Dynamics instability analysis of multi-walled carbon nanotubes conveying fluid A. Azrar 1,a, L. Azrar 1,2,b, A. A. Aljinadi 2,c and M.

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1 Adaned Maerals esearh Onlne: -- ISS: -898, Vol. 8, pp - do:.8/.senf.ne/am.8. Trans Teh Pblaons, Szerland Dynams nsably analyss of ml-alled arbon nanobes oneyng fld A. Azrar,a, L. Azrar,,b, A. A. Aljnad, and M. Hamadhe,d Deparmen of Mahemas, Faly of Senes and Tehnqes of Tanger, Abdelmalek Essaâd Unersy, Tanger, Moroo Deparmen of Mehanal Engneerng, Faly of Engneerng, Kng Abdlazz Unersy, Jeddah, Sad Araba Laboraory LMFA, ECL/UCBL/ISA, Unersy Clade Bernard, Lyon, Frane a abdellhadazrar@yahoo.fr, b l.azrar@fs.a.ma, aljnad@ka.ed.sa, d mahmod.hamadhe@e-lyon.fr Keyords: braon, freqeny response, dynam nsables, mlalled arbon nanobes, nonloal elasy, ylndral shell, an der Waal neraon, fld eloy, nsably, fler. Absra: The dynam nsably analyss of oneyng fld ml-alled arbon nanobes (MWCT s analyzed. Based on he nonloal elasy heory, Donnell s shell model, poenal flo heory and he an der Waal neraon beeen alls, he goernng eqaons are formlaed. The small sale parameer and he nernal fld neraon effes on he dynam behaors of he MWCT-fld sysem as ell as he nsables nded by he fld eloy are nesgaed. The ral eloy and he freqeny-amplde relaonshps are obaned h respe o physal and maeral parameers. I. Inrodon Cylndral shells oneyng fld are fond n nmeros ndsral and engneerng applaons. Therefore, he dynams of shells onanng flo fld has been sded eensely. For hn shells onanng fld, dynam nsably are major problems de o many dfferen loads, nldng fld flo. Fld flos nsde arbon nanobes (CTs hae beome an arae researh op n reen years, and a grea deal of lerare has been pblshed on ops sh as he nonoaal braon of fld-flled ml alled-arbon nanobes, Yan e al []. There are hree major aegores for smlang he mehanal properes of he CTs: eperene, molelar dynam smlaon (MDS and onnm mehans. Consderng he lmed applaon of he MDS, onnm modelng s onsdered o be an approprae mehod of nesgang he mehanal properes of CTs. The heory of nonloal elasy, Erngen [,], allos aonng for he small sale effe ha s ery sgnfan hen dealng h mro and nanosrres. He e al [] and Yan e al [] sded pressre dependene of he nsably of mlalled arbon nanobes oneyng flds. Azrar e al [,7] deeloped hgher order free braon analyses of sngle alled arbon nanobes h aros bondary ondon ypes. Wang [8] and Yan e al [9] sded he dynamal behaor of arbon nanobes oneyng fld. Chang and L [] sed small sale effe on flo nded nsably of doble alled arbon nanobes. Wang e al [] proposed he oplng braon of fld-flled arbon nanobes. Dong e al [] sed he ae propagaon n fld-flled ml-alled arbon nanobes embedded n elas mar. H e al [] proposed he nonloal shell model for elas ae propagaon n sngle and doble-alled arbon nanobes. Wang and [] dsssed braon and nsably of arbon nanobes oneyng fld. Yan e al [] sded nonloal effe on aally ompressed bklng of rple-alled arbon nanobes nder emperare feld. Vbraonal analyss of fld-flled arbon nanobes sng he ae propagaon approah has been proposed by ask e al []. The effe of small sze on dsperson haraerss of ae n arbon nanobes has been sded by Xe e al [7]. In hs paper, he small sale effes on he flo-nded nsably of MWCTs based on Donnell s shell model are nesgaed. The nflenes of he an der Waal neraons beeen alls, he flo eloy and he nonloal parameer on he naral freqenes are obaned by All rghs resered. o par of onens of hs paper may be reproded or ransmed n any form or by any means ho he ren permsson of Trans Teh Pblaons,.p.ne. (#9879, Pennsylana Sae Unersy, Unersy Park, USA-8/9/,::9

2 Maerals and Srres nmerally solng he elaboraed ransendenal nonlnear algebra eqaon. The ral flo eloy as ell he dergene and fler nsably behaors are nesgaed for CT h aros nmber of alls. II. Mahemaal formlaon Le s onsder a mlalled arbon nanobe of lengh L, nnermos rads I, oermos rads o, dameer d and hkness h. The ylndral oordnae sysem (,,z s onsdered here ; Ls he longdnal, ; s he rmferenal and z h / ; h / s he radal oordnaes respeely. The fld nsde he nner be s assmed o be deal nompressble and he flo s dren by pressre. Based on he nonloal shell and an der Waal models he goernng paral dfferenal eqaons ll be gen. II. onloal shell model Based on he Donnell shell heory he sran ensors a eah ylndral all are gen by:,, and ( here s he rads,, and are dsplaemens n he (,,z dreons respeely of he h all. Based on he nonloal heory, he onse eqaons of he arbon nanobe are: [7] ( E e a ( ( a ( E e a ( ( b ( E e a ( here E,, a and e are Yong s modls, Posson s rao of arbon nanobes, he nernal haraers lenghs and he onsan approprae o eah maeral respeely. s he Laplae operaor and and are respeely he normal sress n he and he y dreons and s he shear sress on he plane of he mddle srfae. The eqalen sa sress reslans a he h all are gen by: h/ h/ h/,, dz dz dz, h/ h/ h/ ( a h/ h/ h/ M,, z dz M z dz M z dz h/ h/ h/ ( b The goernng eqaons of moon of he ml-alled arbon nanobes are: h ( a Q h ( b Q Q p (, h (

3 Adaned Maerals esearh Vol. 8 M M Q ; ( d M Q M here p s he an der Waal fore, s he mass densy of he arbon nanobes. Q and ( e Q are he eqalen sa shearng sresses of he h all respeely. The mlalled s assmed o be flled h a mong fld n he nner ylnder. Based on some mahemaal deelopmens, he flo pressre gen by: [] I ( n k f V V ( ( k In k n hh f s he mass densy of he fld, In s he modfed Bessel fnon of order n, prme ( s he derae h respe o he spaal arable and V s he nform mean flo eloy of oneyng fld. Inserng eqaons ( no ( he follong paral deferenal sysem s obaned. ( ( ( ( (, s E : ( a ( ( D( Eh D( ( ( D Eh Eh D( ( (, Eh s E : ( b ( ( ( ( D D Eh Eh D( ( D Eh Eh ( D Eh ( D( ( ( p, : ( s Eh Eh E Eh

4 Maerals and Srres For he dynam nsably analyss h respe o he flo eloy hs paral dfferenal sysem has o be soled. For he sake of smply, only eqaons relaed o he ranserse dsplaemen are reaned. The dynam goernng eqaons of -layered WCTs oneyng fld are hen reded o he follong paral dfferenal sysem: ( D ( ea Eh ( D (, s h p Eh Eh ( D ( ea Eh ( D (,,,..., sh p Eh Eh For onenene, non dmensonal parameers and he follong arables are sed. ( D h, V /,, (, (, r E s r p j j j Eh r L ea,, V L, r,,, r r r r r r r Vr (7 The follong opled dfferenal eqaons are hen obaned. p (, ( p,,,...,. (8 here p (, s he neraon on be de be j and he an der Waal neraon oeffens j are gen by: [] 7 j E j E 9 j j a svr r a svr r (9 here he sbsrps and j denoed he h and j h 7 layers, respeely, and Ej and E are he j ellpal negrals gen by: / ( d j E m m j j and K ( / j ( [ K os( ] m j j

5 Adaned Maerals esearh Vol. 8 7 The reslng flo pressre s gen by: ( f In k V V, ( ( s k In k Vr Vr For he dynam nsably analyss, he PDE (8 has o sole h respe o gen flo eloy V, small sale parameer and oher physal and maeral parameers. The ranserse dsplaemen of he j h be s assmed o be n he form: ( k A e os( n ( j j here he ampldes A j erfy he follong amplde freqeny relaonshp: ( a ( B B V B V A A, j j j ( A a ( A, j j j kk k k,,..., ( n hh for k,,,..., ( n n ( n n n a kk k k j k k k k j k k k k k k I n (k f f I n (k f k I n (k and B (, B (, B ( k I (k V I (k V s n rs n r s I n (k To deermne he nonral solons of A, he follong haraers eqaon has o be soled. ( ( ( a ( B B V B V... ( ( ( a (... de, ( ( ( ( ( a Ths formlaon allos one o ge he naral freqenes (=,,, for alled CT h respe o he onsdered physal and maeral parameers. The dynam nsably behaor relaed o he flo eloy an be nesgaed based on he presened mehodologal approah. III. meral resls and dsssons For nmeral resls, he sed physal and maeral parameers are as follos: he all hkness h. nm, he mass densy of he be s s.g / m, he mass densy of he fld s g / m, he Poson s rao.9, he effee bendng sffness D.8 ev, n plan f sffness Eh J / m, and he parameers sed n he an der Waal neraon oeffen are: a. A,. A,.98 mev and he rads ( - h, =,,...,.

6 8 Maerals and Srres Table : esonan freqenes of a hree WCTs, L= m, e =. Wh fld f, V = Varables Presen [] Presen [] Presen [] Presen [] ( Hz ( H m n / A/A A/A z A/A A/A Table : esonan freqenes of a S WCTs, L= m. e = e =. Wh fld ( Hz, V = m n Fg.. eal and magnary freqeny pars h respe o he fld eloy V of a doble WCTs (L/=, n= and e a= for dfferen ae nmbers m.

7 Adaned Maerals esearh Vol. 8 9 Fg.. eal and magnary freqeny pars h respe o he dmensonless fld eloy V of a DWCTs (L/ =, n= and e a=. Doed lnes, model ho dw, sold lnes, model h dw. Fg.. eal and magnary freqeny pars h respe o he dmensonless eloy V and e a= and e a=nm. Doed lnes, model h dampng, sold lnes, ho dampng of doble WCTs (L/ =, n=, m=. In hs paper, he deeloped mehodologal approah s llsraed for ml-alled CTs and he effe of aros physal and maeral parameers on he dynam nsably an be obaned. Comparsons of he nmerally obaned free braon freqenes for a hree WCTs and he assoaed amplde raos A /A are lsed n able for e =. The frs sh naral freqenes of a s-alled arbon nanobe h and ho nonloal effe are presened n able h dfferen ae nmbers n and m. I s obsered ha hese freqenes derease by nreasng he nnermos rads een for e a. The effes of he aros oher parameers sh as n, m and he an der Waal parameers an be easly analyzed. The dynam nsably an be nesegaed h respe o he fld eloy V for CT h a hosen nmber of alls and he freqenyeloy dependene an be obaned for CT h dfferen nmbers of alls. For a doble WCT, he effe of he ae nmber m s shon n fgre and he effe of an der Waal neraon n fgre for =.9 nm and he rao L/ =. I s obsered, n fgre, ha he nsably ype s frs he dergene and he seond s he fler one. By nreasng V, he dergene and fler nsables or and m has a srong effe on hese nsables. For he an der Waal effe, s obsered n fgre ha he dergene and fler ral eloes are nreased by he an der Waal parameer. The small sale effe on he freqeny-eloy dependene s presened n fgre for e a = and e a=.nm. Inreasng hs parameer leads o a derease of he naral freqenes as ell as he ral dergene and fler eloes as learly presened n fgre. The flo eloy effe s also shon n hs fgre. I s obsered ha hs dampng has no effe on he ral dergene eloy.

8 Maerals and Srres Conlson: Ths paper presens he dynam nsably analyss of MWCTs oneyng fld based on Donnell shell model. The flo eloy-freqeny dependene s analyzed h respe o aros physal and maeral parameers. The obaned resls shoed a srong dependene beeen he fld eloy and he freqeny of MWCTs. The effe of he an der Waal neraon beeen bes s dsssed and resls sho ha he an der Waal neraon and he small sale effes may sgnfanly nflene he sably of mlalled CT. I has o be noed ha he model ho fld dampng effe shos only he fler hoeer, he model h fld dampng effe shos boh dergene and fler nsables. Aknoledgemens The ahors old lke o aknoledge he sppor proded by he onenon CST-CS (Moroo-Frenh, SPM/ and he ehnal and fnanal sppors of he Kng Abdlazz Unersy, nder gran o. (7- - /HC. eferenes: [] Y. Yan, W. Q. Wang and L. X. Zhang, J. Appled Mahemaal Modellng, (, p. [] A.C. Erngen, J. Appl. Phys., (98, p. 7. [] A.C. Erngen and DGB Edelen, In. J. Eng. Sene, [] X. Q. He, C. M.Wang, Y. Yan, L. X. Zhang and G. H. e, J. of Arh Appl. Meh. 78 (8, p. 7 8 [] Y. Yan, X. Q. He, L.X. Zhang and C. M. Wang, Jornal of Sond and Vbraon 9 (9, p. 8 [] A. Azrar, L. Azrar and A. A. Aljnad, sbmed o Physa E, (. [7] A. Azrar, L. Azrar and A. A. Aljnad, ee de Méanqe Théorqe e Applqée, n press (. [8] L. Wang, J. Compaonal Maerals Sene (9, p [9] Y. Yan, W.Q. Wang and L.X. Zhang, J. Appled Mahemaal Modelng (9, p. [] T. P. Chang and M.F. L, Jornal of Mehans A/solds (, p [] X. Y. Wang, X. Wang and G. G. Sheng, J. Phys. D: Appl. Phys. (7, p. 7 [] K. Dong, B.Y. L and X. Wang, J. Compaonal Maerals Sene (8, p. 9 8 [] Y. G. H, K. M. Le, Q.Wang, X. Q. He and B. I. Yakobson, Jornal of he Mehans and Physs of Solds (8, 7 8 [] Wang Ln and Qao, Compaonal Maerals Sene (8, p.99 [] Y. Yan, W.Q. Wang and L.X. Zhang, Jornal Appled Mahemaal Modelng (, p. 9 []T. ask, Q. Q. and M. Endo, J. Appl. Phys. A 9(8, p. [7] G. Q. Xe, X. Han and S. Y. Long, Inernaonal Jornal of Solds and Srres (7, p. -