A New Iterative Method for Multi-Moving Boundary Problems Based Boundary Integral Method

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1 Journal of Appled Maheac and Phyc, 5, 3, 6-37 Publhed Onlne Sepeber 5 n ScRe. hp:// hp://dx.do.org/.436/jap A New Ierae Mehod for Mul-Mong Boundary Proble Baed Boundary Inegral Mehod Kawher K. Al-Swa, Sad G. Ahed Deparen of Maheac and Sac, Faculy of Scence, Taf Unery, Taf, Kngdo of Saud Araba Eal: gahed9@hoal.co Receed 9 June 5; acceped 5 Sepeber 5; publhed 8 Sepeber 5 Copyrgh 5 by auhor and Scenfc Reearch Publhng Inc. Th work lcened under he Creae Coon Arbuon Inernaonal Lcene (CC BY). hp://creaecoon.org/lcene/by/4./ Abrac The preen paper deal wh ery poran praccal proble of wde range of applcaon. The an arge of he preen paper o rack all ong boundare ha appear hroughou he whole proce when dealng wh ul-ong boundary proble connuouly wh e up o he end of he proce wh hgh accuracy and nu nuber of eraon. A new nuercal erae chee baed he boundary negral equaon ehod deeloped o rack he ong boundare a well a copue all unknown n he proble. Three praccal applcaon, one for aporzaon and wo for ablaon were oled and her reul were copared wh fne eleen, hea balance negral and he ource and nk reul and a good agreeen were obaned. Keyword Mul-Mong Boundary Proble, Vaporzaon Proble, Ablaon Proble, Source and Snk Mehod, Fne Eleen Mehod, Hea Balance Inegral Mehod, Boundary Inegral Mehod. Inroducon The dfferenal equaon n a ceran doan afyng oe gen condon referred o a boundary-alue proble. If one or ore of he boundare are no known and ong wh e, he proble hen referred o a ong boundary proble [] []. Furherore, f he goernng equaon e ndependen a well a he boundary condon, hen he proble referred o a free boundary proble [3]. Phae change proble are praccal exaple of free and ong boundary proble, frequenly appear n ndural proce and oher proble of echnologcal nere uch a he deernaon of he deph of he fro or haw peneraon, de- Correpondng auhor. How o ce h paper: Al-Swa, K.K. and Ahed, S.G. (5) A New Ierae Mehod for Mul-Mong Boundary Proble Baed Boundary Inegral Mehod. Journal of Appled Maheac and Phyc, 3, hp://dx.do.org/.436/jap.5.394

2 gnng of he roadway or oher engneerng work n cold regon and he o-called re-enry proble of hyperonc le n aeronaucal cence [4] [5]. When a body expoed o hea flux he urface of he body change phae, he changed phae edaely reoed upon foraon. Th referred o ablaon proble [6]. A ajor dfference beween Sefan and ablaon proble ha he oerall doan n Sefan proble rean fxed n pace whle he doan n he ablaon proble arable and dnhe n ze wh e. Nuercal ehod becoe ore popular raher han analycal ehod. Soe of faou nuercal ehod are fne dfference [7], fne eleen [8], boundary eleen [9] and ore recen eh-le (eh-free) nuercal ehod []. In any apec, he boundary negral equaon ehod for olng boundary alue proble proe o be adanageou oer he conenonal nuercal ehod. The preen paper deal wh ery poran praccal proble of wde range of applcaon. The an arge of he preen paper o rack all ong boundare ha appear hroughou he whole proce when dealng wh ul-ong boundary proble connuouly wh e up o he end of he proce wh hgh accuracy and nu nuber of eraon. A new nuercal erae chee baed he boundary negral equaon ehod deeloped o rack he ong boundare a well a copue all unknown n he proble.. Maheacal Model A e-nfne old nally a unfor eperaure wh he followng conran, here no ub coolng, no uhy zone, old and lqud phae hae equal and conan propere and fnally no conecon. The aheacal forulaon con of hree dfferen age a follow: Heang age (, ) u( x, ) u x =, < x < α u( x,) u (, ) u q = K () = () u x, = u, = u (4) x A x = reache he elng eperaure, he econd age ar appearng. Lqud-old age Ga-lqud-old age (, ) (, ) u x u x =, < x< α (, ) u q = K (, ) (, ) u x u x =, ( ) < x< α (, ) (, ) x u( x, ) = u ( x, ) = u x= x= ( ) (, ) (, ) d u x = u = u (8) (9) u x u x = ρ = () d K K L, x (, ) (, ) u x u x =, < x< α (, ) (, ) u x u x =, < x< α (3) (5) (6) (7) () () 7

3 3. Boundary Inegral Forulaon (, ) (, ) u x u x =, ( ) < x< α (, ) (, ) x= x= ( ) u x u x u (3) = = (4) (, ) (, ) u x = u x = u (5) u d u K K = L d V ( ) ρ (6) (, ) (, ) Sarng by he weghed redual aeen for dffuon equaon a follow: In Equaon (8) u (, x;, ) u x = u x = u (7) τ u u k u ( ξ, x; τ, ) dω d = Ω (8) ξ τ he fundaenal oluon for dffuon equaon and defned a: ( ξ x) ( τ ) r, u ( ξ, x; τ, ) = exp 4πk( τ ) 4k In whch, r ( ξ, x) = X ( ξ) X ( x) and X ( ξ ) he ource pon whle Inegrang equaon (8) wce by par lead o: τ Ω u u k u ( ξ, x; τ, ) dωd ( ξ, ; τ, ) ( ξ, ; τ, ) X x he feld pon. = u x + u x Ω Γ = τ τ k ud d k q ( ξ, x; τ, ) ud d Ω Γ For one-denonal proble, Equaon () ake he followng for: In Equaon () ( ξ, ; τ, ) ( ξ, ; τ, ) u x u x τ τ k + ux dd k q ( ξ, x; τ, ux ) dd = ( ξ, ; τ, ) ( ξ, ; τ, ) u x u x k + = δ (9) () () () δ u = u (3) Makng ue of Equaon () and (3) no Equaon (), he la one ake he followng for: τ k u x q x u x u x x ( ξ τ ) ( ξ τ ), ;, d d,, ;, d = For any pon he negral equaon ake he followng for: τ τ ( ξ τ) = ( ξ τ ) + ( ξ τ ) ( ξ τ ) u, u x, u, x;, d x k u, x;, q x, dxd k q, x;, u x, dxd Dcrezaon of Te Equaon (5) afer he dcrezaon of e ake he followng for: (4) (5) 8

4 (, ) u ξ u x u ξ x x k u ξ x n u x, n =,, ; n, d + (, ; n, ) d = n u ( ξ, x ; n, ) k u( x, ) d n = Aue ha he poenal u and he flux q are conan whn each e ep herefore Equaon (6) can be wren a: F u( ξ, n) = u( x, ) u ( ξ, x ; n,) d x+ q( x, ) u ( ξ, x ; n, ) u( x, ) q ( ξ, x ; n, ) d (7) In Equaon (7), we hae wo dfferen e negral, hey are: The doan negral r a a ( ξ, ; n, ) d e e π { ( ) ( )} F k π I = u x = a a + erf a erf a (8) F I = q ( ξ, x; n, ) d = gn ( r) erf a erf a k (9) a r r = & a = 4k 4k ( ) ( ) n n (,) u x ξ ξ u ( x, ) u ( ξ, x; n,) dx = ( ) erf erf k n k n Now, he fnal for for he negral equaon correpondng o he dffuon equaon defned oer fxed doan wll be: (,) u x u ( ξ, ) erf erf q u d u q d n ξ ξ n = + x= x= k n k n = x = q u u q d n x= d x= = x= Slar procedure can be carred ou akng no conderaon he ong boundare and her noral eloce, herefore and accordng o [] [], and for he econd and hrd age, he negral equaon correpondng o he dffuon equaon wh ong boundare n a general fnal for wll be: ( x) ( ξ, ;, ) k u, u xk c( ξ) u( ξ, k) = α u ( ξ, x ; k, ) u( x, ) + u( x, ) u ( ξ, x ; k, ) Vn d α (33) j In Equaon (33), V n repreen he ouward noral elocy o he urface. In one-denonal proble, a d ( ) d ( ) n our cae udy, h elocy ay be or accordng o he phae underhand. d d 4. New Fxed-Mong-Fxed Algorh In he preen paper, a generalzed nuercal algorh and code for ul-ong boundary proble are deeloped, ung ual Forran 6.6. The an code con of an progra and hree ubroune. The flow char decrbng he an par of he propoed algorh hown n Fgure. x x (6) (3) (3) (3) 9

5 4.. Algorh and Subroune ONEPHASE Fgure. General fxed-ong-fxed negral algorh. In h ubroune, a ngle phae bounded by wo fxed boundare are oled, ung Equaon (3) and he oupu wll be he e a whch elng ar and he correpondng locaon of he fr ong boundary, epa-. rang he lqud and he old, Suggeed Algorh ) Inpu daa, old lengh, paal grd ze x and e ep. ) Apply he boundary negral equaon, gen by Equaon (3) a he end pon of he doan of nere o j j u x =,. j eae j 3) Check u x =,, =,,, N, f ye, hen go o he nex wo-phae ubroune, wh oupu, e of elng and he correpondng poon for he ong boundary eparang lqud and old. If no j+ j updae boh he paal and e arable x = x + x = + and repea ep - 3 ll he re- + +, 3

6 qured oupu of h ubroune acheed. 4.. Algorh and Subroune TWOPHASE Th ubroune concern anly wh wo phae each phae wll be bounded by one fxed and he econd ong are oled and he oupu wll be he e a whch apor ar and he correpondng locaon of he. fr and econd ong boundare and Suggeed Algorh ) The npu un here he oupu fro one-phae ubroune. ) Sole old phae ubjeced o elng eperaure a he ong boundary and nal eperaure a he u fxed end o ge. x ( ) d ( ) u u 3) By knowng and d o eae. j (, ) 4) Check ( j ) ( ) ( ) u x = u V, f ye, hen he oupu of h ubroune wll be he e a whch apor ar appearng wh he new poon of he ong boundary eparang lqud/old and he arng poon of he econd ong boundary eparang apor(ga)/lqud, f no updae he ong boundary eparang lqud/old Algorh and Subroune THREEPHASE Th ubroune for hree phae, he fr one bounded by wo boundare, one fxed and one ong, he econd phae bounded by wo ong boundare, and he hrd phae alo bounded by wo boundare one fxed and one ong. The oupu of h ubroune o rack all ong boundare, deernaon all unknown n all phae up o he end of he proce wh nu nuber of eraon and hgh precrbed accuracy. The flow char of h ubroune hown n Fgure. In h fgure, he abolue error E and E are defned a follow: 5. Nuercal Reul 5.. Vaporzaon Te Proble (, ) (, ) d u x u x E = K K L d ρ (34) ( ) u d u E = K K ρl (35) d In h econ, hree e proble are oled o check he aldy of he propoed algorh and he hgh accuracy expeced. The fr exaple a ul-ong boundary proble [3]. The hero-phycal propere are led below n Table. The old aeral heren of a low heral conducy and o he aporzaon occur before he ong boundary eparang lqud and old reache he adabac boundary. The reul due o he preen algorh hown n Fgure 3. In h fgure, boh apor/lqud and lqud/old ong boundare are ploed boh on he ae fgure. The lqud/old ong boundary n he lef de, whle he oher one on he rgh. Fro h fgure, one can clearly deerne he elng, apor and he e a whch he proce end. Thee e are deerned and uarzed n Table, a wo dfferen e ep. The abolue error beween he reul due o he preen ehod copared wh he fne eleen ehod are alo preened n he ae able. A we ee, he abolue error decreae by decreang he e ep. On he oher hand by decreang he e ep, he nuber of eraon ncreae. The nuber of eraon are hown n Table, correpondng o he wo e ep ued. I clear ha he ncreae n he nuber of eraon no o uch bu he accuracy proed o 3

7 Fgure. Flow char for THREEPHASE ubroune. Fgure 3. Mong boundare locaon for aporzaon e proble. 3

8 Table. Nuercal daa for aporzaon e proble. Paraeer Defnon Nuercal daa c Specfc hea for old c Specfc hea for lqud K Theral conducy for old.59 K Theral conducy for lqud.59 u Melng eperaure 454 u Vaporzaon eperaure 3 L Laen hea for elng 6 L Laen hea for aporzaon 37, u( x,) Inal eperaure 7 q( ) Inpu hea flux a he boundary x = 5 ρ Deny Table. Coparon beween elng, aporzaon and end e. Te ep Type of e FE Preen Abolue error =.5 = e e Aerage nuber of eraon nearly he half. 5.. Ablaon Te Proble- A old edu nally a unfor eperaure, u = 3 K, he boundary x = expoed o wo dfferen 6 4 cae of npu hea flux, conan, Q( ) = 5 and lnear, Q( ) = 3 repecely. The doan n he preen proble ll fxed whle appearng wo ong nerface, old-lqud and apor (ga)-lqud (Ablaon urface). The reul due o he preen algorh are hown n Fgure 4-6, repecely and he reul are copared wh he reul due o he ource and nk ehod [4]. Fgure 4 how he oeen of old-lqud due o conan npu hea flux, and he reulng ablaon urface due o he ae npu hea flux hown n Fgure 5. The ae reul due o lnear cae are ploed on he ae plo a hown n Fgure 6. Fro he copuaon and fgure, found ha he old-lqud nerface ha he ae behaor n boh conan and lnear cae of npu hea flux ha concae upward. In cae of lnear hea flux npu h concay becoe ore apparen han he conan cae. In he conrary, he apor (ga)/lqud nerface behae concae downward bu n lnear cae h concay ncreae Ablaon Te Proble- Th proble for a long enough old old nally a a unfor eperaure. The urface x = expoed o wo dfferen hgh npu hea flux, conan and lnear and he apor reoed a oon a foredablaon proble-. Two pecfc hea flux boundary condon are choen n he preen copuaon, naely, ( ) q =., c and he followng alue ued n he calculaon [5] (Table 3). The ablaon hckne due o he preen ehod copared wh he correpondng fro he hea balance negral ehod hown n Fgure 7. I clear ha he ablaon hckne concae downward n boh cae of 33

9 Sold/Lqud nerface (c) Exaple () Conan npu hea flux Sold-Lqud nerface Preen Mehod Source and Snk Mehod Te (Sec) Fgure 4. Sold/lqud due o Conan hea flux-proble-...9 Ablaed urface (c) Exaple () Conan npu hea flux Ablaed Surface Preen Mehod Source and Snk Mehod Te (Sec) Fgure 5. Ablaed urface due o conan hea flux-proble-. 34

10 Fgure 6. Sold/lqud and Ablaed urface due o lnear hea flux-proble-. Fgure 7. Ablaon hckne for ablaon e proble-. 35

11 Table 3. Thero-phycal paraeer. Sybol Phycal eannng Nuercal alue T α Theral dffuy. f / ec c Characerc e ec L characerc lengh. q r Reference hea flux Bu/ ec f T Teperaure dfference F Inere Sefan nuber T Reference e ec npu hea flux and a he ae e, he ablaon hckne n cae of lnear hea flux hgher han ha for conan cae. There a good agreeen beween he wo ehod n boh cae. 6. Concluon The porance of he preen paper coe fro dealng wh praccal applcaon of wde range of our daly lfe. The boundary negral equaon ehod no o new bu ued a a aheacal ool due o plcy n ue. Baed on h ehod, a generalzed nuercal algorh and copuer code are deeloped o ole uch applcaon. I found fro he copuaon ha he deeloped algorh and he code are o ple o handle and ha an accepable accuracy obaned. Alo by decreang he e ep, and a lle b ncreae of eraon, he abolue error are decreaed o he half nearly. Fnally, he algorh and ubequenly he code can be ealy odfed o coer hgher denonal proble wh accepable accuracy, whch can be proed by decreang he e ep, bu on he oher hand, ably hould be acheed. Reference [] Meaad, N.M.I. (4) Meh-Le Mehod and I Applcaon o Free and Mong Boundary Proble. M.Sc. The, Zagazg Unery, Zagazg. Superon S. G. Ahed. [] Crank, J. (984) Free and Mong Boundary Proble. Clarendon Pre, Oxford. [3] Mohad, N.A.E. () Boundary Eleen Mehod and I Non-Lnear Analy of Hydrofol. Superor: Prof. S. G. Ahed and Prof. A. F. Abd-Elgawad. Zagazg Unery, Zagazg. [4] Nofal, T.A. and Ahed, S.G. (5) An Inegral Mehod o Sole Phae-Change Proble wh/whou Muhy Zone. Journal of Appled Maheac and Maheac, II, [5] Ahed, S.G. and Mekey, M.L. () A Collocaon and Carean Grd Mehod Ung New Radal Ba Funcon o Sole Cla of Paral Dfferenal Equaon. Inernaonal Journal of Copuer Maheac, 87, [6] Mehrf, S.A. and Ahed, S.G. (7) Approxae Inegral Mehod Appled o Ablaon Proble n a Fne Slab. Inernaonal Sypou on Recen Adance n Maheac and I Applcaon (ISRAMA 7), Culcua, 5-7 Deceber 7, -. [7] Zerrouka, M. and Chawn, C.R. (994) Copuaonal Mong and Boundary Proble. Reearch Sude Pre Ld., Baldock. [8] Ruan, Y. and Zabara, N. (99) An Inere Fne Eleen Technque o Deerne The change of Phae Inerface Locaon n Two-Denonal Melng Proble. Councaon n Appled Nuercal Mehod, 7, hp://dx.do.org/./cn.6374 [9] Mohaed, W.A. () Adanced Maheacal Analy for Phae Change Proble wh and whou Muhy Zone. Superor: Prof. S. G. Ahed and Prof. M. E. Mohaed. Zagazg Unery, Zagazg. [] Lu, W., Chen, Y., Jun, S., Chen, J.-S., Belychko, T., Pan, C., Ura, R. and Chang, C. (996) Oerew and Applcaon of he Reproducng Kernel Parcle Mehod. Arche of Copuaonal Mehod n Engneerng: Sae of he Ar Reew, 3, 3-8. hp://dx.do.org/.7/bf7363 [] Ahed, S.G. (5) A New Algorh for Mong Boundary Proble Subjec o Perodc Boundary Condon. Inernaonal Journal of Nuercal Mehod for Hea and Flud Flow, 6, 8-7. [] Abd-El Faah, W.M. and Ahed, S.G. (6) Boundary Inegral Forulaon for Bnary Alloy fro a Coolng Sold 36

12 Wall. Inernaonal Journal of Copuaonal and Appled Maheac, 5, [3] Ahed, S.G. (3) A New Algorh for Fron Trackng of Ablaon Proble n Unbounded Doan. An Sha Unery, Egyp. [4] Mehd, A. and Heh, C.K. (994) Soluon of Ablaon and Cobnaon of Ablaon and Sefan Proble by a Source and Snk Mehod. Nuercal Hea Tranfer, Par A, 6, hp://dx.do.org/.8/ [5] Ahed, S.G. (3) A Nuercal Soluon for Nonlnear Hea Flow Proble Ung Boundary Inegral Mehod. 6h Inernaonal Conference of Copuer Mehod and Experenal Meaureen for Surface Treaen, Cree, March 3. Noenclaure u : Teperaure u : Fundaenal oluon α : Theral dffuy K : Conducy ρ : Deny q( ) : Inpu hea flux : Lqud-old nerface : Vapor-lqud nerface : Truncaed long enough boundary lengh L : Laen hea Subcrp : Sold : Lqud : Inal : Vapor : Melng 37