (0.2.4) , an d (52.1.5) Vi(t)= li m — u(x,t). x-»Xi(()+(-l)'+10 &X Noting tha t x) X £ x <- f IT - -W1 <* - M,t- r) (52.1.6) L * J id (-lYiXM - x) — - - — erf c , , 2dr 2y/t-r we see tha t (S2.1.4 ) ca n b e written i n th e for m »X2(0) 4,{t)E(x-i,t)di JXxW) (-D'iXM-x) 2y/t (S2.1.7) (- l)'(Xj(r) - X) + \ fV,(r)erfc v ^"^L- "' ** 2 Jo 2\/t- T + (- lVJ o Vi(r)E(x - XM.t- r)dr^. Hence w e conclud e tha t -• XX2(0) 2 4(S)EiXiW-s,t)di J'X!«X!(0») (S2.1.8) 2 x^ri//m t i-inxM-Xjit)) t 2y/t 370 S . 2. LIMITIN G BOUNDARY VALUES. GREEN'S FUNCTION + 2 Jo fM erf C WT^r dT + (- 1) ' fo'vM E(Xj(t) - XM,t - r) dr]. Using the formul a o f finite differences , w e find tha t X2<0 u(x,t)-u(Xj(t),t) = r > d_ Jxx(0) OX x-Xj(t) Jx l(o) ^ dx i + £<-i)f/*T,(-l) <\fi(0)E(z v-Xi(fi),t) (S2.1.9) + J^/iM E(zZti - XMJ - r) dr + j\ i(r)^cE(z4-Xi(r)9t-T)drj9 where Zi,z itj(i = 1,2,3,4 ) ar e certai n point s lyin g betwee n x an d Xj(t). Every integra l o n th e righ t sid e excep t th e las t i s a hea t potentia l o f a doubl e layer , convergin g uniforml y fo r — < » < x < o o an d t*z t 0 fo r any valu e t 0 > 0. Usin g thi s an d th e ga p theore m fo r th e doubl e laye r heat potential , w e find tha t th e righ t sid e converge s fo r x—>X,(£ ) + (-l) i+1.0 an d *X2(0) n J (!;)£- E(Xj(t)-t:,t)di; x^o) dx +i; <-1) 1" [/*«>> E(Xj(t) - xm,t) (S2.1.10) I= 1 + fji(r) E(Xj(t) - XiW, * - r ) dr + f X X ^W^^ >W -X f(rU-T)dr] +i 0,(0 . Furthermore, th e sam e resul t i s give n b y differentiatin g (S2.1.7 ) wit h respect t o x fo r X x(0 < x < X2(0 an d t^t 0 an d the n takin g th e limi t for x-+Xj(t). Thu s th e theore m i s prove d unde r th e assumptio n o f existence o f the limi t o f th e valu e Ui(t) of th e derivativ e du/dx a t th e point o f the boundar y x = Xi(t) (i = 1,2) . We not e no w tha t th e equatio n Vi(t) = Vi(t) (i = 1,2 ) ca n b e con - §2. GREEN' S FUNCTIO N 371 sidered a s a syste m o f integra l equation s relativ e t o Vi(t). Setting Wi(t) = \/tVi(t) (i = 1,2) , formin g th e integra l containin g , an d integratin g by parts, w e find that Wi(t) is th e solutio n o f th e syste m o f equation s 4T^W = 2E(- D'f [A(o > - tixmwxjit) - x t(o),o + jj(r) E(Xj(t) - X,(T), * - r ) dr (S2.1.11) + j\((r) A E(Xj(t) - X,(r), t -r)dr} •X2(0) Xx(0) The existence an d uniquenes s o f the solutio n t o thi s syste m i s estab - lished i n exactl y th e sam e wa y a s th e existenc e an d uniquenes s o f th e solution t o th e integra l equatio n (3.3.3 ) o f Par t 2 , Chapte r III . Sinc e the system (S2.1.11 ) i s equivalent t o th e proble m o f definin g u i n th e sense o f Theorem 1 0 o f Chapte r III , Par t 2 , w e find tha t th e limitin g values fro m withi n th e domai n D* for th e boundar y valu e du/dx d o i n fact exist . The theore m i s proved . §2. Green' s function of the double-layer boundary problem on the line and halfline. We examin e th e boundar y proble m (S2.2.1,, ««W0-| £ for a du (S2.2.12) u \ x~0-o = Mx^+o ; k(x) — = k(x) t>0; x^fi-0 OX x~0+O (52.2.13) hu = fi(t) fo r x = a; l 2u = f2(t) fo r x = 0; t>0; (52.2.14) u = (x) fo r t = 0 ; a < x < y. Here 2< \ i Aa i = cons t > 0 ; x < p a( ) = *1 = cons t > 0 ; x>p, (S2.2.15) * Ui = , , . _ ( ki = cons t > 0 ; x < fi X k 2 = const > 0 ; x > 0, 372 S . 2. LIMITIN G BOUNDAR Y VALUES . GREEN'S FUNCTIO N and li an d l 2 are linea r differentia l operator s o f first orde r (S2.2.2) l i = A ~+Bh dx corresponding to boundary condition s o f first, secon d o r third order , such that i f At = 0 the n B , = 0; i f At = 1 , the n B , = 0 or Bt = (- 1 ) \ The Green' s functio n o f the proble m (S2.2.1; ) (i = 1, • • -,5), a s fo r the single-layer problem , is a solution to the equatio n (S2.2.1.!) , define d for t> T , satisfyin g homogeneou s boundar y condition s (S2.2.1 3), initia l conditions (S2.2.1 4) wit h< £ = 8(x — £),where 5(x) i s a Dirac 8- function, and condition s conjugat e to (S2.2.12). Here | i s any poin t o f the inter - val (a , y) differen t fro m 0. It i s easily see n tha t g(x,t,£, r) , considere d a s a functio n o f £ an d T for fixed x and t, satisfies th e conjugat e equatio n (S2.2.3x) a 2(£)-^ + /=0; a < £< y; (*fl; r (S2.2.32) l lg\^a = 0; l 2g\^y==0; g\ T=t=8(x-^) and condition s o f agreement 2 2 dg (52.2.33) a ^li^-o = aWb^+o; a % = a* £=0-0 dk *=0+o Here 2 (52.2.34) M(£ ) = { I ] This assertio n i s proved i n exactly th e same wa y as the analogou s assertion fo r the homogeneou s proble m (se e for example [179]) . From th e equation s (S2.2.3, ) ( i = 1,2,3,4) i t follow s tha t th e solu- tion to the problem (S2.2.1; ) {i = 1, • • -,5) ca n b e expressed i n term s of Green' s functio n an d th e boundar y function s /,(£ ) an d #(x) i n th e usual way , s o that i n the integral representatio n o f the solution the conjugate condition s d o not enter explicitly . We wil l cal l th e Green's functio n a fundamenta l solutio n t o the double-layer problem, if it is constructed for an unbounded line (a = — <» , 7 = + oo) . In this cas e th e boundary condition s wil l b e replaced by the conditio n o f boundedness o f g at infinity . §2. GREEN' S FUNCTION 37 3 We construc t th e fundamenta l solutio n t o th e double-laye r proble m starting wit h 0 = 0 an d th e Green' s function s o f th e first an d secon d boundary problem s o n th e axi s x > 0 , an d startin g wit h 0 = 1 an d some function conjugat e t o i t i n th e sens e o f definitio n (0.2.11) , A 5. It i s obvious that eac h o f thes e function s depend s onl y o n the differ - ence t — r. Therefor e w e wil l writ e g(x,£,t) i n plac e o f g(x,t,£, r) , setting r = 0 . The constructio n i s carried ou t b y mean s o f the Laplace - Carson transform . Th e correspondenc e betwee n th e representatio n and the origina l i s denoted b y th e sig n -r. A. Th e fundamenta l solutio n o f th e double-laye r proble m [137] . Le t G(x,£,p)~£(x,£,0- The n d2G a2(x)—z=pG-p5(x-Z); -oo dG dG (S2.2.4) GU_o=GU+<, ; *(* ) k(x) dx *=-0 dx *=+o ) G\ < o o fo r x — > oo ; £ = const ; p = const . The first o f thes e equation s i s equivalent t o th e syste m d*G a2(x) = pG; -co ^(l + a)exp(-(^-|-)v/p) far-.<«<0<* ; G(x,t,p)=. (S2.2.6) ^(1-S)exp( - — - — k/pj fo r - o o 1-A Mi (S2.2.7) 1 + X ' k xa2' Using th e know n relatio n 1 (S2.2.8) \/pexp( - «v/p)-.-n(\/*e)" exp ( - ^f ) - 2E(a ft)9 we find that !S(x-£,affl+«JB(x + fcai0 fo r - « < x,£ < 0; L±i £^__L, A for-oo<^<0 a2 \a x a 2 / E(x - ^,a|f ) - &E( x + |,a|^ ) fo r 0 < x, £ < oo . Let g* be the function conjugat e t o g, s o tha t (S2.2.10) dg dx' Normalizing g* b y requirin g tha t i t vanis h a t infinity , w e obtain E(x - $,a\t) - 8E(x + $,a$) fo r - o o < x,£ < 0; <£<0 £( ±-ft) for-oo ax Xa j a 2 / E(x - Z fa%) + 8E(x + £,<$ ) fo r 0 < x,£ < oo . B. Th e Green's functio n o f a doubl e laye r fo r th e first boundar y prob - lem o n th e halfline . Agai n le t G(x,£,p)r-g(x,£,t). W e hav e d*G a2(x) pG-p8(x- £) ; 0 G|x==o = 0 ; \G\ < o o fo r x-^oo ; £ = const . Integration give s §2. GREEN' S FUNCTION 375 r i- ^F.{^(_lazfl^)-^(-e±l^) + 5ex p ( ^ \/p) for 0 < x, |< 1 ; forO<£ -„(-[£±t*+l]v5) for 1< x , £ < oo . V. Here (S2.2.13) and 5 is define d i n agreemen t wit h (S2.2.7) . Expanding Fin a serie s wit h respec t t o power s o f 5exp( — 2\/p/ai) 376 S . 2. LIMITIN G BOUNDAR Y VALUES. GREEN'S FUNCTIO N and usin g (S2.2.8) , w e find tha t r £ ( - l) n8n{E(\x - i\ + 2»,af0 - E(x + £ + 2n,a$) n=0 + dE(2n + 2- x- (,a$) 2 -5E(2n + 2-\x-£\,a lt)} for 0 < x, a < 1 ; Z(-l)T(l + ^.J£(^i-f^±^, () „=o ( \ a 2 a x / -£(^i+2t±i±i,,)j \ a 2 a x /) forO<£ "o l \ a 2 a x / (S3.3.14) forO f(_1)Wh-.{,(!£ziL+*!il) feo v \ a 2 « i / \ a 2 a x / ,/|s-f| 2 n + 2 \ + bE\ \ X* 2 a x ' / /*-K-2+2nX| \ a 2 a x /} V for 1< x , £ < oo . It i s easil y see n tha t th e functio n g*(x,£,t), conjugat e t o g(x,£,t) an d normalized b y th e conditio n (S2.2.15) g(x,0,*)=0, K has the for m §2. GREEN' S FUNCTIO N 377 r.£ ( - DVjEflx - £ | + 2n,a\t) + E(x + z + 2n,a& n=0 -5E(2n + 2-\x-$\, aft) - 6E(2n + 2 - x - £ , aft)} for 0 < x , £ < 1 ; 2 £ (-l)T( l + ^{£(^l + ^i^,t) \ a 2 a x I) forO<£ < f (-DT(l-^(E(ill +^±^,t) + JS (LLL+5!±1±£>,)} (S2.2.16) \ a 2 a x / J fcrO f _ -.f*(!iz«LI JC — f I »!.2/i ) ( 1)Wll —++ —f, x + £ax -a 2 2/ x 1 + 2 /x + £-2 2/ 1 + 2 \ + E(-^ +—J—A \ a 2 a x I + *(l£=iL+?I±*,,) \ a^/x + a ^-2 2 I n \) „ ^ + S£ ( 52 + x — ,t) \ fo r Kx, £ < co , V \ a 2 a x /) C. Th e Green' s functio n o f a double laye r fo r th e secon d boundar y problem on the halfline . Omittin g the calculations, whic h ar e completel y analogous t o thos e o f the previou s paragraphs , w e giv e the final result : g(x, {, t) = E(x - & a\t) + E(x + & a?* ) + £ 5»[£( x - £ - 2/i , a\t) + j£( x - £ + 2/*, afO + E(x + |-2/i, fl#) + £( x + f + 2n , aft)] fo r 0 < x, £ < 1 ; 378 S . 2. LIMITIN G BOUNDARY VALUES. GREEN'S FUNCTION I + ,(£^i + l±i±U,,)l I \ a 2 a x / J for 0 < { < 1 < X < o o ; *(*,&*)« < »To L \ a 2 a x / (S2.2.17) forO i£i L \ a 2 a x / + ,(£±1^+5S±»,,)1 \ a 2 a x / J V for 1 < x , £ < oo . Finally, th e functio n g*(x,£,t) f conjugat e t o g(x,£,t) an d normalize d by th e conditio n (S2.2.18) g(x,0,t)=0 9 has th e for m **(*,&*)= £(x-£,af*)-2S( x + £,af O + £ 6"[JS( x - £ - 2n , a?* ) - £( x + £ - 2n , aff l »=i + E(x - t + 2n, aft)-E(x + £ + 2n,a\t)] for 0 < x, £ < 1; §2. GREEN' S FUNCTIO N 379 r „-.o + «i;i-r*(i^. +^i±*!,<) »=o L \ a 2 a x / _£(i^.+i+i±^.,)] \ a 2 a j / J forO<£ -'(ST1*1-*!**")] (S2.2.19) forO<.x \ a 2 a x I £i L \ a 2 a x / • /« + «-2 g » + 2 X T + £ | \ a 2 ^ / J I for 1 < x, £ < oo . SUPPLEMENT II I NUMERICAL ILLUSTRATIONS §1. Exampl e of the numerical solution of the integral equations of the Stefan problemlf We will no w illustrat e th e solutio n o f the single-phas e Stefa n proble m by the metho d o f integra l equations , usin g the followin g proble m a s a n example. We consider a problem o f heat transfe r i n a mediu m whic h i s initiall y in a two-phas e state , an d occupie s th e laye r 0 < x < 1 ; w e assum e tha t the phas e 2 materia l lie s i n th e regio n y x(t) < x (S3.1.1) u| x_0= -1; K|#- O = 4(*-0.25) ; u\ x^yit) = 0; y(t)=v(t); y(0)=0.25 ; v =-^u(y(t),t). The calculatio n o f th e temperatur e u afte r determinin g v an d y i s o f no interest . W e limi t ourselve s t o th e calculatio n o f v an d y o n th e interval (0 , T), wher e T i s th e tim e interva l i n whic h phas e 2 exists , defined b y th e equatio n y(T) = .5. See Par t 2 , Chapte r VIII , §3 . I n th e followin g w e describ e method s whic h ar e oriented toward s numerica l han d computation . I n usin g th e fiVSM computer , th e methods o f Par t 2 , Chapte r VIII , § 3 ar e employed . 380 §1. INTEGRA L EQUATIONS 38 1 In the case under examination the integral equations (2.2.3, ) (i = 1,2,3) assume th e followin g form : (S3.1.2) -- L ['W(t,T\v(T),y(t),y(r))-i=; y(t) = 0.25 +Cv(T)d T, where W=viT) l^t^T exp L 4(T^ H (S3.1.3) exp 4(t - r) L—w^r\\ We begin th e calculation s wit h a preliminary estimat e o f the lengt h T o f th e process , whic h ca n b e don e b y th e us e o f Lemm a 1 of Part 2, Chapter II. We set (S3.1.4) Ul =-(l-^^); 2 l = 2VF and choos e y suc h tha t (^ 1 K\ 2 - y ^ dUl I - ex P(~72) \/t v* \x= zi y/irterfy It is obvious that, a s they ar e defined, u x and zx satisfy ever y conditio n of the lemma . Thu s z x(t) ^y(t) fo r ever y *E(0 , T). Calculatio n show s that on e ca n tak e 7 = 0.6. Thi s give s z x(t) =0.5 fo r *=* 0.174. Thu s T < 0.174. On the othe r hand , i n orde r t o determin e a majorant o f y it suffice s for u s to tak e (53.1.6) u 2 = 4(x - z 2(t)); z 2(t) = \/0.0625 + 2t. This give s T > 0.093 . Thu s (53.1.7) 0.09 3 < T < 0.174. We pos e th e problem o f determinin g y(t) wit h precisio n 0.0025 , 382 S. 3. NUMERICA L ILLUSTRATIONS which corresponds to th e maximu m relativ e erro r o f 1% . Fro m (S3.1.7 ) and (S3.1.1) it follow s that fo r thi s it automaticall y suffice s t o evaluat e v(t) with an accuracy o f 0.014. This requirement can be met b y choosin g the mesh length appropriately. TABLE 1 . Supporting Mesh Method of Remark s value t length Interpolation 0.00001 0.00001 Linear 0.00002 0.00004 0.00007 0.00010 0.00020 0.00005 0.00040 0.0001 0.0010 0.0002 0.0025 0.0005 0.0050 0.010 0.0010 Interval 0.018—0.020 with mesh 0.001 0.020 0.0020 0.028—0.030 with mesh 0.001 0.030 0.038—0.040 with mesh 0.001 0.040 0.055—0.060 with mesh 0.0025 Parabolic 0.060 0.0050 3 Points 0.080 0.01 0.070—0.080 with mesh 0.0025 0.100 0.090—0.100 with mesh 0.0025 0.125 0.120—0.125 with mesh 0.0025 We shall cal l the valu e t fo r whic h v(t) i s evaluate d b y mean s o f a n iteration process , a supportin g value , an d th e value s fo r whic h v(t) is defined by means of interpolation, intermediate. In Table 1 we indicate the supporting value s t, the lengt h o f th e mes h fo r calculatin g v(t) fo r supporting value t, an d th e metho d o f interpolatio n use d fo r evaluatin g v(t) a t intermediat e value s t. The choice of the mesh length and the method o f interpolation ensures the attainment o f the require d accurac y i n ever y case . Here w e do no t examine the proble m o f finding th e greates t permissibl e mes h lengt h §1. INTEGRA L EQUATIONS 383 or an optimal method of numerical integration. The essential requiremen t which we make i s that o f simplicit y o f computatio n o n a calculator . The integral on the right side of (S3.1.2 ) i s calculated i n th e followin g way. The interval (0 , t) is divided by a point t* < t into two parts. Alon g the interva l 0 S r S t* th e integra l i s evaluate d b y Simpson' s rul e with the mes h length s indicate d i n Tabl e 1 . Alon g th e interva l (t*,t), where Simpson' s rul e i s inapplicable , th e integra l i s evaluate d b y th e formula (8.3.19) . In the last colum n o f Table 1 we indicate an additiona l subdivision o f th e integratio n interva l whic h i s chose n s o finely tha t the magnitude o f the oscillatio n o f W(t, T ) o n eac h o f th e interval s ob - tained b y th e subdivisio n i s smal l enoug h t o ensur e tha t th e erro r o f computation b y formul a (8.3.19 ) doe s no t excee d a give n degre e o f tolerance.2) The choic e o f th e "zeroth " approximatio n v 0(ti) i s carrie d ou t b y means o f graphica l extrapolatio n fo r t > 0.01. Fo r 0.0000 2 ^ t ^ 0.0 1 we use the formul a p p (S3.1.8) Bbtt +1) = v(td - tt ft-i>y fc> (t .+1 _ ti ), H ~~ H- l where the correction coefficient a is introduced wit h the aim o f calculatin g the decreas e |i>(0| . W e tak e 0. 5 S « ^ 0.75. 3) Finally , MO.OOOOl ) i s taken equa l t o i;(0 ) = 4 . I n Tabl e 2 an d Figur e 7 th e result s o f cal - culating v n(t) an d y n(t) fo r supportin g valu e t ar e given . Her e n i s th e number o f iterations . As is see n fro m th e table , th e erro r o f iteratio n fo r evaluatin g u(t) i s not greater than 0.001 for n S 3 for ever y value o f t except t = 0 , t = 0.0 2 and t = 0.06 . For * = 0.0 2 thi s erro r i s not greate r tha n 0.00 1 fo r n ^ 4 , and for t = 0.0 6 it i s less than 0.00 2 fo r n ^ 5 . Obviously fo r these point s the graphica l extrapolatio n fo r chose n t k i s les s satisfactory . W e not e that i;(0.006 ) = 1.944 . Thus th e absolut e erro r equal s 0.00 2 whil e th e corresponding relativ e erro r equal s 0.1%. If the number of subintervals in (0 , t*) i s odd , the n w e appl y Simpson' s rul e t o th e interval (t% tt*)t wher e h i s th e neares t subdivisio n poin t t o t = 0 . Withi n th e interva l (0,^) th e integra l i s evaluate d b y th e trapezoida l rule . In practic e th e introductio n o f th e correctio n coefficien t a i s convenient , althoug h the numerica l valu e o f a i s generall y chose n withou t theoretica l justification . Makin g use of the Taylor formul a fo r small t would b e highl y inconvenien t becaus e o f th e rapi d variation o f v(t). I n orde r t o guarante e sufficien t closenes s t o th e actua l valu e v 0(ti) with th e ai d o f Taylor' s formul a woul d requir e difference s o f orde r highe r tha n tha t which is possibl e wit h th e numbe r o f point s an d amoun t o f computatio n feasible . 1 p pO o O o o o o o o p p P p P p p o b b b b to o oo 8 2 co to §88 g 8 8 8 8 8 8 r*. C71 cn to o o 4^ to H* cn H* 1—i 1—' to to to to to CO oo CO CO CO co CO CO CO 4^ 4^ co 4^ o CO 4^ <1 Cn •"* 00 00 4^ <1 o 4 c: 9 cn II © 05 V SNOixvHxsrmi ivoiaawrm e s *8S §1. INTEGRA L EQUATIONS 385 TABLE 3 a = 0.0000 1 « = 0 <^; 0 = 0.0000 9 a0* 0 = 0.1 2 Trapezoidal Simpson's Formula Trapezoidal Simpson's Formula formula formula (8.3.19) formula formula (8.3.19) 0.05418 0.05367 0.05367 0.30539 0.30070 0.30100 As an illustration, i n Table 3 we give the error o f the approximation of the valu e (S3.1.9) I a0 = ~ f ' W(t,r\y n, v n) -—==- for t = 0.0001 fo r n = 0 and t = 0.12 5 fo r n = 2. Iafi is evaluated b y the trapezoidal rule , b y Simpson' s formul a an d by formul a (8.3.19) . Her e (a,p) i s the largest interva l alon g whic h the integration ca n be carrie d out by Simpson's rule if we use the network show n i n Table 1 . We thus see that th e erro r o f th e approximatio n i n calculatin g #(0.0001 ) an d i;(0.125) i s not greate r tha n 0.005 . I n additio n th e estimated approxi - 2,5 2.4 / 2.3 — /7 = 2,2 ±az2 /7*H 2.1 >-n~0 20 \-n=2 n*3y- 1.9 \ n*1 A id f risO^ &* 1.7 1.6 n*i - Jtf 1.5 I i /^0N > 0 0,02 Q(fr 0.06 0,08 0,10 0.12 t 0 WK 0.008 0.012 0.016 *.j±ft FIGURE 7 . 386 S. 3. NUMERICA L ILLUSTRATIONS mation error on every interva l examine d varie s wit h time . But togethe r with the estimate s o f th e iteratio n erro r give n below , thi s prove s tha t the erro r o f determinatio n doe s no t i n fac t excee d th e tolerabl e valu e 0.014 o n ever y interva l 0 < t < 0.125. TABLE 4 A — "Exact" solution (method of integral equations) B — Quasi-stationary approximation of Leibenzon C — Modified method of Leibenzon t ^A yB yc Remarks 0.00001 0.250040 0.250040 0.250040 y(T) - 0.5 0.00010 0.250389 0.250400 0.250394 0.00100 0.253673 0.253968 0.253820 0.00250 0.258760 0.259808 0.259293 0.00500 0.266690 0.269258 0.268041 5TC= (l~|y 100 0.01000 0.281347 0.287228 0.284637 0.02000 0.307925 0.320156 0.315603 0.03000 0.332077 0.350000 0.344085 0.04000 0.354519 0.377492 0.370676 0.06000 0.395471 0.427200 0.419298 0.08000 0.432581 0.476990 0.441749 0.10000 0.466754 0.512348 0.503410 0.12500 0.506240 0.559017 0.551453 0.12092* 0.5 0.09375* 0.5 * Obtained by 0.10726* 0.5 linear interpolation 6TB = 22.5 % ; 5T C = 10.8 % In Table 4 we giv e the "exact " solutio n value s correspondin g t o th e quasi-stationary approximatio n o f Leibenzo n an d wit h modification s used by us in §§1 and 2 of Chapters II and III, Part 1 . Namely, Leibenzon's quasi-stationary approximatio n i s determine d a s th e solutio n o f th e problem 2 2 d u/dx = 0; 0 s d*u/dx = du/dH 0 (S3.1.11) u|,_ 0 = 4(x - 0.25)i,(y(A ) - JC)„(0.2 5 - x); y\t) = du(y(AM)/3x| w- v(t); y(0 ) = / = 0.25 . Using th e expressio n ipo.i.lz; *«_« , + (-lYE(x + $-2ny(\),t)] for th e Green's function s o f th e first and second boundar y problem s on the interval (0,y(X)) , w e find withou t difficult y tha t (38.1.18) y W=4i:[erf—-^ er f ^ J . Equation (S3.1.13 ) wa s integrated o n th e BfiSM-2 M b y th e metho d of Runge-Kutta usin g a standar d program , ensurin g a resul t wit h the accuracy o f si x figures. In th e last lin e o f the table w e find the siz e o f the relative erro r of determining th e final momen t o f th e proces s (i.e . th e momen t whe n the boundary reache s y = 0.5). Assum e tha t thi s momen t i s determined " exactly'' b y the method o f integral equations . A s we see, on changin g from th e quasi-stationar y metho d t o th e modifie d quasi-stationar y method w e decreas e th e erro r o f thi s determinatio n b y bette r tha n a factor of two, and hence we can use this metho d fo r empirical justifica - tion o f the modifie d metho d o f Leibenzon . §2. Crystallizatio n of a melt under insertion of a plate or a sphere4) We examin e th e process o f crystallizatio n o f a mel t upo n insertio n of a plate or a sphere of more refractory material , describe d in dimension- less variable s b y th e equation s (5.1.1 ) —(5.1.7) an d (6.1.7) , (6.1.8x ) — (6.I.85) o f the second part , respectively . W e pas s fro m dimensionles s to dimensioned variable s by formulas (5.5.1 ) o f Part 2 , with / replace d by RQ in the spherica l case , wher e R 0 i s th e radiu s o f th e immerse d sphere, an d u = (T - T k)/(T0 - T k) b y u = x(T - T k)/(TQ - T k). The value s o f w(t), f(t) an d y(t) ar e determined, i n the cas e o f an See Part 2, Chapter V, §§ 1 and 2; Part 2, Chapter VI, § 1 and Part 1 , Chapter II, §3 . 388 S. 3. NUMERICA L ILLUSTRATION S immersed plate , a s th e solutio n o f th e syste m o f integra l equation s (5.2.6), (5.2.7 ) o f Part 2 , and i n the cas e o f a n immerse d sphere , a s the solutio n o f the system (6.1.17) , (6.1.19 ) an d (6.1.20 ) o f Part 2 . The numerical algorith m whic h w e use is identica l t o tha t describe d earlier (se e Part 2 , Chapter I , §6A) , except tha t th e value s w, / an d y on the interval 0 S t S k obe y th e identitie s (S3.2.1) w(t)^w(0); f(t)^f(0); y(t) » 1 - at + 2pw(0)s/T. Here w(0) i s the negative roo t o f equation (6.1.27) , and /(0) i s the root of equation (6.1.25) . This corresponds to the case of an immersed sphere . For an immersed plat e th e computatio n o f f(t) i s unnecessary . The integral s encountere d i n th e evaluatio n ar e o f the for m *7 dr r y dr (S3 J 0ft, r)-=; J 2= I 2 a h Ca e/i = 2>/T (a + ^ by + ~ CT ) - 2\/ a \ + \ <* + \ J 5 2 (S3.2.3!) J 2 = (2a + bt + - ct J . farcsin J| - arcsi n J^J + (b + Ct) (x/WT^a - VWt^y) + % (vW* ~ j(t - 2y) - VWt - a(t - 2a)) , 4 where a = Tia M" + ?)0(« ) — 4ay h = T_Z_^ ; $ = 2_2 ; ^( a) s <£(*,«) , .. .,0(T) = For the numerical solution o f the problem o n th e machin e BESM-2 M a progra m wa s composed , permittin g th e calculatio n withou t passag e to the external memory. I n thi s connectio n w e note th e following . A s i s seen fro m th e descriptio n o f th e algorithm , th e siz e o f th e calculatio n increases strongly wit h th e increas e o f tim e t, sinc e i n passin g fro m t k to t k+x an integra l o f th e for m k jQ F(tk,Tfw(r)9y(tk),y(r))dr must agai n b e computed . Durin g th e calculatio n th e memor y o f th e machine mus t stor e th e value s o f w, f an d y fo r ever y valu e t k; thi s imposes a limitatio n o n th e lengt h o f th e tim e interva l o n whic h th e calculation ca n b e conducte d withou t passag e t o externa l memory , fo r a required degre e o f accuracy . Th e compositio n o f the progra m permit s a calculatio n fo r 54 5 and 34 4 value s t k o n th e lin e an d respectivel y fo r the sphere. I n th e printou t w e ar e give n th e dimensione d tim e r and , corresponding to it , th e dimensionles s y, w, and , i n th e sphere , / . The momen t a t whic h th e iteratio n cycl e i s ende d i s fixed wit h th e aid o f th e inequalitie s \ y*+i(tk) - y H(tk) \ < ey, \w m+1(tk) - w m(tk)\ The choice o f the mes h i s made by the machine without changin g th e basi c progra m determined b y th e chose n system . Essentiall y ou r ai m i s th e step-by-ste p enlargemen t 390 S. 3. NUMERICA L ILLUSTRATION S TABLE 5 Characteristic for m Machine time of the calculation sec. ymax sec. spent on calculation Without iteration 38.98 1.873 129.9 3'30" Without iteration 40.60 1.836 131.5 9'45" Without iteration 41.41 1.814 132.4 36'30" Without iteration 42.22 1.800 130.7 1*24'10" A* = 2~ 6 With iteration 41.93 1.795 130.8 Variant A With iteration 42.82 1.798 131.3 Variant B As w e se e fro m Tabl e 5 , th e magnitud e y^ i s determine d wit h a n accuracy o f 0.01 . Th e sam e accurac y i s obtaine d fo r y a t al l times . The moment o f attainment o f the maximum o f y i s fixed with equivalen t accuracy, whic h i s roughl y 1% o f r^. Th e momen t o f completio n o f the proces s i s fixed wit h grea t relativ e accuracy. 6) Below we give the results o f calculation fo r the dynamic crystallizatio n of a mel t o f carbo n unde r insertio n i n i t o f a plat e an d spher e o f iro n and molybdenum . Th e therma l characteristic s o f th e resultin g carbon , of the iron and o f the molybdenum, ar e given in Table 7 . The calculatio n is carrie d ou t fo r T 0=293°K fo r / = 0.02 m i n th e linea r cas e an d R0 = 0.005 , 0.0 1 an d 0.05 m i n th e spherica l case . (I n th e linea r cas e the calculatio n i s carrie d ou t onl y fo r iron ; i n th e spherica l case , fo r iron an d molybdenum. ) Th e siz e o f 10" 6 • q i s take n a s = 0.06 , 0.12 , 0.36, 0.6 0 an d 1.00J/m 3sec. of th e ste p siz e i n th e initia l calculation , an d th e successiv e reductio n o f th e distanc e from th e boundar y t o th e poin t y = 1 , a n indicatio n o f th e terminatio n o f th e meltin g process o f th e crus t whic h i s crystallize d fro m th e melt . Tables 5 an d 6 refe r t o th e cas e o f immersio n o f a plate . An analogou s resul t ca n b e obtaine d fo r th e spherica l problem . §2. CRYSTALLIZATIO N OF A MELT 391 TABLE 6 Variant A Variant B Mesh Range Number At Mesh Range Number A* of meshes of meshes 1—22 22 2-12 1—8 8 2-i3 11 23—51 28 2" 9—20 12 2-i2 11 52—65 15 2-io 21—43 23 2" 9 66—124 59 2~ 44—61 18 2-io 125—149 25 2~8 62—80 19 2"9 150—186 37 2"7 81—102 22 2"8 187—218 32 2~6 103—143 41 2"7 219—344 126 2"5 144—170 27 2~6 345 1 2~6 171—283 113 2~5 346 1 2"7 284—309 26 2"6 347 1 2"8 310—332 23 2"7 333—339 7 2-8 340—344 5 2"9 345—348 4 2-io TABLE 7 Parameter Carbon phase Iron Molybdenum i = 2 i = 1 i = 1 n-°K 1803 1803 2893 k'J/m • sec • grad 33 70 109 p-kg/m3 7200 7050 10100 C-J/kg • grad 0.69 -103 0.645 • 10 3 0.31 • 103 T-J/kg. 275 • 103 275 • 103 209 • 10 3 The result s o f determinin g y^, r^ an d r k ar e give n i n Table s 8 and 9. Parallel t o th e solutio n o f th e proble m b y thi s precis e treatment , one may use an approximate treatmen t propose d b y M . Ja. Antimiro v (see Part 1 , Chapter I , §3 , and [3]). In the case o f a melt the solutio n is determined b y the equations (1.3.14) , (1.3.10 ) an d (1.3.11)—(1.3.13 ) of Part 1 . In the case of a sphere the solution to the problem was carried S. 3. NUMERICA L ILLUSTRATIONS 200 600 1000 14-00 1800 2200 t FIGURE 8. Cas e of an Iron Plate. 1—" Exact'' solution; 2—quasi-stationar y approximation . S i Q4- I i 0,3 xV^7 \v^ i f;2\ I \ 4r\ /*Ns . q*0.06 ^'j/m 3-sec 0,1 ji*mio9 \w^s 212/0* 01 V ^ 25 50 75 100 125 150 t FIGURE 9. Molybdenu m Sphere, R = 0.0 1 mm. 1—"Exact" solution; 2—quasi-stationar y approximation . §2. CRYSTALLIZATIO N O F A MELT TABLE 8 Iron plate; / = 0.02 m 6 10~ -q ^max ^max Tk 10 ~* -q ymax ^"max n J/m2 • sec sec sec J/m2 • sec sec sec 0.06 3.61 440 2420 0.60 2.10 70 218 0.12 3.12 275 1187 1.00 1.80 42 111 0.36 2.42 110 374 TABLE 9 Sphere of molybdenum and iron Molybdenum Iron 6 R0,m q • 10~ ^max ^~max n J^max Tmax n J/m2 • sec sec sec sec sec 0.005 1.00 1.387 0.8 5.0 1.437 1.3 6.5 0.60 1.417 1.1 8.4 1.517 1.6 10.8 0.36 1.440 1.4 14.0 1.552 2.1 18.0 0.12 1.469 1.8 42.0 1.600 2.9 54.0 0.06 1.493 2.2 84.0 1.631 3.6 108 0.01 1.00 1.338 2.3 10.1 1.413 3.5 13.0 0.60 1.374 3.1 16.8 1.455 4.6 21.6 0.36 1.405 4.0 28.0 1.504 6.2 36.0 0.12 1.440 6.0 84.0 1.574 9.0 108 0.06 1.488 7.5 168 1.601 12.0 216 0.05 1.00 1.191 19.0 58.0 1.211 25.0 76.0 0.60 1.243 28.0 95.0 1.275 40.0 127 0.36 1.290 42.0 153 1.343 60.0 208 0.12 1.380 | 76. 0 i 42 7 | 1.46 5 115 568 1 0.0 6 1.421 108 845 1.528 1 168 1105 out b y Antimiro v b y integratio n o f a syste m o f ordinar y differentia l equations. The dependenc e o f th e thicknes s o f th e crus t o n th e tim e i s foun d for th e "exact " solutio n o f th e proble m i n th e quasi-stationar y treat - ment represente d i n Figure s 8 and 9 . W e se e tha t afte r th e maximu m thickness is attained, the crust curve x = y(t) turn s ou t t o b e essentiall y linear, an d th e tangen t o f it s angl e o f inclinatio n coincide s wit h th e 394 S. 3. NUMERICA L ILLUSTRATIONS corresponding valu e a , define d b y (5.5.1 ) o f Par t 2 . Thus , t o withi n sufficiently hig h accuracy , w(t) vanishes . B y ou r assumption s du/dx\ x=0 = 0 , which shows that t o withi n sufficientl y hig h accurac y i t i s possibl e to assum e that durin g som e time afte r th e passag e o f the maximu m o f the boundary o f the thir d phas e th e temperatur e o f th e immerse d bod y and th e crystallize d crus t upo n i t ar e practicall y dependen t onl y o n time, remainin g spatiall y constant . I t i s natura l tha t th e duratio n o f the tim e interva l o n whic h thi s take s plac e i s smalle r fo r a large r flux of heat fro m th e bod y o f th e plat e imbedde d i n th e mel t o f th e body . At th e sam e tim e i t increase s wit h th e thicknes s o f th e plate , o r wit h the radius o f the sphere . We also note the a priori evident fac t that th e duratio n o f th e growt h of the crust o n the bod y immerse d i n th e mel t i s considerabl y les s tha n the duratio n o f it s melting . Thi s property , togethe r wit h th e vanishin g of w immediatel y afte r th e passag e o f th e maximu m boundar y valu e x = y(t), permit s a n importan t reductio n i n th e computin g time . By lookin g a t th e graph s w e se e that i n al l problem s o f relevanc e t o engineers the quasi-stationary approximatio n give s a satisfactor y result , which i s the mor e accurat e th e smalle r th e hea t flux a t th e immerse d body an d th e large r th e dimension s o f thi s body ; thi s justifie s th e us e of th e quasi-stationar y approximatio n fo r approximat e calculation . §3. Dynamic s of growth of a bubble of air in water7' Below w e examin e wit h precisio n th e quasi-stationar y statemen t o f problem (2.1.4x ) —(2.1.44) o f Par t 1 fo r initia l condition s (2.1.11) . I n the quasi-stationary formulation , th e proble m i s reduced t o a n integra l equation (2.1.13 ) o f th e first kin d wit h identifyin g paramete r A and time t. I n th e exac t treatmen t i t i s easil y show n tha t th e proble m i s reduced by the method o f Part 1 , Chapter I to the solution o f the syste m of integra l equation s TABLE 1 0 6 3 3 9 p0 = 1 • 10 dyn/cm; Po = 1.29 3 • 10" g/cm ; P o = aPo; a = 1.29 3 .10~ g/cm; 3 a = 70dyn/cm ; R 0 = M0" cm; 0 = 2a/p 0R0 = 10.14 ; b = 2/8/ 3 = 0.0933; 4 a = b/a = 0.0247 ; c = 0.00346 ; c,(0 ) = 8(p 0 + 2a/R 0) = 3.5 5 - 10" ; (c0 - c 8(0))/8p0 =< y = \-l-0; \ = 0.50; 0.75 ; 1.00 ; 1.25 ; 1.50 . See Par t 1 , Chapte r II , §1 . §3. GROWT H OF A BUBBLE 39 5 TABLE 1 1 (Solution of the system (S3.3.1) — y; solution of equation (2.1.13) — y*) y = - 0.64 ; A = 0.5 0 y = - 20.39 ; X = 0.7 5 y = _ 0.14 ; X = 1.0 0 t y y* t y | y* t y y* 0 l l 0 l l 0 I l 1.0 0.972 0.97 1.0 0.982 0.989 1.0 0.996 0.993 2.0 0.950 0.951 2.0 0.969 0.970 2.0 0.991 0.989 5.0 0.892 0.894 5.24 0.931 0.940 5.0 0.978 0.978 8.2 0.830 0.831 9.24 0.885 0.895 10.0 0.958 0.956 12.2 0.794 0.751 12.24 0.850 0.857 20.0 0.918 0.920 17.0 0.643 0.654 16.24 0.802 0.819 30.0 0.876 0.877 22.6 0.495 0.510 18.24 0.777 0.795 40.0 0.831 0.844 27.4 0.313 0.316 23.24 0.711 0.724 50.0 0.782 0.798 29.0 0.210 0.213 27.24 0.653 0.678 60.0 0.726 0.746 29.6 0.151 0.142 31.24 0.589 0.615 70.0 0.662 0.688 30.0 0.077 0.028 39.24 0.428 0.446 80.0 0.585 0.626 43.24 0.304 0.392 90.0 0.484 0.538 45.99 0.130 0.207 100.0 0.315 0.420 46.24 0.082 0.187 110.0 0.009 0.002 46.50 0.019 0.134 47.14 0.000 1 0.01 8 y = 0.11 ; X = 1.2 5 y = 0.36 ; X = 1.5 0 Remark t y y* t y y* 0 l l 0 l l t i s dimensionless time: 5 1.02 1.01 1.06 1.06 5 D 10 1.03 1.02 10 1.11 1.12 20 1.06 1.11 20 1.20 1.21 -IT 30 1.09 1.16 30 1.29 1.30 50 1.15 1.25 50 1.46 1.46 100 1.30 1.47 100 1.83 1.81 200 1.58 1.85 200 2.47 2.39 300 1.85 2.18 300 3.01 2.86 400 2.10 2.39 400 3.53 3.28 500 2.34 2.55 500 4.00 3.65 396 S.3. NUMERICA L ILLUSTRATIONS 2\ >exp(/ J(yW-i) \ i-y(0\ w(t) = T=A + yy/t (1 - erf- + 20\/ * J^ y(r)E(y(t) - y(r),t- r)dr (S3.3.1) - 2J^w(r) ^-j- x E(y(t) - y(r), t - r)dr; The algorith m fo r th e solutio n o f th e syste m (S3.3.1 ) coincide s wit h the algorith m fo r th e solutio n o f th e proble m o f crystallizatio n o f a melt unde r immersio n o f a plate , investigate d i n th e previou s section . The initia l valu e w is define d a s the roo t o f th e equatio n 2 (S3.3.2) w(0) = -^ exp( - n w\0)) + ^(0)erf( M^(0)), V*" where n = a/(I + 6 ) an d a an d 6 are define d i n agreemen t wit h (2.1.3 ) of Part 1 . The parameter s definin g th e proces s are show n i n Tabl e 10 . TABLE 1 2 y = - 0.14 ; X = 1.0 0 Constant mesh Variable mes h At= 1 i h On th e tim e y(td At y(td interval fo r t 1 1 0.9956 0.9935 0.01 0— 1 2 2 0.9910 0.9888 0.1 1— 5 3 5 0.9784 0.9760 0.5 5— 1 0 4 10 0.9584 0.9559 1.0 10— 4 0 5 20 0.9184 0.9157 5.0 40—100 6 40 0.8314 0.8281 7 50 0.7819 0.7788 8 60 0.7264 0.7240 9 70 0.6625 0.6605 10 80 0.5853 0.5842 11 90 0.4838 0.4859 12 100 0.3151 0.3184 §3. GROWT H O F A BUBBLE 39 7 TABLE 1 3 Component Parameter Air "C5H12 ttC6H14 /1C7H16 Initial i= - 1 i = 0 ; = I i = 2 opening DH • 105cm2/sec 6.9 5.8 4.9 2 2 Di2 • 10 cm /sec 8.4 7.6 7.0 2 dix - g/em 0.630 0.660 0.684 0.647 3 3 di2 .10 g/em 1.293 3.27 3.90 4.55 M g/mol e 29 72 86 100 75.5 3 i>imcm /mole 114 130 146 aL • 106mole/cm3 32.4 10.2 3.2 Xj • 103cal/mole 6.15 6.82 7.41 epical/mole • grad 30.0 34.5 40.8 31.6 el2eal/mole • grad 69 31.4 29.6 Cio/Cm = N i0 0.8 0.1 0.1 4 3 ci0 • 10 mole/cm 71 7 7 dg/cm3 0.647 lj • 104cal/cm • sec • grad 2.94 \ - 104cal/cm • see • grad 5.8 b{ • 103cm2/sec 1 2 2 b2 - 10 em /sec 2 3 Qi = Q2 = Q* = 0 ; a u = 1.0 ; a 21 = 0.919 ; a 12 = 36.2 ; 02 2 = 34.7 ; 00 2 = 38. 0 398 S . 3. NUMERICA L ILLUSTRATIONS The similarit y solution 8) u = A x ( 1 + erf--— -= J ; v = A 2erfc ——-=; (S3.4.1) A ( x \ x ct = A ix ( 1 + erf - -=. ) ; TABLE 1 4 Ax = - 0.01651 , A 2 = - 0.01616 , A u = 0.06990 , A 2i = 0.1104 , Aog = 0.003018 , A 12 = 0.001250 , A& = 0.0004013 , 0 = - 0.064806 . The result s o f th e computatio n ar e give n i n Tabl e 11 . Th e actua l computations o f y ar e fo r th e illustratio n give n i n Tabl e 12 . The computatio n o f y(t) an d y*(t) fo r X > 1 results i n les s accuracy ; this is due to the necessity o f using, together with t, a rapidl y increasin g sequence o f variable mesh lengths, with the ai m o f decreasing th e outla y of machine time . It follow s that th e reductio n i n th e siz e o f th e mes h for X > 1 leads to the approach o f y(t) an d y*(t) t o one another. Grantin g this, and examinin g Table 11 , we find that th e quasi-stationar y approxi - mation y* approximate s the limit y with an error o f not mor e than 10 % for ever y cas e examine d an d o n ever y interva l o f tim e include d i n Table 11 ; thus it s us e i s full y justified . §4. Dynamic s of evaporation of a solution of normal sextane and heptane in normal pentane9 The dynamic s o f evaporatio n o f a syste m #C 5H12: wC 6H14: nC 7Hm can be examined approximatel y (se e Part 1 , Chapter II, §2) . The param - eters o f the proces s an d initia l compositio n o f th e syste m ar e give n i n Table 1 3 [145] . The solution i s constructed b y neglectin g th e term s CJ Q comparabl e t o 1 . 9) See Part 1 , Chapte r II . §4. EVAPORATIO N O F A SOLUTIO N 399 In th e cas e unde r examinatio n th e approximatio n determine d b y equations (2.2.25) , (2.2.26) , (2.2.28 ) an d (2.2.29 ) o f Par t 1 lead s t o the relation s u = friV^ioerfcl ^^— ^ -a^ 0J J ; 2y/t / x v = - 6 2V ^erfc^^—| -a^ 0j J ; (S3.4.2) Ci = a tl\/^i*erfc I — ^— = - a^0 oJ J ; *i = - a^v^Mrfe l — (TT/ T ~ " a°^o)02#0 J/ ; y = 2al2 TABLE 1 5 4>10 = _ 0.002238 , (f> = 0.0005003 , * u = 0.03907 , 4>u = 0.06721 , 4 5 0 = _ 0.446 8 • 10~\ 4>i = - 0.195 6 • 10~ , <£ 2 - 0.656 5 • 10~ , ao20o = - 0.06470 . TABLE 1 6 y, cm. y,cm. l t, days (S3.4.2) (S3.4.1) days (S3.4.2) (S3.4.1) 2 —53.79 —53.88 12 —131.8 — 132.0 4 —76.08 —76.20 14 —142.3 —142.5 6 —93.17 —93.32 16 —152.2 — 152.4 8 —107.6 —107.8 18 — 161.4 — 161.6 10 —120.3 —120.5 20 — 170.1 — 170.4 In Table 1 6 we give the value s o f y calculate d fro m formula s (S3.4.1 ) and (S3.4.2 ) fo r t varyin g fro m 0 t o 2 0 days . Thu s th e relativ e erro r in determining y i n accordance with (S3.4.2 ) compare d to the determina - tion of y i n accordance with (S3.4.1 ) doe s no t excee d 0.2% ; thi s testifie s to th e complet e suitabilit y o f th e quasi-stationar y approximatio n fo r 400 S. 3. NUMERICA L ILLUSTRATION S practical calculatio n o f th e dynami c evaporatio n o f a polycomponen t mixture i n th e atmospher e fo r distributio n b y diffusio n o f th e vapo r concentration. §5. Zona l noncrucible melting of a cylindrical rod.10) We no w presen t th e numerica l solutio n o f th e proble m o f zona l noncrucible meltin g o f a cylindrica l rod , briefl y formulate d i n Par t 2 , Chapter I , §5 . We recal l tha t th e ro d i s movin g forwar d i n a field o f high frequenc y inductio n melting . Th e proble m wil l b e solve d b y th e method o f finite difference s wit h th e us e o f th e metho d o f Oleini k fo r the constructio n o f a generalize d solutio n fo r th e Stefa n proble m (se e Part 2, Chapter VII, §2); we do not rais e the problem o f justifying th e result.n) We wil l writ e z, x , r i n plac e o f z, x, r, puttin g th e ba r ove r th e dimensioned coordinates . W e introduc e th e dimensionles s variable s a s _ r _ z ^x _ a ? r~/r 2~ir x~ir r~]T2'; (83.5.1!) ^ = -12^ b== T¥~; y== '~h—; 1; l< 0 M*-i\ .._"-"n . - (k tT fo r T (53.5.13) a(u)={k x . . 1 r-u + 6 fo r u > 0. In term s o f th e variable s (S3.5.1! ) — (S3.5.13) an d wit h th e ai d o f th e method describe d i n Part 2 , Chapter VII , §2 , problem (1.5.3; ) o f Par t 1 result s i n th e proble m d2u 1 du d 2u _ da(u) (S3.5.20 ar 5" + r Tr + ~dz*~ ~ dr ' 0 10) See Part 1 , Chapte r I , §5 ; Part 2 , Chapte r VII, § 2 and Chapter VIII , §2. Such a justification i s necessary , sinc e th e domai n o f definitio n o f th e solutio n i s unbounded and the boundary conditio n at the surface of the rod is nonlinear . §5. ZONA L NONCRUCIBLE MELTING 401 du 4 (S3.5.22) = y[k(u)u + l] - f(x); x = z + fi T; ~dr du (S3.5.23) lim — = 0 ; O^r^l . i*i— dz The solution u shoul d b e bounde d fo r r = 0 . The conditio n o f bounded - ness ca n b e replace d b y th e conditio n du (S3.5.24) = 0; oo <2 < oo; ^0. ~dr The initial condition is not written, sinc e our interes t i s in the asymptoti c behavior o f th e solutio n fo r r— » oo. We pas s t o th e stationar y syste m o f coordinate s x f r, r , fixed t o th e inductor, an d simultaneousl y w e not e tha t th e problem , pose d fo r a n unbounded domain , ca n b e formulate d fo r a bounde d domai n wit h the additiona l boundar y conditio n (S3.5.25) — =0 fo r Ix l = £ > 0 ; 0 S r S 1 ; r > 0 . dX It i s obviou s tha t suc h a reductio n o f th e domai n — o o < x < o o i s permissible i f B > 0 i s sufficientl y large . Thus w e arriv e a t th e proble m 2 u 1 du d u I d d \ , x l l (S3.5.3X) dr" r dr dx \dr dx) 0 du 4 (S3.5.32) = y(k(u)u + l) - $(x); -B du du (S3.5.33) - n - = 0 fo r r ^ 0 . ~dr r=0 ' dX 1*1-* \f/(x) i n (S3.5.3 2) i s assume d give n a s T . 5 + a/2 x-a/21 R p| arctg — — arctg 1 — (S3.5.4) Hx) = 2 t x + a/ 2 x-a/2\ ,- ' 2TTR arctg - j arct g ; ) ax /:.( d * d Here p = cons t > 0 i s th e powe r radiate d b y th e inductor , a i s th e 402 S. 3. NUMERICA L ILLUSTRATIONS diameter of a cross-sectio n o f th e inducto r an d d i s th e magnitud e o f the tolerance between the inductor and the rod. 12) Followin g the metho d of Oleihi k (se e Par t 2 , Chapte r VII) , w e smoot h th e jum p functio n a(u), i.e . we replac e i t b y th e polygo n U(u)+/S ; ue(-I,i ) by using a smoothing o f it i n a neighborhood o f the verte x (se e Par t 2 , Chapter VIII , §2C) . W e se t da* (S3.5.6!) c(u)-w; u*±L, so that outsid e o f a smal l neighborhoo d o f th e poin t u— ± L 1; u< -L; (S3.5.62) c(w) «2 Thus we arriv e a t th e proble m /d2 Id d 2\ , . / d d\ for 0 < r < 1 ; - £ < x < B; r > 0 ; 4 (S3.5.7) ^ =[ 7(*(u)u + l) -^(x)]r==1; -JB (S3.5.4) give s th e powe r o f th e energ y bein g radiate d ove r th e conductin g surfac e [5], whic h fo r sufficientl y larg e inducto r radiu s satisfactoril y approximate s th e energ y distribution a t th e surfac e o f th e rod . §5. ZONA L NONCRUCIBLE MELTIN G 403 is necessary by virtue o f the nonlinearity o f the problem) i n combinatio n with th e "matri x sweep " method . I t i s know n tha t a pur e iteratio n scheme fo r th e solutio n o f a n ellipti c equatio n converge s extremel y slowly. Th e "matri x sweep " metho d impose s rigi d limitation s o n th e maximum number o f mesh points which can b e use d withou t conversio n to exterior memory o n the fiVCM. Fo r the many-invariant computation , when i t i s required tha t w e obtai n a s larg e a reductio n i n th e machin e time a s possible , a computin g algorith m i s desire d whic h ca n b e use d without passage to the external memory o f the fiVCM. I n ou r cas e thi s goal i s attaine d b y th e solutio n o f th e finite-difference analo g o f th e system (S3.6.7 ) wit h th e us e o f on e o f th e version s o f th e metho d o f splitting o f a two-dimensiona l differenc e paraboli c operato r int o one - dimensional operators . Two schemes have been tested: the scheme o f Peaceman an d Rachfor d [ill] an d th e schem e recommende d b y A . A . Samarski i [152] , [153] , The first o f thes e schemes , especiall y intende d fo r th e solutio n o f th e linear stationary problem , seem s suitabl e whe n th e mes h lengt h i n tim e is chosen no t to o large . I n correspondenc e t o th e theor y o f stationar y conditions, thi s asymptoticall y define d solutio n prove s to b e practicall y independent o f the choic e o f the siz e o f the tim e mesh . A limitatio n o n the value o f the tim e ste p i s still neede d fo r th e sak e o f stability . An additional check on the applicability o f the method i s the following . In the case ^ = 0 the stationary temperature distribution is not dependen t on th e choic e o f th e valu e o f L enterin g int o th e expressio n (S3.5.6 2) for the heat content. In the method o f Peaceman-Rachford thi s conditio n is satisfie d (se e Tabl e 18 , below) . The stationar y condition s attaine d b y th e us e o f Samarski f s schem e are found t o be dependent o n the value o f the time step, and thus fo r it s application on e must necessaril y us e a muc h smalle r mes h tha n i s ac - ceptable fo r th e Peaceman-Rachfor d scheme , i n spit e o f th e greate r stability o f Samarski f s schem e a s compare d t o thei r scheme . A t th e same time , th e stationar y condition s attaine d b y usin g Samarski f s scheme turn out to be essentially dependen t o n th e valu e o f L fo r n = 0 (see Table 18) . Thu s th e schem e o f Samarski i i s unacceptabl e fo r th e solution o f th e proble m unde r consideration . Therefor e th e calculatio n is conducte d b y th e Peaceman-Rachfor d method . We pass to the details of the algorith m an d th e resultin g calculations . We introduc e a differenc e mes h wit h th e spac e node s a t th e point s (Xi,rj) and the time nodes at the points r k, wit h i = 0 , • • -, M; JC 0 = — B; xM = B; j = 0 , • • -,N; r Q = 0 , r N = 1 ; * = 0,1, • • •; r 0 = 0 . W e se t 404 S . 3. NUMERICA L ILLUSTRATION S k k u itj = u (xif rjt tk); cfj = c (u itJ); (S3.5.8) ht = Xi — Xi-i; g = Tj — ry_ x = const ; l k = r k — T^. In additio n t o th e tim e step s wit h integra l indices , w e introduc e tim e steps wit h semi-integra l indices . I n agreemen t wit h th e Peaceman - Rachford schem e th e first o f th e equation s (S3.5.7 ) i s replace d b y th e equations -J- \U>iJ — "ij) - h T——rW+U U i~U > u 2u w u u (53.5.91) ^^-= z?( u-i1-2u - ht ++ ^t+i+1) + ^rr(2 ^ u+i ~ u-i) r*+l/2 „^*+l/ 2 V<^' - "*/ 1/2) + m- « +#2 - "<*-# > 2 (53.5.92) -p-(^- i " ^ + ujtfi ) + ^>^ - a^ii ) 2u\H'J 2u\ll'J 2utf l/2 + hi+i(hi + hi+ x) hi(hi + A I+1) hihi+x The boundar y conditio n o n th e plan e r = 1 i s approximate d b y th e condition (S3.5.9 ) - (aft 1 - uftLi ) = fc - 7(*(^)^+ D'^G^ft 1 + 1) . 3 g The conditio n du/dr = 0 fo r r=0 i s replace d b y th e conditio n d2u d 2u _ _ du du (S3.5.94*) 2 — + — 1 = — + n— for r = 0 , implied b y th e equatio n (S3.5.7) , since , b y virtu e o f th e fac t tha t du/dr = 0 fo r r = 0 , w e hav e 2 :1 du d u lim - - — , o2 r-o r dr dr r=0 The conditio n (S3.5.9* ) i s approximate d b y a differenc e expressio n 1 analogous t o th e expressio n (S3.5.9 2) bu t wit h u*t i replace d b y a** . The analogou s conditio n du/dx\ j x(=B = 0 i s replace d b y th e conditio n §5. ZONA L NONCRUCIBLE MELTIN G 405 (S3.5.9i) fo r i = 0 an d i = N, bu t wit h th e replacemen t i n i t o f u kJ$2 and i4tY 2 b y u\f 12 an d u k_\]j2 respectively . The syste m (S3.6.9; ) i s solve d b y sweepin g i n th e directio n o f th e x-axis i n passag e fro m a tim e plan e wit h a n intege r inde x t o on e wit h a fractiona l index , and fo r passag e fro m a fractiona l plan e t o a n intege r plane i t i s solved b y sweepin g i n th e directio n o f the r-axis . On every interval o f the variabl e x w e have use d 8 0 steps . The valu e B i s chose n suc h tha t th e shap e o f th e ro d doe s no t lea d t o essentiall y distorted isotherm s i n th e domai n o f influenc e o f th e inductor . Finall y B i s taken equa l t o 13.5 . The interva l — 13. 5 < x < 13. 5 i s subdivide d by point s x t suc h tha t 1 fo r i = 1,2 , ,6; 1/2 fo r i = 7,8 , ,16; 1/6 fo r i = 17,18 ; ,22 hi ={ 1/1 2 fo r i = 23,2 4 ,58 1/6 fo r i = 59,6 0 ,64 1/2 fo r i = 65,6 6 ,74 1 fo r i = 75,7 6 ,80 The ste p siz e i n r i s constant , an d take n a s g = 1/12 . Th e valu e L , which determines the character o f th e approximatio n a(u), i s subject t o change. L wa s chose n a s L = 0.0125 , whic h guarantee s tha t eac h pat h is located i n an interval o f 2 or 3 mesh nodes. I n Figur e 1 0 w e sho w th e form o f th e isother m obtaine d i n on e o f th e typica l calculation s o f variants fo r L = 0.02 5 an d L = 0.0125 . Fo r compariso n w e sho w i n r I 1,0 2 Ay' VI as FIGURE 10 . Meltin g isotherms. 1—L = 0.025 ; 2— L = 0.0125 ; 3—Withou t smoothin g a(u), and with an explicit difference scheme . 406 S. 3. NUMERICA L ILLUSTRATIONS the sam e figure th e isother m fo r th e meltin g foun d b y mean s o f th e solution t o th e proble m wit h th e us e o f a n explici t differenc e scheme . All three curves are sufficiently simila r for us to regard the approximatio n L a s satisfactor y fo r carryin g ou t a n engineerin g calculation . TABLE 1 7 Parameters Remarks a = 0. 4 cm; d = 0. 5 cm See (S3.5.4) . Th e paramete r p change s from cas e t o case . c = 0.9 6 volt • sec/g • grad See definitio n (S3.5.1) . Th e radiu s R o f kx = 0. 3 volt/cm • grad the rod , it s velocit y v and th e rati o o f th e P = 2. 3 g/cm3 coefficients o f conductivit y o f th e soli d A = 180 0 volt - sec/g and liqui d phase s ki/k 2 chang e fro m case to case . ff = 3. 6 • 10 "12 volt/cm2 • grad a include s a correctio n fo r dullnes s TABLE 1 8 L = 0 L = 0.0 5 Remarks uA uB Uc uA UB Uc 0 0.031 0.025 0.028 0.006 0.006 0.027 The calculatio n 1/12 0.028 0.022 0.026 0.005 0.004 0.026 is performe d 2/12 0.022 0.016 0.022 0.000 0.000 0.022 for tim e ste p 3/12 0.012 0.007 0.015 —0.007 —0.008 0.014 lk = 0.025 ; 4/12 —0.001 —0.004 0.005 —0.017 —0.018 0.005 5/12 —0.015 —0.018 —0.007 —0.030 —0.030 —0.007 ki/k2 = 1 ; 6/12 —0.031 —0.033 —0.020 —0.044 —0.044 —0.020 7/12 —0.048 —0.050 —0.035 —0.056 —0.057 —0.035 R = 1. 5 cm; 8/12 —0.066 —0.067 —0.051 —0.073 —0.074 —0.051 9/12 —0.084 —0.084 —0.068 —0.090 —0.091 —0.068 p = 170 0 volt All the calculation s hav e bee n conducte d fo r silicon . Th e numerica l parameter value s enterin g int o th e calculation s ar e give n i n Tabl e 17 . In Tabl e 1 8 w e sho w th e result s o f compariso n o f th e effec t o f in - clusion i n th e calculatio n o f th e smoothe d uni t functio n a(u) i n th e schemes o f Samarski i an d Peaceman-Rachford . I n th e tabulatio n UAf U B an d U c denot e th e value s o f th e stationar y temperatur e o n the axi s o f th e rod , evaluate d b y Samarskii' s schem e (U A an d U B) §5. ZONA L NONCRUCIBLE MELTIN G 407 and b y th e Peaceman-Rachfor d schem e (U c). Her e U A i s obtaine d a s the limit fo r k— > < » of the values Uit0 for k a n integer , an d U B, th e limi t k +1/2 13) of th e half-su m \(u ip + uf 0 ). Th e colum n L = 0 correspond s t o the truncatio n fo r smoothe d a(u). Th e column s L ^0 ar e truncated. 14) As we se e fro m th e table , th e truncatio n fo r th e smoothe d functio n is not foun d t o hav e a n effec t o n th e valu e o f U c withi n thre e digits . Conversely, the value s U A an d U B chang e ver y muc h unde r a chang e of L = 0 t o L = 0.05 . I n particular , th e chang e i n L displace s th e position of the isotherm o f the melt, which reall y exhibit s th e inadmissi - bility o f usin g Samarskii' s schem e fo r th e solutio n o f th e proble m o f zonal noncrucibl e melting . In Tabl e 1 9 we have tabulate d th e value s o f th e temperatur e u(x,r) computed b y th e Peaceman-Rachfor d metho d wit h th e mes h node s belonging t o a domai n occupie d b y th e melt . Th e calculatio n i s con - ducted fo r R = 1. 5 cm, p = 170 0 volts , v = 5mm/mi n an d kjk 2 = 1. In carryin g ou t th e computatio n i t i s assumed tha t th e stead y stat e is reached i f the maximu m differenc e i n th e value s o f th e temperatur e for two successive integer meshes is not greater tha n 10~ 3. To determin e the form of the isotherm of melting (u = 0) , a linear interpolation betwee n two adjacent lattic e points is constructed. Thi s introduce s a n additiona l error i n determinin g th e unknow n isotherm , whic h differ s fro m cas e t o case. In those case s when th e minimu m distanc e betwee n th e isotherm s of th e mel t an d o f th e crystallizatio n i s sufficientl y large , th e exac t construction o f the isotherm turns ou t t o b e satisfactory . I n thos e case s when the isotherms o f the liquid and soli d come together, i.e. in th e cas e similar t o th e appearanc e o r vanishin g o f th e continuou s mel t o f th e rod, th e accurac y sharpl y decreases . Thi s indicate s th e necessit y o f concentrating th e mesh , particularl y i n th e domai n o f th e phas e transition. The character o f dependence o f the form an d position o f the isotherm s of meltin g an d o f crystallizatio n o n th e speed o f motio n o f th e ro d i s shown in Figure s 1 1 an d 12 . Qualitativel y th e pictur e obtaine d i s th e For th e Dirichle t proble m o f Laplace' s equatio n i n a rectangula r domain , th e value UB automatically give s a mor e accurat e resul t tha n doe s U A. In carrying out the calculation o f the smoothed a{u) i n a neighborhoo d o f a verte x of the polygon , (S3.5.5 ) i s no t used . Instead , w e tak e c(u) = c(u — 0). I n this regard , the probabilit y o f assumin g th e value s u = zf c L a t a nod e o f a smal l mes h fo r suc h a smoothing i s practicall y zero ; henc e thi s ha s n o effec t o n th e resultin g computations . 408 S. 3. NUMERICA L ILLUSTRATIONS 50iOCOQOlfliOK3^Nfl5«Ofl)05«H(NOOOO OXCOCOOCNCOO^iOiOCOOiCO^CNOCN^iO rHOOOOOOr-ITHTHI—IrHOOOOOOOO ^CO OOOCO^rHOCOlOt>000500COlOCO rH C O C O " ^ T H 0> 0 5 i O iO 0 0 _ _ _ O 0 0 C O ^ * C N rH O O O O T-»COiCCOCOCOlC^COr-lOrH OOOOOOOOOOOOOO CO oooooooooooooooooooo H ^ * l O ^00Wi0C005t^HOkO?0OiCW^H^ ^w^O!HONP3^iOiO^«(NHOHrtTji co O oqqooqqqqqqqqqqoqo00 CO q CN o o d d d d d d d d d d d d d d d d d d OOOCOlOCOrHOOCNCOCOCOCOCNCNrHQrHCO'** T-t i-ioqqoooqoooqqqooqqoq CN U 5 O ^ 0 0 lO C O i-li-fOC000t>C00005t-lC0THC0 O 0 0 t ^ i O « N H rHtNCNCNCNCNrHOOrHCO^ Of t rH O O O O O O St c> a COt^«05ifJW(NWii5WO)CO^COO^U50lO 0 0 0 C N C O O00O*C^C0CNrHOOOrHrHrHrHOOrH co <*• o o TH q q q q q q q q q q q q q q q q q ^OO^rHI>iCLOC005 ^O>lOCN00I>t>00rH Tf* t > t > L O r H t> © C O l > OOO^CO^MINHH O © O O ~ ~ C N C O ^ THO qqooqqq © © © © 88 o o o dddddddd d ooooooooo i0OC0C0O0000OC0l>C^(N'^iO^rH00OC0t> ooit>colOcocN(MrHQOOOOOooc^^co^t, rHqqqoooqoqqqqqoooqoq lOOC0C0O0i00OC000C0 CN FIGURE 11 . Meltin g isotherms: FIGUR E 12 . Meltin g isotherms: R in cm; R in cm» V i n mm/min; k = k x/k2. ^i n mm/min; k = k x/k2. BIBLIOGRAPHY [l] P . B . Afanas'ev , Ja . B . Zel'dovi c an d O . M . Todes , On the spatial distribution of deposits in crystallization of mutually diffusing substances, Z . Fiz . Him . 2 3 (1949) , 156-179. (Russian ) [2] E . L . Albasiny , The solution of nonlinear heat conduction problems on the pilot ACE , Proc. Inst . Electric . Engrg . 10 3 (1956) , Par t B , Suppl . 1 , 158-162 . [3] M . Ja . Antimirov , Approximate solution of the problem of crystallization of a melt with an imbedded plate, cylinder or sphere, Latv . Mat . Ezegodni k 1965 , Izdat . "Zinatne", Riga , 1966 , 175-195 . (Russian ) [4] V . C. Arpaci, I . A . Clark an d P . S . Larsen , The dynamics of gas-vapour bubbles in binary systems, Proc . Roy . Soc . Londo n Ser . A 28 3 (1965) , 50-63 . [5] G . I . Babat , Induction heating of metals and its applications in industry, Gosener - goizdat, Moscow , 1946 . (Russian ) [6] R . D . Baceli s an d V . G . Melamed , Solution of a limiting boundary-value problem to which the generalized Stefan problem can be reduced, Sibirsk . Mat . Z . 5 (1964) , 738-745. (Russian ) M R 2 9 #4942 . [7] M . Bentwich , Ablation of a two-dimensional semi-infinite ice block in a uniform stream of water, Proc . Fift h Israe l Annua l Conferenc e Aviat . an d Astronaut. , Jerusalem Academi c Press , Te l Aviv , 1963 , pp . 82-86 . [8] I . V . Bocarova , On the asymptotic solution of a problem with a free boundary for the heat conduction equation, Dokl . Akad . Nau k SSS R 14 3 (1962) , 259-26 1 = Sovie t Math. Dokl . 3 (1962) , 365-367 . [9] B . A . Boley , A method of heat conduction analysis of melting and solidification problems, J . Math . Phys . 4 0 (1961) , 300-313 . [10] , The analysis of a problem of heat conduction and melting, Proc . Thir d Sympos. o n Nava l Structura l Mechanics , Internat . Conferenc e o n Elevate d Temperature Mechanics , Pergamo n Press , Ne w York , 1963 . [ll] , Upper and lower bounds for solution of a melting problem, Quart . Appl . Math. 2 1 (1963) , 1-11 . M R 2 6 #4642 . [12] V . T . Borisov , The two-phase zone for crystallization of a melt in a nonstationary condition, Dokl . Akad . Nau k SSS R 14 2 (1962) , 581-58 3 = Sovie t Physic s Dokl . 7 (1962) , 70-71 . [13] M . Brillouin , Sur quelques probldmes non-re'solues de la physique mathematique classique. Propagation de la fusion, Ann . Inst . H . Poincar e 1 (1931) , 285-308 . [14J B . M . Budak , E . N . Solov'e v an d A . B . Uspenskii , A difference method with smoothing of coefficients for the solution of the Stefan problem, Z . Vycisl . Mat . i Mat . Fiz. 5 (1965) , 828-84 0 = USS R Comput . Math , an d Math . Phys . 5 (1965) , no . 5 , 59-76. M R 3 3 # 8109. [15] B . M . Budak , F . P . Vasil'e v an d A . B . Uspenskii , Difference methods for solving certain boundary value problems of Stefan type, Numerica l Method s i n Gas Dynamics , Izdat. Moskov . Univ. , Moscow , 1965 , pp . 139-183 . (Russian ) M R 3 4 #7126 . [16] A . Buikis , L . Rubinstei n an d A . Skroman , Crystallization of a melt with imbedded plate or sphere, Latv . Mat . Ezegodni k 1965 , Izdat . "Zinatne" , Riga , 1966 , 155-173 . (Russian) 410 BIBLIOGRAPHY 411 [17 I. A . Carnyi , On the advance of the boundary of change in aggregate state under cooling and heating of bodies, Izv . Akad. Nau k SSS R Otd . Tehn . Nau k 1948 , no . 2 , 187-202. (Russian ) [18 P. L . Chambre , On the dynamics of phase growth, Quart . J . Mech . Appl . Math . 9 (1956), 224-233 . MR 18 , 358 . [19 S. J. Citron, Heat conduction in a melting slab, J. Aero . Space Sci . 2 7 (1960) , 219-228 . 20 , On the conduction of heat in a melting slab, Proc . Fourt h U . S . Nat . Congress, Appl . Mech . (Univ . o f California , Berkeley , 1962) , vol . 2 , Amer . Soc . Mech. Engrs. , Ne w York , 1962 , pp . 1221-1228 . [21 I. Crank , Two methods for the numerical solution of moving-boundary problems in diffusion and heat flow, Quart . J . Mech . Appl . Math . 1 0 (1957) , 220-231 . [22 A. B. Dacev , On the linear problem of Stefan, Dokl . Akad . Nau k SSS R 5 8 (1947) , 563-566. (Russian ) M R 9 , 513 . [23 , Sur le probldme liniaire de Stefan, Ann . Univ . Sofi a Fac . Sci . Math . Phys. 4 5 (1949) , 321-352 . M R 12 , 504 . [24 , On the linear problem of Stefan. The case of two phases of infinite thickness, Dokl. Akad . Nau k SSS R 7 4 (1950) , 445-448 . (Russian ) M R 12 , 263. [25 , On the linear problem of Stefan. The case of alternating phases, Dokl . Akad . Nauk SSS R 7 5 (1950) , 631-634 . (Russian ) M R 12 , 710 . [26 , On the two-dimensional Stefan problem, Dokl . Akad . Nau k SSS R 10 1 (1955), 441-444 . (Russian ) M R 17 , 624 . [27 , On the three-dimensional problem of Stefan, Dokl . Akad . Nau k SSS R 10 1 (1955), 629-632 . (Russian ) M R 17 , 624 . [28 P. V . Danckwerts , Unsteady-state diffusion or heat-conduction with moving boundary, Trans. Farada y Soc . 4 6 (1950) , 701-712 . [29 V. L . Danilov , Integro-differential equations of motion for the interface between two fluids in a porous medium, Izv . Kazan. Fil . Akad. Nauk SSS R Ser . Fiz.-Mat . Tehn . Nauk 1 1 (1957) , 99-133 . (Russian ) [30 V. A . Ditki n an d P . I . Kuznecov , Handbook of operational calculus. Fundamentals of the theory and tables of formulas, GITTL , Moscow , 1951 . (Russian ) [31 J. Douglas , Jr. , A uniqueness theorem for the solution of a Stefan problem, Proc . Amer. Math . Soc . 8 (1957) , 402-408 . M R 19 , 1060 . [32 J. Douglas , Jr. an d T . M . Gallie , On the numerical integration of a parabolic differ- ential equation subject to a moving boundary condition, Duk e Math . J . 2 2 (1955) , 557-571. M R 17 , 1241 . [33 P. S . Epstei n an d M . S . Plesset , On the stability of gas bubbles in liquid-gas solu- tion, J . Chem . Phys . 1 8 (1950) , 1505-1509 . [34 G. W. Evan s II , A note on the existence of a solution to a problem of Stefan, Quart . Appl. Math . 9 (1951) , 185-193 . M R 13 , 243 . [35 G. W. Evan s II , E . Isaacso n an d J. K . L . MacDonald , Stefan-like problems, Quart . Appl. Math . 8 (1950) , 312-319 . M R 12 , 263 . [36 H. K . Forster , Diffusion in a moving medium with time-dependent boundaries, A. I. Ch . E . J. 3 (1957) , 535-538 . M R 2 0 #654 . 37 H. K . Forste r an d N . Zuber , Growth of a vapor bubble in a superheated liquid, J . Appl. Phys . 2 5 (1954) , 474-478 . 38 K. E. Forster , Growth of a vapor-filled cavity near a heating surface and some related questions, Phys . Fluid s 4 (1961) , 448-455 . [39 F. C . Frank , Radially symmetric phase growth controlled by diffusion, Proc . Roy . Soc. Londo n Ser . A 20 1 (1950) , 586-599 . M R 12 , 184 . [40 A. Friedman , Free boundary problems for parabolic equations. I , Melting of solids, J. Math . Mech . 8 (1959) , 499-517 . M R 2 6 #1626 . 412 BIBLIOGRAPHY , Free boundary problems for parabolic equations. II , Evaporation or con- densation of a liquid drop, J . Math . Mech . 9 (1960) , 19-66 . M R 2 6 #1627 . , Free boundary problems for parabolic equations. Ill , Dissolution of a gas bubble in liquid, J . Math . Mech . 9 (1960) , 327-345 . M R 2 6 #1628 . , Remarks on Stefan-type free boundary problems for parabolic equations, J. Math . Mech . 9 (1960) , 885-903 . M R 2 6 #1629 . , Remarks on the maximum principle for parabolic equations and its appli- cations, Pacifi c J . Math . 8 (1958) , 201-211 . MR2 1 #1444 . H. L . Frisch , Durch Diffusion kontrolliertes Phasenwachstum, Z . Electrochem . 5 6 (1952), 324-326 . H. L. Frisch and F. C . Collins , Diffusional processes in the growth of aerosol particles, J. Chem . Phys . 2 0 (1952) , 1797-1803 . I. V . Frjazinov , On the Stefan problem for inhomogeneous media, Z . Vycisl . Mat . i Mat. Fiz . 1 (1961) , 927-93 2 = USS R Comput . Math , an d Math . Phys . 1 (1962) , 1097-1103. M. G . Gadzieva , On an approximative method for solution of the one-dimensional Stefan problem, Ucen . Zap . Azerbaidzan . Gos . Univ . No . 4 (1955) , 3-5 . (Russian ) D. Geis t an d U . Dehlinger , Warmeleitung bei Phasenumwandlungen, Z . Natur . 6 (1949) , 415-423 . M. Gevrey , Equation aux dirivies partielles du type paraboUque, J . Math . Pure s Appl. (6 ) 9 (1913) . T. R . Goodman , Aerodynamic ablation of melting bodies, Proc . Thir d U . S . Nat . Congress Appl . Mech. , Ne w York , 1958 , pp . 735-745 . T. R . Goodma n an d J . J . Schea , The melting of a finite slab, J . Appl . Mech . 2 7 (1960), 16-24 . MR2 8 #1873 . E. Goursat, Cours d'analyse. Vol . 3 , par t I , 5t h ed. , Gauthier-Villars , Paris , 1956 ; Russian transl. , GITTL, Moscow , 1933 , p. 245 ; English transl. , Dover , Ne w York , 1963. E. Gra y an d G . B . Mathews , A treatise on Bessel functions and their applications to physics, Macmillan , Ne w York , 1895 ; Russia n transl. , IL , Moscow , 1949 . I. R. Griffi n an d D. R . Coughanowr , Analysis of phase-boundary motion in diffusion- controlled processes. I ; Solution of the diffusion equation with a moving boundary. II; Application to evaporation from a flat surface. Ill ; Penetration of metal ions into cellulose xanthate fibers and growth of sulfuric acid droplets in humid air, A . I . Ch . E. J . 1 1 (1965) , 133-137 , 151-154 , 246-253 . S. S.Grigorjan , On heating and melting of a solid body owing to friction, Prikl . Mat . Meh. 22 (1958), 577-585 = J . Appl. Math. Mech . 2 2 (1958) , 815-825. MR 2 1 # 7730 . L. K . Gusacenko , On a problem with nonstationary rate of heating (melting) of a semi-infinite solid body, Inz . Fiz . Z . 9 (1962) , 99-101 . (Russian ) R. Gutowski , The problem of motion of the freezing front in liquids, Arch . Mech . Stos. 1 5 (1963) , 167-182 . The motion of the freezing front in a liquid taking into account the de- pendency of thermal parameters on the temperature, Arch . Mech . Stos . 1 6 (1964) , 49-63. R. Hicklin g an d M . S . Plesset , Collapse and rebound of a spherical bubble, in water, Phys. Fluid s 7 (1964) , 7-14 . A. Huber, Hauptaufsdtze uber das Fortschreiten der Schmelzgrenze in einem linearen Leiter, Z . Angew . Math . Mech . 1 9 (1939) , 1-21 . A. M . Il'in , A . S . Kalasniko v an d O . A . Oleinik , Second-order linear equations of parabolic type, Uspeh i Mat . Nau k 1 7 (1962) , no . 3 (105) , 3-14 6 = Russia n Math . Surveys 1 7 (1962) , no . 3 , 1-143 . M R 2 5 #2328 . BIBLIOGRAPHY 413 [63] A . Ju. Islinski i an d G . I . Barenblatt , On the impact of a viscous-plastic rod against a rigid barrier, Dokl . Akad . Nau k SSS R 14 4 (1962) , 734-73 7 = Sovie t Physic s Dokl. 7 (1962) , 515-517 . [64] G . P . Ivancov , The temperature field around a spherical, cylindrical or pointed crystal growing in a cooling solution, Dokl . Akad . Nau k SSS R 5 8 (1947) , 567-569 . (Russian) M R 9 , 439 . [65] , Solution of the cooling problem for a half-space filled with a material with n critical points and differing thermal properties in the intervals between them, Dokl . Akad. Nau k SSS R 5 9 (1947) . (Russian ) [66] , ''Diffusion" supercooling with crystallization of a binary alloy, Dokl . Akad. Nau k SSS R 8 1 (1951) , 179-181 . (Russian ) [67] E . A. Ivanov, Remarks on the article by I. G. Portnov "Exact solution of a problem of freezing with arbitrary change of temperature on a fixed surface", Izv . Akad . Nauk SSSR Otdel . Tehn. Nau k Meh . Masinostr . No . 1 (1963) , 129-131 . (Russian ) [68] N . S . Ivanov , On modelling of heat processes in frozen strata of the earth's crust, Processes o f Hea t an d Mas s Exchang e i n Froze n Rocks , "Nauka" , Moscow , 1965. (Russian ) [69] V . V . Ivanov , On the growth of drops under condensation, Izv . Vyss . Skol y Fiz . 3 (1962) , 62-64 . (Russian ) [70] I . L. Ivanova , The heat-conduction equation for a rod of bounded length with movable boundaries, Trud y Meh . Mat . Fak . Kazah . Univ . 1 (1960) , no . 2 , 60-69 . (Russian ) [71] Jiang Li-shang , The two phase Stefan problem. I , Chines e Math . Act a 4 (1964) , 686-702; II, ibid . 5 (1964) , 36-53 . (Englis h transl . o f Act a Math . Sinic a 1 3 (1963) , 631-646; ibid . 1 4 (1964) , 33-49. ) [72] , The proper posing of free boundary problems for nonlinear parabolic differential equations, Chines e Math . Act a 3 (1963) , 399-418 . (Englis h transl . of Act a Math . Sinic a 1 2 (1962) , 368-388. ) [73] S . L. Kamenomostskaja , On Stefan's problem, Nauen . Dokl . Vyss . Skol y 1 (1958) , no. 1 , 60-62; se e also : Mat . Sb . 5 3 (95 ) (1961) , 489-514 . (Russian ) M R 2 5 #5292 . [74] L . I . Kamynin , On the existence of the solution of Verigin's problem, Z . Vycisl . Mat. i Mat . Fiz . 2 (1962) , 833-85 8 = USS R Comput . Math , an d Math . Phys . 2 (1963) , 954-987 . M R 2 8 #2364 . [75] A hydraulics problem, Dokl . Akad . Nau k SSS R 14 3 (1962) , 779-78 1 = Soviet Math . Dokl . 3 (1962) , 480-482 . M R 2 4 #A2758 . [76] , On the linear Verigin problem, Dokl . Akad . Nau k SSS R 15 0 (1963) , 1210-1213 = Sovie t Math . Dokl . 4 (1963) , 847-851 . M R 2 7 #1717 . [77] O . D . Kellog , Foundations of potential theory, Springer , Berlin , 1929 . [78] A . G . Kolesnikov , On a change in the mathematical formulation of the problem of soil freezing, Dokl . Akad . Nau k SSS R 8 2 (1952) , 889-891 . (Russian ) [79] A . N . Kolmogoro v an d S . V . Fomin , Elements of the theory of functions and of functional analysis. II . Measure, Lebesgue integral, Hilbert space, Izdat . Moskov . Univ., Moscow , 1960 ; Englis h transl. , Grayloc k Press , Albany , N . Y. ; Academi c Press, New York , 1961 . MR 2 2 #9565 ; 2 2 # 9566a,b . [80] I . I . Kolodner , Free boundary problem for the heat equation with applications to problems of change of phase, Comm . Pur e Appl . Math . 1 0 (1957) , 220-231 . [81] S . S . Kovner , On a problem of heat conduction, Z . Geofiz . 3 (1933) , no . 1 , 32-41 . (Russian) [82] , On the freezing of a sphere, Transaction s o f th e Firs t Geological-Surve y Conference o f th e Principa l Norther n Se a Route . 3 . Permanen t frost , Leningrad , 1936. (Russian ) [83] F. Kreith an d F . E. Romie , A study of the thermal diffusion equation with boundary 414 BIBLIOGRAPHY conditions corresponding to solidification or melting of material initially at fusion temperature, Proc . Phys . Soc . Londo n Sect . B 6 8 (1955) , 277-291 . Lien-kun Ku , The behavior of the solution of Stefan's problem when time increases unboundedly, Dokl . Akad . Nau k SSS R 13 8 (1961) , 263-26 6 = Sovie t Math . Dokl. 2 (1961) , 570-573 . M R 29 #380 . W. T. Kyner , On a free boundary value problem for the heat equation, Quart . Appl . Math. 1 7 (1959) , 305-310 . M R 23 #A1165 . , An existence and uniqueness theorem for a nonlinear Stefan problem, J. Math . Mech . 8 (1959) , 483-498 . M R 2 6 #1630 . K. Lachmann , Zum Problem des Erstarrens fur den durch zwei parallele Ebenen begrenzten Korper, Z . Angew . Math . Mech . 1 5 (1935) , 345-358 . G. Lam e an d B . P . Clapeyron , Mimoire sur la solidification par refroidissement d'un globe solide, Ann . Chem . Phys . 4 7 (1831) , 250-256 . H. G . Landau , Heat conduction in a melting solid, Quart . Appl . Math . 8 (1950) , 81-94. M R 11 , 441 . W. E . Langlois , Similarity rules for isothermal bubble growth, J . Flui d Mech . 1 5 (1963), 111-118 . P. S. Larsen , J. A . Clark , W . O . Randolph an d I . L . Vaniman , Cooling of cryogenic liquids by gas injection, Advance s i n Cryogeni c Engineering , vol . 8 , New York , 1963 . L. S . Leibenzon , Handbook on petroleum mechanics, GNTI , Moscow , 1931 ; se e also: Collected works. Vol . 3 , Izdat . Akad . Nau k SSSR , Moscow , 1955 , pp . 435-439 . (Russian) , On the hardening of the sphere of the Earth from its original fused state, Izv. Akad . Nau k SSS R Ser . Geograf . i . Geofiz . 1933 , no. 6 ; se e also : Collected works. Vol. 4 , Izdat . Akad . Nau k SSSR , Moscow , 1955 , pp . 317-359 . (Russian ) V. Li , On closed solutions of the many-dimensional Stefan problem, Akad . Nau k Kazan. SS R Trud y Sekt . Mat . Meh . 2 (1963) , 71-72 . (Russian ) M R 32 #2747 . N. M . H . Lightfoot , The solidification of molten steel, Proc . Londo n Math . Soc . (2) 3 1 (1930) , 97-116 . M. Lotkin , The calculation of heat flow in melting solids, Quart . Appl . Math . 1 8 (1960/61), 79-85 . M R 2 2 #4120 . G. A . Mart y no v, On propagation of heat in a two-phase medium for given law of motion of the phase boundary, Z . Tehn . Fiz . 2 5 (1955) , 1754-1767 . (Russian ) MR 19 , 710 . On the solution of the inverse problem of Stefan for the semispace when the phase boundary moves according to a linear law, Dokl . Akad . Nau k SSS R 10 9 (1956), 279-282 . (Russian ) M R 18 , 314 . V. G . Melamed , Reduction of the Stefan problem to a system of ordinary differential equations, Izv . Akad. Nauk SSSR Ser. Geofiz. 1958,848-869. (Russian) MR 23 # A1163 . Solution of the Stefan problem in the case of the second boundary problem, Vestnik Moskov . Univ . Ser . Mat . Meh . Astronom . Fiz . Him . 1959 , no . 1 , 17-22 . (Russian) M R 23 #A1164 . J. W . Miles , A note on Riemann's method applied to the diffusion equation, Quart . Appl. Math . 8 (1950) , 95-101 . M R 11 , 521. W. L . Miranker , A free boundary value problem for the heat equations, Quart . Appl . Math. 1 6 (1958) , 121-130 . M R 20 #656 . W. L . Miranke r an d J . B . Keller , The Stefan problem for a non-linear equation, J. Math . Mech . 9 (1960) , 67-70 . M R 22 #1331 . W. D. Murra y an d Fre d Landis , Numerical and machine solution of transient heat- conduction problems involving melting or freezing. I , Trans . ASM E Ser . C = J . Heat Transfe r 8 1 (1959) , 106-112 . BIBLIOGRAPHY 415 [105] V. V . Nemyekii , The fixed-point method in analysis, Uspeh i Mat . Nau k 1 (1936) , 141-174. (Russian ) [106] L. Nirenberg , A strong maximum principle for parabolic equations, Comm. Pur e Appl. Math . 6 (1953) , 167-177 . M R 14 , 1089 ; 16 , 1336 . [107] O . A . Oleinik , A method of solution of the general Stefan problem, Dokl . Akad . Nauk SSS R 13 5 (1960) , 1054-105 7 = Sovie t Math . Dokl . 1 (1960) , 1350-1354 . MR 23 #A2644 . [108] , Solution of fundamental boundary value problems for second order equations with discontinuous coefficients, Dokl. Akad . Nau k SSS R 12 4 (1959) , 1219-1222 . (Russian) M R 21 #1442 . [109] , The Stefan problem, Firs t Summe r Schoo l i n Math. , vol . 2 , "Naukov a Dumka", Kiev , 1964 . (Russian ) [110] O . A . Oleini k an d S . N . Kruzkov , On some nonlinear problems for equations of elliptic type, Uspeh i Mat . Nau k 1 5 (1960) , no . 5 , 203-205 . (Russian ) [ill] D . W . Peacema n an d H . H . Rachford , The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math. 3 (1955), 28-41. MR 17 , 196. [112] I. Petrovsky , Zur ersten Randwertaufgabe der Warmekitungsgkichung, Compositio. Math. 1 (1935) , 383-419 . [113] B. Pini , Maggioranti e minoranti delk soluzioni delk equazioni paraboliche, Ann. Mat. Pur a App l (4 ) 3 7 (1954) , 249-264 . M R 16 , 593 . [114] M . S . Plesse t an d S . A . Zwick , The growth of vapor bubbks in superheated liquids, J. Appl . Phys . 2 5 (1954) , 493-500 . [115] G . Poots , An approximate treatment of a heat conduction probkm involving a two- dimensional solidification front, Internat. J . Hea t Mas s Transfe r 5 (1962) , 339-348 . [116] , On the application of integral methods to the solution of problem involving the solidification of liquids initially at fusion temperature, Internat. J . Hea t an d Mass Transfe r 5 (1962) , 525-531 . [117] I. G . Portnov, On a method of solving certain boundary value probkms with unknown boundary, Vestnik Moskov . Univ . Ser . I Mat . Meh . 1962 , no. 3 , 67-70. (Russian ) MR 27 #1719 . [118] , An exact solution of the freezing probkm with arbitrary variation of the temperature on a fixed boundary, Dokl. Akad . Nau k SSS R 14 3 (1962) , 559-56 2 = Soviet Physic s Doklad y 7 (1962) , 186-189 . [119] D. Quilghini , Sul comportamento asintotico delk soluzioni di un probkma del tipo di Stefan, Atti Sem. Mat. Fis. Univ. Modena 1 2 (1962/63), 107-120 . MR 28 #2786. [120] , Su di un nuovo probkma del tipo di Stefan, Ann . Mat . Pur a Appl . (4 ) 62 (1963) , 59-97 . M R 28 #3240 . [121] , Un teorema di unicita per un probkma del tipo di Stefan, Boll. Un . Mat . Ital. (3 ) 1 8 (1963) , 270-278 . M R 28 #1414 . [122] H. Reis s an d V . K . L a Mer , Diffusional boundary value probkms involving moving boundaries, connected with the growth of colloidal particks, J. Chem. Phys. 18 (1950), 1-12. [123] M . E. Rose , A method for calculating solutions of parabolic equations with a free boundary, Math. Comp . 1 4 (1960) , 249-256 . M R 22 #6085 . [124] L. I . Rubinstein , On the solution of Stefan's probkm, Izv . Akad . Nau k SSS R Ser . Geograf. Geofiz . 1 1 (1947) , 37-54 . (Russian ) M R 8, 516 . [125] , On the determination of the position of the boundary which separates two phases in the one-dimensional probkm of Stefan, Dokl . Akad . Nau k SSS R 5 8 (1947), 217-220 . (Russian ) MR9 , 287 . [126] , On the stability of the boundary of the phases in a two-phase heat-conduction medium, Izv. Akad. Nauk SSS R Ser . Geograf. Geofiz. 1 2 (1948), 557-560. (Russian) MR 10 , 458 . 416 BIBLIOGRAPHY [127] , On the initial velocity of the front of crystallization in the one-dimensional problem of Stefan, Dokl . Akad. Nauk SSSR 6 2 (1948), 753-756. (Russian) M R 10 , 254. [128] , On the propagation of heat in a two-phase system having cylindrical symmetry, Dokl. Akad. Nau k SSS R 7 9 (1951) , 945-948 . (Russian ) M R 13 , 243. [129] , On the propagation of heat in a stratified medium with varying phase state, Dokl. Akad. Nau k SSS R 7 9 (1951) , 221-224 . (Russian ) M R 13 , 134 . [130] , Concerning the existence of a solution of Stefan's problem, Dokl . Akad . Nauk SSS R 6 2 (1948) , 195-198 . (Russian ) M R 10 , 196 . [131] , On some nonlinear problems arising from the Fourier equation, Dissertation , Moscow Stat e University , Moscow , 1957 . (Russian ) [132] , On the asymptotic behavior of the phase separation boundary in the one- dimensional problem of Stefan, Dokl . Akad. Nau k SSS R 7 7 (1951) , 37-40 . (Russian ) MR 12 , 710 . [133] , On the uniqueness of solution of the homogeneous problem of Stefan in the case of a single-phase initial condition of the heat conducting medium, Dokl . Akad . Nauk SSS R 7 9 (1951) , 45-47 . (Russian ) M R 13 , 243. 134] , On the question of the uniqueness of the solution of the one-dimensional problem of Stefan in the case of a single-phase initial condition of the heat conducting medium, Dokl . Akad . Nau k SSS R 11 7 (1957) , 387-390 . (Russian ) 1135] , A numerical solution of the integral equations of the Stefan problem, Izv . Vyss. Ucebn. Zaved . Matematik a 1958 , no. 4 (5) , 202-214 . (Russian ) M R 2 5 #768 . 136] , On a case of Stefan's problem, Dokl . Akad. Nauk SSS R 14 2 (1962), 576-577 = Soviet Physic s Dokl . 7 (1962) , 21-22 . M R 2 5 #4258 . Se e also : On a case of the one-dimensional Stefan problem with forced nonlinearity, Ucen . Zap . Latv . Gos . Univ. P . Stuck i 47 , Trudy Vycisl . Centr . no . 1 , 1963 , 163-217 . (Russian ) 137] , On frictional heating and melting of a solid, Dokl . Akad . Nau k SSS R 142 (1962) , 1061-106 4 - Sovie t Physic s Dokl . 7 (1962) , 106-109 . 138] , On the solution of Verigin's problem, Dokl . Akad . Nau k SSS R 11 3 (1957) , 50-53. (Russian ) M R 19 , 774. Se e also : On determining the position of the boundary between two slightly compressible liquids filtering through a deformable porous medium, Trudy Ufim . Neft . Inst . No . 1 (1956) , 75-108 . (Russian ) 139] , On a case of filtration of two slightly compressed fluids through a deformed porous medium, Izv . Vyss . Skol y Matematik a 1959 , no . 1 (8) , 174-179 . (Russian ) 140] , On determination of the boundary separating phases in a two-phase heat- conducting medium in a steady heat regime, Dokl . Akad . Nau k SSS R 10 5 (1955) , 437-438. (Russian ) M R 17 , 625 . 141] , The uniqueness of the solution to a Stefan type two-layer single-phase problem, Dokl . Akad . Nau k SSS R 16 0 (1965) , 1019-102 2 = Sovie t Math . Dokl . 6 (1965), 248-252 . MR3 0 #3311 . Se e also : Latv . Mat . Ezegodni k 1965 , Izdat . "Zinatne", Riga , 1966 , 131-154 . (Russian ) 142] , On the temperature field of a geological stratum under heat injection, Ucen . Zap. Kazan . Gos . Univ . 12 1 (1961) , no . 5 , 129-156 . (Russian ) 143] , The two-phase Stefan problem on a line-segment with one-phase initial state of the heat-conduction medium, Ucen . Zap . Latv . Gos . Univ . P . Stuck i 5 8 (1964), Trud y Vycisl . Centr . No . 2 , 111-147 . (Russian ) 144] , On the question of the process of propagation of freezing in frozen soil, Izv. Akad. Nauk SSSR Ser. Geograf. Geofiz. 1 1 (1947), 489-496. (Russian) M R 9 , 287. 145] , On the dynamics of one-dimensional anisothermal evaporation of ideal fluid mixtures, Ucen . Zap . Latv . Gos . Univ . P . Stuck i 5 8 (1964) , Trud y Vycisl . Centr. No . 2 , 177-226 . (Russian ) 148] A . L . Ruoff , An alternate solution of Stefan's problem, Quart . App L Math . 1 6 (1958), 197-201 . M R 19 , 1230 . BIBLIOGRAPHY 417 147] A . I . Safroncik , Nonstationary flow of a visco-plastic material between parallel walls, Prikl . Mat . Meh . 2 3 (1959) , 925-93 5 = J . Appl . Math . Mech . 2 3 (1959) , 1314-1327. [148] , Rotation of a cylinder with a variable angular velocity in a visco-plastic medium, Prikl . Mat . Meh . 2 3 (1959) , 1051-105 6 = J . Appl . Math . Mech . 2 3 (1959) , 1504-1511. M R 2 2 #11685 . 149] E . M . Sahov , One-dimensional nonstationary heating and melting of a solid body under its motion in a gas along one of its plane surfaces, Inz . Z . 1 (1961) , no . 3 , 46-59. (Russian ) 150] , On the Stefan problem, Akad . Nau k SSSR , Trud y Inst . Istor . Estest . i Tehn. 3 4 (1960) , 512-526 . (Russian ) 151] , Establishment of a constant rate of melting of a semi-infinite solid body, Inz. Z . 2 (1962) , no . 4 , 237-244 . (Russian ) [152] A. A . Samarskii , An efficient difference method for solving a multi-dimensional parabolic equation in an arbitrary domain, Z . Vycisl . Mat . i Mat . Fiz . 2 (1962) , 787-811 - USS R Comput. Math, and Math . Phys . 2 (1963) , 894-926. MR 3 2 #609 . 153] A . A. Samarski i an d V . D . Moiseenko , An efficient scheme for the through compu- tation in a many dimensional Stefan problem, Z . Vycisl . Mat . i Mat . Fiz . 5 (1965) , 815-827 = USS R Comput . Math , an d Math . Phys . 5 (1965) , no . 5 , 43-58 . MR 3 4 #3807 . 154] R . W . Sanders , Heat conduction in a melting slab: an exact solution, AR S J . 3 0 (1960), 1030-1031 . 155] L . E. Scriven , On the dynamics of phase growth, Chem . Engrg . Sci . 1 0 (1959) , 1-13 . 156] F . Seling , Bemerkungen zum Stefanschen Problem, Osterr . Ingnr.-Arch . 1 0 (1956) , 277-280. 157] G . Sestini , Esistenza di una soluzione in problemi analoghi a quello di Stefan, Rivista Mat . Univ . Parm a 3 (1952) , 3-23 . M R 14 , 381. 158] , Esistenza ed unicita del problema di Stefan relativo a campi dotati di simmetria, Rivist a Mat . Univ . Parm a 3 (1952) , 103-113 . M R 14 , 476 . 159] , Problemi di diffusione lineari e non lineari analoghi a quello di Stefan, Conf. Sem . Mat . Univ . Bar i 55-5 6 (1960) . 160] , Sul problema unidimensionale non lineare di Stefan in uno strato piano indefinito, Ann . Mat . Pur a Appl . (4) 5 1 (1960) , 203-224 . M R 2 3 # B804 . 161] , Sopra un teorema di unicita in problemi unidimensionali analoghi a quello di Stefan, Boll . Un . Mat . Ital . 1 2 (1957) , 516-519 . M R 2 0 #1086 . [162] , Ancora su di un teorema di unicita in problemi unidimensionali analoghi a quello di Stefan, Boll . Un . Mat . Ital . (3 ) 1 4 (1959) , 373-375 . M R 2 2 #5806 . 163] , Sul problema non lineare di Stefan in strati cilindrici o sferici, Ann . Mat . Pura Appl . (4) 5 6 (1961) , 193-207 . M R 2 5 #4259 . L16%4] L . A . Skinner an d S . G . Bankoff , Dynamics of vapor bubbles in spherically symmetric temperature fields of general variation, Phys . Fluid s 7 (1964) , 1-6 . M R 2 8 #5658 . [165] P. V . Solovieff , Fonctions de Green des Equations paraboliques, C . R . (Dokl. ) Acad . Sci. URS S 2 4 (1939) , 107-109 . M R 2 , 55 . 166] G . S . Springe r an d D . R . Olson , Method of solution of axisymmetric solidification and melting problems, Pape r Amer . Soc . Mech . Engrs . 1962, N. Wa.-246 . 167] J. Stefan , Uber einige Probleme der Theorie der Warmeleitung, S.-B . Wien . Akad . Mat. Natur . 9 8 (1889) , 173-484 . [168] , Uber die Diffusion von Sauren und Basen gegen einander, S.-B . Wien . Akad. Mat . Natur . 9 8 (1889) , 616-634 . [169] , Uber die Theorie der Eisbildung insbesondere uber die Eisbildung im Polarmeere, S.-B . Wien . Akad . Mat . Natur . 9 8 (1889) , 965-983 . [170] , Uber die Verdampfung und die Auflbsung als Vorgange der Diffusion, S.-B. Wien . Akad . Mat . Natur . 9 8 (1889) , 1418-1442 . 418 BIBLIOGRAPHY [171] "W. Sternberg, Uber die Gleichung der Warmeleitung, Math . Ann. 10 1 (1929), 394-398 . [172] G . W . Sutton , The hydrodynamics and heat conduction of a melting surface, J . Aeronaut. Sci . 2 5 (1958) , 29-32 , 36 . [173] I . Tadjakhs h an d W . Lininger , Free boundary problems with regions of growth and decay. An analysis of heat transfer in the dip soldering process, Quart . J . Mech . Appl . Math. 1 7 (1964) , 141-155 . [174] D . E . Temkin , The temperature field in a crystallizing ingot of cylindrical form, Inz.-Fiz. Z . 1962 , no . 4 , 89-92 . (Russian ) [175] A . N . Tihonov , On regularization of incorrectly posed problems, Dokl . Akad . Nau k SSSR 15 1 (1963) , 501-50 4 = Sovie t Math . Dokl . 4 (1963) , 1035-1038 ; Dokl . Akad. Nau k SSS R 15 3 (1963) , 49-5 2 = Sovie t Math . Dokl . 4 (1963) , 1624 - 1627. M R 2 8 #5576 ; 2 8 #5577 . [176] , Functional equations of Volterra type and their applications to certain problems of mathematical physics, Bjull . Moskov . Gos . Univ . Ser . A Mat . Meh . 1 (1938), no . 8 . (Russian ) [177] , On the heat-conduction equation in several variables, Bjull . Moskov . Gos . Univ. Ser. A Mat . Meh . 1 (1938) , no . 9 . (Russian ) [178] , Boundary conditions containing derivatives of order higher than the order of the equations, Mat . Sb . 2 6 (68 ) (1950) , 35-56 ; Englis h transl , Amer . Math . Soc. Transl . (1) 4 (1962) , 440-466 . M R 11 , 440 ; 14 , 377 . [179] A . N. Tihono v an d A . A. Samarskii, The equations of mathematical physics, 2n d ed. , GITTL, Moscow , 1953 , p . 266 . M R 16 , 364 . [180] A . N . Tihono v an d E . G . Svidkovskii , On the theory of a discontinuous ingot, Z . Teoret. Fiz . 1 7 (1947) , no . 2 , 161-176 . (Russian ) [181] G . A . Tirskii, Melting of a body in the neighborhood of a critical point and lines in a dissociated flow of air with evaporation of a film of the melt, Z . Prikl . Meh . i Tehn . Fiz. 1961 , no . 5 , 39-52 . (Russian ) [182] , Sublimation of a blunt body near the critical point in a plane or axi- symmetric stream of a mixture of gases, Z . Vycisl. Mat. i Mat. Fiz. I (1961) , 884-902 = USSR Comput . Math , an d Math . Phys . 1 (1962) , 1050-1070 . M R 2 5 #3667 . [183] , Melting of a heat-conducting wall behind a moving discontinuity of density, Dokl. Akad. Nau k SSS R 12 9 (1959) , 989-99 2 = Sovie t Physic s Dokl . 4 (1959) , 1202-1206. [184] , Two exact solutions of Stefan's nonlinear problem, Dokl . Akad . Nau k SSSR 12 5 (1959), 293-296 = Sovie t Physic s Dokl. 4 (1959) , 288-292. MR 2 1 #7728 . [185] W . F . Trench , On an explicit method for the solution of a Stefan problem, J . Soc . Indust. Appl . Math . 7 (1959) , 184-204 . M R 2 2 #1087 . [186] |D. L . Turcott , The melting of ice in a hot humid stream of air, J . Flui d Mech . 8 (1960), 123-129 . M R 2 2 #7670 . [187] F . P . VasiFev , On the method of finite differences for solving a one-phase Stefan problem, Z . Vycisl . Mat . i Mat . Fiz . 3 (1963) , 861-87 3 = USS R Comput . Math . and Math . Phys . 3 (1963) , 1175-1191 . MR 3 1 #897 . [188] F. P . VasiFe v an d A . B . Uspenskii , A difference method for solving a two-phase Stefan problem, Z . Vycisl . Mat . i Mat . Fiz . 3 (1963) , 874-88 6 = USS R Comput . Math, an d Math . Phys . 3 (1963) , 1192-1208 . M R 3 1 #1784 . [189] V . P . Veinberg , Ice, GITTL , Moscow , 1940 . (Russian ) [190] T . D. Ventcel', A free boundary problem for the heat equation, Dokl . Akad. Nauk SSS R 131 (1960) , 1000-100 3 = Sovie t Math . Dokl . 1 (1960) , 359-361 . M R 2 2 #8225 . [191] N. N . Verigin , On the pressurized forcing of binder solutions into rocks in order to increase the strength and imperviousness to water of the foundations of hydrotechnical installations, Izv . Akad. Nau k SSS R Otd . Tehn . Nau k 5 (1952) , 674-687 . (Russian ) BIBLIOGRAPHY 419 [192] , Diffusion at the surface of a solid immersed in a fluid, Z . Fiz . Him . 3 2 (1958), 2097-2106 . (Russian ) [193] , On the shifting of gas-bearing contours by exploitation of deposits of natural gas, Izv . Akad. Nauk SSSR Otd. Tehn. Nau k 1958 , no. 3, 169-171 . (Russian ) [194] , On the kinetics of dissolving and carrying off salts in the presence of water- filtration in soils, Dissolvin g an d Dealkalinizin g o f Rocks , Gosstroiizdat , Moscow , 1957. (Russian ) [195] S . S. Vjalo v an d L . G . Kogan , Solution of some heat engineering problems by methods of electrical modelling, Izv . Akad . Nau k SSS R Otd . Tehn . Nau k 1951 , 39-52 . (Russian) M R 16 , 527 . [196] R . O . Vyborny , Properties of the solutions of certain boundary problems for equa- tions of parabolic type, Dokl . Akad . Nau k SSS R 11 7 (1957) , 563-565 . (Russian ) MR 2 3 #A3918 . [197] S . P. Vyrodov , Diffusion problem with a moving boundary, Dokl . Akad . Nau k SSS R 147 (1962) , 68-7 0 = Sovie t Physic s Dokl . 7 (1963) , 972-973 . MR2 7 #1265 . [198] H. Weber , Die partiellen Differentialgleichungen der mathematischen Physik. Ban d 2, 6-t e Aufl. , Braunschweig , 1919 . [199] Ja. B . ZeFdovic , G . I . Barenblat t an d R . L . Salganik , On quasi-periodicity of precipitation for the mutual diffusion of two substances (Lizegang rings), Dokl . Akad . Nauk SSS R 14 0 (1961) , 1281-128 4 = Sovie t Physic s Dokl . 6 (1961) , 869-871 . [200] Ja. B . ZeFdovi c an d O . M . Todes , On the mathematical formulation of the theory of periodic precipitation, Z . Fiz . Him . 2 3 (1949) , 180-191 . (Russian ) [201] I . A . Zolotar' , Calculation of the amount of freezing and heaving of soil, with ac- count taken of the movement of the liquid, Processe s o f Hea t an d Mas s Exchang e i n Frozen Rocks , "Nauka" , Moscow , 1965 . (Russian ) Supplementary bibliographica l informatio n an d certai n approximativ e solution s fo r a problem o f Stefan type can be found i n the monograp h b y H . S . Carsla w an d J . C . Jaeger , Conduction of heat in solids, 2n d ed. , Clarendo n Press , Oxford , 1959 ; Russia n transl. , "Nauka", Moscow , 1964 .