A Polynomial Expansion of Axial Velocity Profiles. to Solve Transient Laminar Flow in Elastic Pipe

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1 Aed Mhemc ceces, o. 9, 5, o. 9, HIKAI Ld, h://d.do.org/.988/ms A Poyom Eso of A eocy Profes o ove Trse Lmr Fow Esc Pe H. mr Lborory of Mechcs & Eergy, Uversy Mohmed Agd - Fcuy of ceces, b, Morocco B. Bhrr Fud Mechcs Grou, Eergy & Evrome, Hss II Uversy - Lborory, DIA ENET, Mohmmed, Morocco K. Guerou Lborory of Mechcs & Eergy, Uversy Mohmed Agd - Fcuy of ceces, b, Morocco F. Akef Fud Mechcs Grou, Eergy & Evrome, Hss II Uversy - Lborory, DIA ENET, Mohmmed, Morocco Coyrgh 4 H. mr, B. Bhrr, K. Guerou d F. Akef. Ths s oe ccess rce dsrbued uder he Creve Commos Arbuo Lcese, whch erms uresrced use, dsrbuo, d reroduco y medum, rovded he org work s roery ced. Absrc Ths sudy rovdes heorec d umerc modeg of sher sress due o he frco of rse mr fow o e w. Ths work s smfed mode of Prdo e. []. I s bsed o he eso of he seous veocy rofes of he fow oyom seres me d rd vrbe cross he seco of e. The se r dervves equos obed from he coservo of mss d he heorem of momeum s he soved by he mehod of chrcerscs. The resus obed re good greeme wh hose of Homboe d oueu [], sedy se Newo mr fow.

2 448 H. mr e. Keywords: Trse her sress, Newo mr fow, eocy rofes, Poyom eso, Mehod of chrcerscs Noos A : seco of e : ceery of wer hmmer : coeffce of oyom eso j D (,) : er dmeer of he e D (, : er dmeer of he e me e : hckess of he e w D m D e : verge dmeer E : Youg s moduus of he e w g : cceero of grvy G : mr j H j : mr H : ezomerc hed e he reservor : de j: de, P : ressure, verge ressure q : he fu r : rd vrbe r : dmesoess vrbe s :eroy : source erm v : seous veocy : verge veocy : wegh verge veocy : buk moduus of wer :dymc vscosy of he fud : kemc vscosy : chored coeffce of he e : desy of wer : gob sher sress : me T: emerure Iroduco I rse e fow, he esse r of he eergy dsso comes from he ressure oss due o frco of he fud o he w of he e. The frco rse e fow dffers from he frco for sedy se e fow, reeer d Wye [3, 4]. The dscreces re roduced by dfferece veocy rofe, urbuece d rso from mr o urbue fow d vce vers. There re umber of usedy frco modes whch hve bee roosed he erure. We c, rcy, ce he works of Zeke [5], Trkh [6], Kgw e. [7], Brow [8], uzuk e. [9], rdy e. []. I hs works, he frco erm s deede o seous me fow veocy d weghs for s veocy chges. I dffere wy, Bruoe e. [], hs eressed he frco erm deede fuco of seous me fow veocy, seous oc cceero d seous covecve cceero. Geery, dffcues rse yss of rse urbue fow. Modes from he erure re cbred for cer fow codos wheres he deveome of geer frco mode rse urbue fow s subjec of esve reserch wordwde. Comred o hese oe-dmeso modes rdy e. [] deveoed wo-dme-

3 A oyom eso of veocy rofes 449 so mode. I hs bee show h, corry o he qus-sory regme, he veocy rofes re o rboc. ecey, o evue he w sher sress rse mr fow es, Prdo e. [] hve deveoed sem-yc mehod bsed, essey, o he oyom seres eso of seous veocy rofes s fuco of rd vrbe d me seco of e. Ths mehod s ssoced wh he mehod of chrcerscs. Ths work s, essey, devoed o revey sme do of hs mode o ccue he veocy rofes, ressure, me fow veocy d w sher sress he rse e. The se dffere equos obed re of hyerboc ye d sube o be resoved by he mehod of chrcerscs. Assumos d bsc equos Ths sudy s coduced uder he ssumo of symmerc usedy fow of Newo, seroc d comressbe fud. The deformo of he e w s of ow mude. The er erms re eggbe d he e s modeed by juoso of deede rgs whou mss. The rdus of he e s suffcey eggbe comred wh s egh eough h he curre es of fud re srgh Assume, furhermore, h he ogud veocy grdes re very ow comred wh he rsverse grdes. The bsc reos for he fud re: Coservo of mss: v d () momeum equo: d v d z v ( r ) r r r () (3) r eo (3) shows h he ressure s subsy cos ech seco of he e d s equ o s verge vue. The seous deformo of he e s reed o he ressure by he Hooke s w [4]: D(, D(,) D(,) D(,) e E By roducg he verge veocy of he fow cross seco of e: vda/ da vrdr / rdr v r dr A A Where we oed by he reduced vrbe. r r /

4 45 H. mr e. The egrg of equos () d () o cross seco A of he e rovdes he foowg sysem of r dffere equos: ( A) ( A ) 4 D The w sher sress deeds o he veocy rofes (4) (5) Eergy equo By usg eroy s se vrbe o eress he eergy of he fud, we hve he foowg reo dffere: ds q 4 (6) d T D T The hermodymc behvour of he fud resus he reosh ( s, ) Whch, by dffereo c be wre he form: Where we u: d d d T ds c (7) s T c P s P, c P d re resecvey, he fud comressby, coeffce cos eroy, he secfc he d coeffce of voumerc eso cos ressure. I he cse of seroc fow, we c ob he foowg reo: secfc he d coeffce of voumerc eso cos ressure. I he cse of seroc fow, we c ob he foowg reo: d d () Ths eds, fy, o he foowg dffere sysem o sove: (8) (9) (4)

5 A oyom eso of veocy rofes 45 (4b) Where he source erm s reed o he w sher sress by: 4 / D / / D / ee werhmmer ceery m 3 Modeg he w sress The sysem (4) d (5) of r dffere equos s hyerboc d s erfecy sube o be resoved by he mehod of chrcerscs. To sove, s frs ecessry o kow he w sher sress, whch sef deeds of he veocy rofes. The echque of ccuo roosed by Prdo e.[] s o mode he veocy rofe ech seco of e d ech me by oyom erm of he dmesoess vrbe. r r / he form: v(, r, j j (, ( r ) jj () Where he summo eeds o rbrry se of eger umbers greer or equ o. The seous veocy s defed, verfy, uomcy, he codo of dheso he w rereseed by he reduced vrbe r e shoud be be o rerese, so, he codo of he sory mr fow of Poseue - Hge: 3. Deermo of he coeffces HP ( r ) () By roducg he weghed verge veoces such s: v (, r, r dr v (, r, r A (, ( ) v (, r, r d r A r da r For,... dm( J ), he eressos re obed for he coeffces j : d r d r dm( J ) j Gj (3) Where, cosderg he eresso () of he veocy rofes,

6 45 H. mr e. j Gj ( ) ( r ) r d r (4) j From he eresso () veocy rofe, s so ossbe, o deduce for mr fow of Newo fud, he w sress he form: v J (,, dm( ) jgj (5) r Therefore, Ths mode cosders he w sress,, o oy, ccordg o he verge seed s he cse of qus-sory mode, bu so fuco weghed verge veoces,,,,...dm( J). 3. Ccuo of weghed verge veoces Iegrg equo () o rgh seco of he e fer muco by for,...dm( J), eds o dffere sysem: r, Where he source erms A v ( r ) r r r A r dr dr (6) (7) ( ) r r v r r d r jj dm( J ) k H j G jk k The coeffces of he mr H re defed by: ( ) j H j ( j) : Kemc vscosy of he fud, he weghed w sress d w sher sress he w s: (9) I summry, we hve o sove he foowg se of dervve r equos: (8)

7 A oyom eso of veocy rofes 453 () () () 4 Modeg he w sress The umerc souo of hyerboc sysems (9), () d () ssoced o d boudry codos c esy be obed by he usu mehod of chrcerscs [6], [7]. The sysem c be esy egred by he mehod of fe dffereces og he chrcersc curves soe e. Ths eds o he foowg gebrc reosh, o ccue he vues of he vrous rmeers me + s fuco of hose me : P P ) ( (3) Aog he chrcersc of soe d d / P P ) ( (4) Aog he chrcersc d d / d, (5) for =,,.dm(j)-, og he chrcersc / d d Where he dces, re resecvey he dscrezo me d sce. 5 I d boudry codos The codos s fuy esbshed sedy d mr fow of eyods umber ess h, d he bce for he e w. A me =, he vve s suddey cosed. The boudry codos re ddo o he ressure mosed by he k o he usrem ed, he seous cosg of vve o he dowsrem.

8 454 H. mr e. 6 Aco d resus Fg. : dgrm of he sysem suded I order o usre he resus d vde hs mode, we cosder he reev rmeers of he eerece crred ou by Homboe d oueu [], corresodg o he suo show fgure, bove, whch: Pe egh, 36m Pe rdus,. 7m 5 Kemc vscosy, 3.96 m / s ( 7C) Wer hmmer ceery, 34.4m / s The sedy se veocy before cosg he vve,.8m / s eyods umber, e 8. The hegh of he fud he reservor, suosed cos, H ( m) The robem ws soved for hree oos of he se J. / J, 6,, dm(j )- = 3, =,,,3 b/ b, 3, 4, 6,8,9,, c/, 3, 4, 5,6,7,8,9,, J dm(jb)=8, =,,,3,4,5,6,7 J dm(jb)=, =,,,3,4,5,6,7,8,9, c The eressos for he sher sress he w for he hree cses re, resecvey: oo : (-793, , , ,6973) 3 (-6, , , , ) (-594, ,386-49, ,3953) (-984, , ,3498+9,9573) (-396, , , ,635483) Due o he comey of he eressos of he erms for he oher oos, we hve med ourseves rbrry o he eresso of w sher sress d hs eds o:

9 A oyom eso of veocy rofes 455 oo: b (-66,4849+3, , ,853+3, , , ,48377 ) Oo c: (-365, , , , , , , , , ,6699 ) I hese cos, we yss he resose o fs cosg of he vve he dowsrem of he e. The fgures, 3 d 4 corresod o he evouo versus me, of he o-dmeso reseos of he ressure d he me veocy of he fow he e mdo d he vve dowsrem of he e. The grhs he fgure 5 usre, he sme codos, he evouo of he o-dmeso sress he med of he e. The resus for ech fgure show he sueroso of dffere oos gs he qus sory se. Oe c oce he cer dfferece bewee he qus-sory mode d he mode reseed. However, we c see from he grhs h he verge veocy d ressure re o deede uo he order of he oyom romo of rofes veocy. These resus re, quvey, comred o hose foud by Homboe e. d re erfec greeme. The fgure 6 shows h hs mode, ows, ddo o he ressure, verge veocy d sress o hve more formo o he veocy rofes rse mr fow e. P(3)/(ro*c*HP) ressure =L/,5, ,5 - *c/l ressure =L/ oo [ DIM4 (,6,,)] ressure =L/ oo b [ DIM8 (,3,4,6,8,9,,)] ressure =L/ oo c [ DIM (,3,4,5,6,7,8,9,,)] ressure =L/ QUAI-TATIONNAIE Fgure.: Pressure e mdo P(L)/(ro*c*HP) ressure he vve =L,5, ,5 - -,5 *c/l Fgure.3: Pressure he vve ressure he vve oo [ DIM4 (,6,,)] ressure he vve oo b [ DIM8 (,3,4,6,8,9,,)] ressure he vve oo c [ DIM (,3,4,5,6,7,8,9,,)] ressure he vve QUAI- TATIONNAIE

10 456 H. mr e. me veocy =L/,,,9,8,7,6,5,4,3,, -, -, ,3 -,4 -,5 -,6 -,7 -,8 -,9 - -, (3)/HP *c/l Fgure.4: Me veocy e mdo me veocy oo [ DIM4 (,6,,)] me veocy oo b [ DIM8 (,3,4,6,8,9,,)] me veocy oo c [ DIM (,3,4,5,6,7,8,9,,)] me veocy QUAI-TATIONNAIE Tw(L/,/Tw(L/,) w sher sress =L/ *c/l w sher sress =L/ oo [DIM4 (,6,,)] w sher sress =L/ oo b [ DIM8 (,3,4,6,8,9,,)] w sher sress =L/ oo c [DIM(,3,4,5,6,7,8,9,,)] w sher sress =L/ qussore Fgure.5: ress he e mdo veocy rofes seeced mes oo [DIM 4 (,6,,)], r/,8,6,4 =L/* =L/*+T/4 =L/*+T/ =L/*+3*T/4 =L/*+T, -,5 - -,5,5,5 v(r,l/)/hp Fgure.6: Profes veocy he e mdo 7 Cocuso Ths sudy o ou h, comred o he qus-sory fow mode, hs mode kg o ccou he vro of veocy rofes hs dvge o roduce sgfc correco o he rse sress of fow e. However, we fd h he cresg he degree oyom eso of veocy rofes hs effecs o he veocy rofes bu o o he me veocy, ressure d sress of fow o e. A resus for ech of he chose ses re cose o ech oher. Ths mode hs he dvge o requre e ddo erms o gve beer rereseo of he w sher sress d, hus, c esy be used ogsde he esg codes o ccue he mr rse fow es.

11 A oyom eso of veocy rofes 457 efereces [] A. E. Lrreeguy,. A. Prdo, A Trse sher sress mode for he yss of mr wer-hmmer robems, o er IAH Jour of Hydruc eserch, (). h://d.do.org/.8/ [] E. L. Homboe, W. T. oueu, The effec of vscous sher o rse qude es, Jour of Bsc Egeerg AME, 89 (967), h://d.do.org/.5/ [3]. L. reeer, E. B. Wye, Fud Trses, McGrw-H, New York, UA, 978. [4] E. B. Wye, Advces he use of MOC he usedy fow, Proc. 4 h I. Co/. O Pressure urges, BHA, Bh, Egd, (983), [5] W. Zeke, Frequecy-deede frco rse e fow, Jour of Bsc Egeerg, AME, 9 (968), 9 5. h://d.do.org/.5/ [6] A. K. Trkh, A effce mehod for smug frequecy-deede frco rse qud fow, Jour of Fuds Egeerg, AME, 97 (975), [7] F. T. Brow, O weghg fucos for he smuo of usedy urbue Fow, Foru o Usedy Fow, AME, New Ores, UA, FED, 5 (984), 6-8. [8]. Jg-Cho, C. Ygg, A effce rome eresso for rse fow of hgh vscous fud hydruc ees, Proc. 6h I. Co/. o Pressure urges, BHA, Crfed, Egd, (989), [9]. o, K. uzuk, d T. Tkeom, Imrovg Zeke's mehod of smug frequecy-deede frco mr qud e fow, Jour of Fuds Egeerg, AME, 3 (99), h://d.do.org/.5/.9656 [] K. L. Hwg, A. E. rdy, A chrcerscs mode of rse frco es, Jour of Hydruc eserch, IAH, 9 (99), h://d.do.org/.8/ [] B. Bruoe, U. M. Go, d M. Greco, ome remrks o he momeum equo for fs rses, I. Meeg o Hydruc Trses wh Coum ero. 9h oud Tbe, IAH, ec,, (99), eceved: November 4, 4; Pubshed: Jury 7, 5