GREEN S FUNCTION FOR HEAT CONDUCTION PROBLEMS IN A MULTI-LAYERED HOLLOW CYLINDER

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1 Joual of ppled Mathematcs ad Computatoal Mechacs 4, 3(3), 5- GREE S FUCTIO FOR HET CODUCTIO PROBLEMS I MULTI-LYERED HOLLOW CYLIDER Stasław Kukla, Uszula Sedlecka Isttute of Mathematcs, Czestochowa Uvesty of Techology Częstochowa, Polad staslaw.kukla@m.pcz.pl, uszula.sedlecka@m.pcz.pl bstact. I ths pape, devato of the Gee s fucto fo the heat coducto poblems a fte mult-layeed hollow cylde s peseted. Fomulato ad soluto of the poblem cludes a abtay umbe of the cylde layes chaactezed by vaous themal popetes. t the tefaces pefect themal cotact was assumed. The Gee s fucto fo the thee-dmesoal heat coducto poblems the cyldcal coodate has bee peseted the fom of a poduct of two othe Gee s fuctos. Keywods: Gee s fucto, heat coducto, mult-layeed composte cylde Itoducto The Gee s fucto (GF) method has bee wdely used the soluto of heat coducto poblems fo basc geometes the cosdeed egos the cases of classcal bouday codtos. pplcato of the Gee s fucto to such poblems ca be foud the book by Beck et al. [], the book by Özşk [] ad the book by Duffy [3], as well as seveal papes, fo example efeeces [4-7]. The advatage of the Gee s fucto method les the fact that the soluto of the o-homogeeous poblems ca be expessed tems of the Gee s fucto. I the soluto to the poblem the tems whch fluece the tempeatue dstbuto ca be dstgushed: () the tem whch gves the cotbuto of the tal codto, () the tem whch epesets the cotbuto of the eegy geeato ad (3) the tem whch epesets the cotbuto of the o-homogeeous bouday codtos. I the books [-3] ad the papes [4-7] devatos of the Gee s fuctos fo the heat coducto equato fo vaous bouday codtos ae peseted ad examples of applcatos ae gve. The poblems peseted the efeeces [-7] coce the systems chaactezed by costat themomechacal quattes. I the book by Özşk [3] ad the pape by Mllga ad Ka [4] a applcato of the GF method to the oe-dmesoal heat coducto a layeed medum s also gve. The pupose of ths study s to develop a aalytcal soluto of the thee- -dmesoal heat coducto poblem a hollow multlayeed cylde by usg

2 6 S. Kukla, U. Sedlecka the Gee s fucto method. The fomulato ad soluto to the poblem coce the fte cylde wth the bouday codto of a thd kd ad pefect themal cotact at the tefaces.. Fomulato of the poblem Cosde a fte, -layeed hollow cylde as s show Fgue. I ode to solve the heat coducto poblem the cylde by usg the GF method, a auxlay poblem s fomulated []. Ths poblem les o the devato of the ecessay Gee s fuctos G j,, j,,...,. The fuctos G j the cyldcal coodates satsfy the followg dffeetal equato G j G j G j Gj G j δ ( ρ) δ( ϕ ϕ) δ( z ζ) δ( t τ) ϕ z α t [ ] [ ] z, L,,, ρ j, j, j,,..., () whee α ad ae the themal dffusvty ad themal coductvty, espectvely,, ϕ, z, ρ, ϕ, ζ ae the cyldcal coodates, δ s the Dac delta fucto. L Fg.. The sketch of the -layeed hollow cylde We assume zeo tal codto ad covectve codtos (the thd kd of bouday codtos fo the tempeatue dstbuto) at the boudaes of the hollow cylde. The Gee s fuctos Gj( t,, ϕ, z; τ, ρ, ϕ, ζ ) assocated wth such a heat coducto poblem satsfy the followg homogeeous bouday codtos α + * j G j fo * j α G j fo () (3)

3 Gee s fucto fo heat coducto poblems a mult-layeed hollow cylde 7 j γ γ Gj fo z j ν ν Gj fo z L (4) (5) * * whee,, γ, ν ae the themal coductvtes, α+, α, γ, ν ae the heat tasfe coeffcets. Moeove, the codtos of pefect themal cotact at the tefaces holds [] Gj G+ j fo,, j,,..., (6) j + j + fo,, j,,..., (7) The GF fo heat coducto a cylde wth costat themal popetes (dffusvty ad coductvty) ca be peseted the fom of a poduct of two Gee s fuctos []. Smlaly, the case of -layeed cylde, each fucto G j ca be expessed the fom of the poduct j (,, ϕ, ; τ, ρ, ϕ, ζ) (,, ϕ; τ, ρ, ϕ ) (, ; τ, ζ) G t z G% t G t z (8) j The Gee s fuctos fo the adal decto heat coducto G % j ad fo oedmesoal heat coducto G ae solutos of the dffeetal equatos % % % % G j G j G j G j δ ( ρ) δ( ϕ ϕ) δ( t τ) ϕ α t, j,,..., (9) G + δ ( z ζ) δ( t τ) () z t The bouday codtos fo the fuctos the poduct (8) equatos ()-(7). G% adg ae obtaed by substtutg j. Devato of the Gee s fuctos The fuctos sees: G % j occug equato (9) we fd the fom of the cose

4 8 S. Kukla, U. Sedlecka G % t g t () j (,, ϕ; τ, ρ, ϕ ) (, ; τ, ρ) cos ( ϕϕ ) j κ whee κ π ad κ π fo,,.... Takg to accout the sees () the equato (9), we obta a equato fo the adal decto Gee s fuctos the fom g + g j + δ( ρ) δ( t τ) α t [ ] j,, ρ j, j,, j,,...,, ().. Devato of the adal decto Gee s fuctos j t τ ρ satsfy the dffeetal equato (), the zeo tal codto ad the bouday codtos whch follow fom codtos ()-(3): The fuctos g (, ;, ) g α + * j g j fo g * j α g j fo (3) (4) Moeove, o the bass of equatos (6)-(7), usg (8) ad (), we obta: gj g+ j fo,,,..., (5) g g fo,,,..., j + j + The fuctos g (, ;, ) j t τ ρ we peset the fom of a expaso (6) whee ( ) m ( ) ( ) φ φ ρ g t t (7) m jm j (, ; τ, ρ) Θm( ; τ) m m φ ae the egefuctos of the followg bouday poblem + +, [, ],,,..., m φ m (8) dφ * m ( ) α + φm( ) (9) d

5 whee Gee s fucto fo heat coducto poblems a mult-layeed hollow cylde 9 γ m m α. dφ d ( ) ( ) φm φ + m,,,..., () dφ +,,,..., () d + ( ) ( ) m m dφ * m ( ) α φm( ) () d The geeal soluto of equato (8) has the fom ( ) ( ) ( ) [ ] φ m c J m + c Y m,,,,,..., (3) whee J ad Y ae the Bessel fuctos of the fst ad secod kd, espectvely. Substtutg the fuctos (3) to bouday codtos (9)-() we obta a system of homogeeous equatos. We have wtte the equato system the matx fom, j C (4) whee, j C c c c c... c, c,. The o-zeo elemets of the matx ae C c c c c... c, c, α J J ( ) + ( ) + * m m m m α Y Y ( ) + ( ) + * m m m m ( ), ( ) J Y,, m, m ( ), ( ) J Y,,+ + m,+ + m J J Y Y, ( ) ( ), ( ) ( ) +, m + m +, m + m m m J J ( ) ( ) + + m +,+ + m + + m m + m, T T,

6 S. Kukla, U. Sedlecka Y Y ( ) ( ) + + m +,+ + m + + m m + + m fo,,...,, + α J J ( ) ( ), * m + m m m + α Y Y ( ) + ( ), * m m m m The o-zeo soluto of the system (4) exsts fo values of γ m fo whch the detemat of the matx s equal to zeo:. ad det (5) The equato (5) wth espect to the egevalues γ m s the solved umecally. The egefuctos φ m coespodg to the calculated egevalues wll be fully detemed f the costats c, c, c, c,..., c,, c,, occug equatos (3) wll be appoted. I detemg these costats we assume c, equato (4) ad delete the last ow the matx ths equato. s a esult we obta the matx equato the fom C B (6) whee the matx ases fom the matx by deleto of the last ow ad the last colum, C c c c c... c, ad B m... Y( m ) Y( m ) Y+ ( m ) m m The fuctos φ m( ), m,,, gve by equato (3) wth c, c detemed fom equato (6) ceate a othogoal set of fuctos fo ay,,,.e. the followg codto holds α T Φm( ) Φ m' ( ) d (7) m The dffeetal equato fo the fucto ( tτ ; ) fo m' m fo m' m Θ m occug equato (7) s obtaed by substtutg the sees (7) to equato () ad usg the othogoalty codto (7). s a esult we obta the equato the fom Θ m ( t; τ) t m m ( t; ) ( t ) + γ Θ τ δ τ (8) T

7 Gee s fucto fo heat coducto poblems a mult-layeed hollow cylde Solvg equato (8) takg to accout the zeo tal codto oe obtas ( t; τ) exp γ ( t τ) H( t τ) Θ m m (9) Hece, the adal decto Gee s fucto souds lke ( ) ( ) φ φ ρ g t H t t (3) m jm j (, ; τ, ρ) ( τ) exp γ m( τ) m m whee fuctos ( ) φ ae gve by equato (3). m.. Gee s fucto fo oe-dmesoal heat coducto G t z τ ζ occug equato (8) s a soluto of the dffeetal equato () ad satsfes the zeo tal codto ad the bouday codtos whch follow fom codtos (4)-(5): The Gee s fucto (, ;, ) γ γ G z fo z ν ν G z fo z L (3) (3) Ths fucto s gve the book [3] the fom of fte sees ad t s peseted below completo of the deved Gee s fucto fo heat coducto the fte hollow cylde. The fucto s gve by whee ( k ) (, ; τ, ζ) exp β ( t τ) G t z Ψ ( z) Ψ ( ζ) k k (33) z k k ( ) ψ z γ β cos β z+ γ s β z (34) k k k k ( ( ) ) ( ) ( ) cos s γ β k γ γ γ γ β γ γ γ β γ β s β z k + k+ L k L + k z+ k L 4β ad β k ae oots of equato ( ) L ( ) γ ν γ ν β cos β + γ ν β + γ ν s βl (35) k k k k k

8 S. Kukla, U. Sedlecka Fally, the Gee s fucto fo the heat coducto poblems the fte mult-laye hollow cylde wth bouday codtos of the thd kd ad pefect cotact at the tefaces has the fom j (,, ϕ, ; τ, ρ, ϕ, ζ) G t z ( )( t ) ( ) ( ) Ψ ( z) Ψ ( ) ϕ ϕ ρ ζ γ + ϑ τ cos ϕ ϕ m jm k k exp m k z k m m κ Coclusos ( ) The Gee s fucto fo thee-dmesoal heat coducto poblems a fte hollow -layeed cylde wth pefect themal cotact at tefaces has bee deved. The fucto s ecessay solvg a ohomogeous heat coducto poblem the cylde by usg the Gee s fucto method. The fucto s peseted the fom of a poduct of two Gee s fuctos. To deteme these fuctos two egepoblems must be solved. Use of the package Mathematca was a geat assstace symbolc computatos to deve the Gee s fucto. The use of the Gee's fucto defed hee wll assst the study of heat flow composte cyldes. (36) Refeeces [] Beck J.V., Cole K.D., Haj-Shekh., Ltkouh B., Heat Coducto Usg Gee s Fuctos, Hemsphee, Washgto DC 99. [] Özşk M.., Heat Codto, secod edto, Joh Wley & Sos, Ic., ew Yok 993. [3] Duffy D.G., Gee s Fuctos wth pplcatos, Chapma&Hall/CRC, Washgto DC. [4] Mllga K.B., Ka V.K., Elastothemodyamc dampg of fbe-efoced metal-matx compostes, Joual of ppled Mechacs 995, 6, [5] Lu X., Tevola P., Vljae M., Taset aalytcal soluto to heat coducto composte ccula cylde, Iteatoal Joual of Heat ad Mass Tasfe 6, 49, [6] ezhad Y.R., sem K., khlagh M., Taset soluto of tempeatue feld fuctoally gaded hollow cylde wth fte legth usg mult layeed appoach, Iteatoal Joual of Mechacs ad Mateals Desg, 7, 7-8. [7] Haj-Shekh., Beck J.V., Tempeatue soluto mult-laye bodes, Iteatoal Joual of Heat ad Mass Tasfe, 45,