Exact Solution of Unsteady Tank Drainage for Ellis Fluid

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1 Joual of Applied Fluid Mechaics, Vol, No 6, pp 69-66, Available olie a wwwjafmoliee, IN 75-57, EIN DOI: 95/jafm69 Exac oluio of Useady ak Daiae fo Ellis Fluid K N Memo,, F hah ad A M iddiqui Depame of B, MUE, Jamshoo, idh, Pakisa Pesylvaia ae Uivesiy, Yok Campus, Edecombe 7, UA Depame of Mahemaics ad aisics, QUE, Nawabshah, idh, Pakisa Coespodi auho kamaaimemo@mailcom eceived Febuay 5, ; acceped Apil, ABAC I his wok, we ivesiae he he poblem of a useady ak daiae while cosidei a isohemal ad icompessible Ellis fluid Exac soluio is oe fo a esuli o-liea PDE paial diffeeial equaio- subjec o pope bouday codiios- he special cases such as Newoia, Powe law, ad as well as Biham soluio ae eieved fom his suesed model of Ellis fluid Expessios fo velociy pofile, shea sess o he pipe, volume flux, aveae velociy, ad he elaioship bewee he ime vay wih he deph of a ak ad he ime equied fo complee daiae ae obaied Impacs of diffee developi paamees o velociy pofile v ad deph ae illusaed aphically he aaloy of he Ellis, powe law, Newoia, ad Biham Plasic fluids fo he elaio of deph wih espec o ime, ufold ha he ak ca be empy fase fo Ellis fluid as compaed o is special cases Keywods: ak daiae, Ellis fluid; Exac soluio NOMENCAUE A fis ivli Eickse es d diamee of he pipe maeial deivaive f body foce aviaioal acceleaio deph of fluid i he ak a ay ime iiial deph of he fluid leh of he pipe powe law idex p dyamic pessue p pessue ile p pessue oule Q flow ae adius of he pipe adius of he ak V yield adius i cicula pipe flow adial coodiae exa sess eso yield sess maeial cosa ime velociy veco V aveae velociy axial coodiae Geek lees maeial cosa scala quaiy maeial cosa cosa desiy INODUCION I ece yeas, o-newoia fluids have emaied he focus of mode eseach due o hei umeous bioloical, idusial ad echoloical applicaios such as ooh pase, dilli mud, eases, pais, blood, polyme mels, clay coais ec No- Newoia fluid is a expasive class of fluids; so, ulike Navie-okes equaio fo Newoia fluids, hee is o a sile model ha ca delieae all he popeies of o-newoia fluids Chhaba & ichadso, 999 ece, seveal cosiuive equaios e, Powe law fluid model, secod ode fluid model, hid ode fluid model, isco fluid model, eyi fluid model, ad Pha-hie-ae fluid model have bee poposed o pedic he physical sucue ad behavio of vaious ypes of o-newoia fluids Memo e al, ; Du & ajaopal, 995 Fo such models, he exac soluios ae ae o be obaied fo he equaios of moio specially fo o-newoia fluids, because of he oliea aue of hose equaios Faooq, ahim, Islam, &

2 K N Memo e al / JAFM, Vol, No 6, pp 69-66, iddiqui, ; Faooq, ahim, Islam, & Aif, ; a few umbe of exac soluios ae foud i he exisi eview of lieaue pecially whe he cylidical coodiaes ae used hese ypes of soluios come eve o be ae, fo he easo ha of he olieaiy i he hihe ode viscosiy pa ad i ieial pa Faooq e al, Numeical esuls of he diffeeial equaios, o poblem how coec hey ae, sill have o bee he exac soluios fo he easo ha of he paamees ivolved i equaio, which should have o be ive values fo each esul hee ae ceai facos ad easos fo which he exac soluios ae cosideed o be impoa Wa, 99; Wa, 99; Behabi & iddiqui, 99; iddiqui & Kaloi, 96; ajaopal & Gupa, 9; Wa, 966, such as; i hese exac soluios siify basic fluiddyamic flows Cosequely, hee is possibiliy, fo he fudameal episodes defied by he Navie- okes equaios, o be examied moe hoouhly due o he uifom aioaliy of exac soluios ii hese soluios Exac help i scuiii phase fo empiical, umeical, expeimeal, ad asympoic mehods Eve houh, he ieaio of he equaios of moio is compleely made feasible houh compue echiques, wheeas he accuacy of he oucomes is possible o be esablished by compai hem wih a exac soluio A deailed ad excepioal eview of exac soluios of he Navie-okes equaio is povided by Wa Wa, 99 udy of ak daiae flow has eceived siifica aeio due o he focus upo he pacical applicaios of hese flows i he coempoay scieces ice he fomulaio of hese ypes of flows, hee have bee may eseach aemps fo hei aalysis he Newoia fluid has bee used fo ak daiae flow by Papaasasiou, 99 ad powe law fluid by Bid, ewa ad ihfoo, 96 o ivesiae ad solve he poblem exacly he heoy descibi he efflux ime of a ak has bee deived by Cosby Cosby, 96ad by Bid, ewa, ad ihfoo Bid, ewa ad ihfoo, 96, ad fuhe exeded o sysems wih he isalled fiis by aesia aesia, 9 I is a foudi fac ha, whe he ak is daied by a hole, oicelli s equaio is used o descibe he dischae velociy ad flow ae which is ive i De Neves, 5; Bid, ewa ad ihfoo, 96hese poblems ae evisied i Joye & Bae, Ude codiios of ubule flow i he exi pipe, elaioship bewee he efflux ime ad heih of he liquid o he boom of he exi pipe is calculaed by Wilkes, 6, fuhe he Mechaics of he slow daii fo a lae ak ude aviy is biefly explaied i Va Doe & oche, 999 Useady daii flows fom a ecaula ak wo dimesioal ad wo layeed is ive by Fobes & ocki, 7, ad fo cicula aks a hee dimesioal daii flow fo wo-fluid sysem is sudied by Fobes & ocki, Efflux ime ad compaiso of a cylidical ak wih diffeeial fom is ive i ubbaao, ; Devi, ih, eddy, Dhawal & ubbaao, ; eddy & ubbaao, ; Devi, Padma & ubbaao, low daii of lae spheical ak ude he acio of aviy is sudied by ubbaao, ao, aju & Pasad,, i which mahemaical ad expeimeal values have bee compaed, ad foud o be i ood aeeme wih he model Usae of polyme soluios fo da educio i aviy dive flow sysems is ive i ubbaao, Madhavi, Naidu & Ki, ; ubbaao, Yadav & Ki,, ad exac soluio of ak daiae fo Newoia fluid wih slip codiio have bee solved by Memo, iddiqui, & hah, 7 I his mauscip, we sudied ak daiae poblem of Ellis fluid Bid, ewa ad ihfoo, ; Afaasiev, Müch, & Wae, 7; Ali, Abbasi, & Ahmad, 5, esuli o-liea paial diffeeial equaios-subjec o bouday codiios-ae acquied aalyically wih exac soluios ad also we have eieved he special cases, such as Newoia, Powe law ad Biham plasic fluid Fo he vey hih yield sess soluio of he poblem eieve fo Newoia case Papaasasiou, 99, o eplaceme,, we e he soluios fo Powe law case Bid, ewa ad ihfoo, ad whe we subsiue we e he soluio fo Biham plasic fluid ubsequely, expessios fo velociy pofile, shea sess o pipe, flow ae, aveae velociy, deph of fluid i he ak, ad he ime equied fo complee daiae ae obaied As pe he bes of ou isih, he soluio of he poblem has o bee accoued fo i he exisi lieaue I secio of his pape, he ovei equaios of Ellis fluid model ae specified ecio povides fomulaio ad soluios of he ak daiae poblem, such as velociy pofile, shea sess o he pipe, flow ae, aveae velociy, shea sess o he pipe, elaioship bewee he vaiaio i ime ad i he deph of fluid of he ak, ad also bewee he ime equied fo complee daiae ecio deals wih is special cases of Ellis fluid model he esuls ad discussio ae icluded i secio 5, while secio 6 cocludes he sudy BAIC EQUAION Esseial ovei equaios fo icompessible Ellis fluid flow, diseadi hemal effecs ae: V, DV p f, D he symbol V epeses velociy veco, deoes he cosa desiy, p be he dyamic pessue, f is he body foce ad he exa sess eso he opeao deoes he maeial deivaive he exa sess eso descibi a Ellis Bid, ewa ad 6

3 K N Memo e al / JAFM, Vol, No 6, pp 69-66, ihfoo, ; Afaasiev, Müch, & Wae, 7; Ali, Abbasi, & Ahmad, 5 is made by: A, Whee he paamee is defied as: :, A V V 5 ee A is fis ivili- Eickse eso, is a scala quaiy ad Eq coais hee cosas,, which may be deemied expeimeally fo each fluid ad also he symbol epese i Eq is : ace 6 pecial cases ae esseial fo Ellis fluid model, a low shea sess, his model cools o a Newoia fluid model, ad appopiaely a hih shea aes he poposed model coves io he Powe law fluid model ANK DAINAGE e suppose a cylidical ak is of he adius coaii a isohemal, icompessible Ellis fluid ad he pipe of he diamee d is aached a he cee of he boom of he ak he iiial deph of he fluid is ake o be he fluid i he ak is daied dow houh by a a pipe havi leh ad adius Addiioally moe, he deph of fluid i he ak a ay ime is assumed o be i he ak, flow of he fluid i he pipe is due o hydosaic pessue of he fluid i he ak ad aviy ou plae is o deemie velociy pofile, shea sess o he pipe, flow ae, aveae velociy, elaioship bewee he vaiaio i ime ad i he deph of fluid of he ak, ad also wih he ime equied fo complee daiae ee we have use cylidical coodiaes, wih -axis omal o he pipe, ad -axis alo he cee of he pipe i veical diecio As he flow is idividual i he -diecio, ad he ad compoes of velociy veco V ae equal o eo,,, v,,, V, [ v, v, v ] 7 Fi ak daiae flow dow by mea of cicula pipe Papaasasiou, 99 By meas of Eq 7, he equaio of coiuiy is idisiuishably fulfilled; ad we have elec he coveced pa of he acceleaio, he momeum Eq dimiishes owads compoe of momeum : p p, compoe of momeum: compoe of momeum :, 9 v p he flow i he pipe of adius is due o boh hydosaic pessue ad aviy A he eace ad exi i he pipe, he pessues ca be descibed as: a, p p, a, p p, so ha, p he velociy i he pipe flow almos cosa wih ime, due o slow daii, so ha we may elec he ime deivaive v he equaio fo momeum fo his combied flow is: he associaed bouday codiio s a daiae pipe a,, ceelie 5 a, v o-slip 6 Accodi o he ceelie bouday codiio 5 shea sesses ae eo i he middle of he pipe as a esul of he maximisaio of he velociy of he fluid O he ohe had, as pe No-lip he bouday codiio 6 he velociy of he fluid paicles is eo a he walls of he pipe By iiai Eq wih espec o, keepi as a cosa, we obai f 7 f is abiaay fucio of,by usi ee ceelie bouday codiio 5, we e Now by usi Eq i he cosiuive equaios, he aaeme of Eqs ad is, v 9 We coside as posiive because of daiae flow poblem, heefoe fo elaxi o he absolue codiio ad by usi Eq i Eq 9, we e 6

4 K N Memo e al / JAFM, Vol, No 6, pp 69-66, 6 v o ieai wih espec o, we have: v f Whee f is a abiaay fucio of, afe usi o slip cosiio, we e f ece Eq educes o v Flow ae, Aveae Velociy, hea ess o he Pipe ad he elaio of ime wih Deph of he Fluid fo Complee Daiae he flow ae Q pe ui widh is specified houh he fomula,, d v dd v Q Usi velociy pofile i Eq, he flow ae ca be calculaed Q 5 We deemie he aveae velociy, V by usi he followi fomula 6 Q V 6 o by he use of Eq 5 i Eq 6, aveae velociy of he fluid ca be V 7 hea sess o he pipe is ive by Mass balace ove he eie ak is Q d d 9 ubsiui flow ae fom Eq 5 io Eq 9, ad ewie he diffeeial equaio as followi d d Iiae o Eq o boh sides, we e l C By meas of iiial codiio a fo ak daiae, we acquie ha l C he fluid elevaio i he ak he desceds slowly aeei o e ad he ime equied fo complee daiae fo Ellis fluid is obaied by aki, i l PECIA CAE: NEWONIAN, POWE AW, BINGAM PAIC FUID Newoia Fluid Whe yield sess is vey hih, mahemaically fo, Ellis soluios educes o he

5 K N Memo e al / JAFM, Vol, No 6, pp 69-66, Newoia fluid Papaasasiou, 99, ha is v Q V e l 9, which povides a easoable esimae fo he ak daiae fo Newoia fluid Powe-law Fluid Fo subsiuio, Ellis soluio ives he esul fo Powe law fluid Bid, ewa ad ihfoo,, v Q V By usi Eq i 9, we will e, ad ime equied fo complee daiae fom a ak fo Powe law fluid is by aki i Biham Plasic Fluid his fluid is defied by he model: v, fo 5 v, fo 6 hus, fo sesses lae ha Eq 9 is simplified o 5 by sei, while fo 6 is obaied fom fo he easo hihly viscous ad also yield sess is vey hih Fom Eq, which upo subsiuio of fo yields he disace bewee he adius of he pipe beyod which Eq 6 holds: v fo, 7 v fo 9 he volume flow ae is ive by Q v d v d 5 houh usi equaio umbe 7 ad, i equaio umbe 9, we will e oal flow ae Q By usi Eq 5 i Eq 6, he aveae film velociy will be, V Maki he use of Eq 5 i 9 we will e 5 5 e 5, ad he ime equied fo complee daiae fom a ak fo Biham plasic fluid is by aki i v, 6 l 5 5 Fi Effec of o velociy pofile fo Ellis fluid, whe poise, / cm, 5 cm, 55 dy / cm, cm, cm 9 9 6

6 K N Memo e al / JAFM, Vol, No 6, pp 69-66,, v 6 6 Fi Effec of o velociy pofile fo Ellis fluid, whe poise, / cm, 5 cm, 55 dy / cm, Fi 6 Effec of o deph fo Ellis fluid, whe cm, / cm, 5cm poise, 55 dy / cm, cm, 5cm 7, v Fi Diffeece of o he velociy pofile fo Ellis fluid, whe, 7 / cm,, v poise, 55 dy / cm, cm, 5cm 5 5 Fi 5 Diffeece of o he velociy pofile fo Ellis fluid, whe, 7 /cm, poise, 55 dy / cm, cm, 5cm 6 5 Fi 7 Effec of o deph fo Ellis fluid, whe, / cm, 9 6 poise, 55 dy / cm, cm, 5cm able elaio of deph wih espec o ime fo Ellis fluid ad is special cases whe 5 6 cm, 7 / cm, 7, 5 poise, 55 dy / cm, cm, 5 cm, 5 cm, cm, 9 Ellis Newoi a Powe aw Biham Plasic

7 K N Memo e al / JAFM, Vol, No 6, pp 69-66, 5 EU AND DICUION I his wok, we examied useady ak daiae poblem by meas of a isohemal, icompessible Ellis fluid, houh which exac esuls fo he oliea diffeeial equaio wee obaied he vaiaio of velociy pofile v ad deph has bee ivesiaed o diffee paamees he effecs of he Ellis idex, leh of pipe, viscosiy, deph of he ak o velociy pofile ae deeced houh Fis -5, ad effec of he,adius of he ak, ad he deph is show i he Fis 6 ad 7 I he Fi, i is obseved ha he maiude of velociy iceases wih he icease i Ellis idex his explais ha he maiude of velociy decease wih he decease of fluid viscosiy he effec of ad o velociy pofile v is show i he Fis -5 I hese fiues, i ca be oiced ha as ad icease, he maiude of he velociy disibuio also iceases; ad vice vesa he velociy disibuio, viscosiy, adius of he pipe, ad he deph of he ak ae foud o be ieliked wih each ohe he leh of he pipe is also foud o be ivesely popoioal o he velociy pofile, which ca be see i Fi 5 he effecs of Ellis idex ad adius of he ak o heih of he fluid i ak, ae show i he Fis 6 ad 7 A icease i causesdecease i he deph of fluid ; ad he deph iceases whe adius of ak is iceased A compaiso of deph of Ellis fluid wih is special cases is peseed i able - wih fixed paamees-, which ae meioed i capio of he able he esuls peseed i able ae compued umeically abulaed daa showi deph a vaious imes idicaes ha he deph of Ellis fluid is lowe ha he deph of is special cases his also explais he easo ha velociy pofile of Ellis fluid is hihe ha is special cases 6 CONCUION Cosidei equaio fo useady ak daiae flow fo icompessible ad isohemal Ellis fluid, we have obaied exac soluios fo he poposed model Also we have exacly eieved he special cases fo Ellis model, such as Newoia, Powe law, ad Biham plasic A elaioship bewee,, ad he vaiaio i ime ad he deph is deived I is, oed ha as he fluid becomes hicke, he velociy of he fluid iceases; ad i is also impoa o oe ha Ellis fluid dais quickly as compaed o is special cases EFEENCE Afaasiev, K, A Müch ad B Wae 7 adau-evich poblem fo o-newoia liquids Physical eview E 76, 67 Ali, N, A Abbasi ad I Ahmad 5 Chael flow of Ellis fluid due o peisalsis AIP Advaces 59, 97 Behabi, A M ad A M iddiqui 99 Ceai soluios of he equaios of he plaa moio of a secod ade fluid fo seady ad useady cases Aca Mechaica 9-, 5-96 Bid, B 96 ewa, WE, ad ihfoo, EN aspo pheomea, Bid, B, W E ewa ad E N ihfoo aspo pheomea JohWiley & os, New Yok Chhaba, P ad J F ichadso 999 No- Newoia Flow: Fudameals ad Eieei Applicaios Buewoh-eiema Cosby, E J 96 Expeimes i aspo pheomea Joh Wiley & os De Neves, N 5 Fluid mechaics fo chemical eiees McGaw-ill ciece/eieei/mah Devi, A, P V ih, Gopal, G V K eddy, J Dhawal ad C ubbaao, A eview o Efflux ime Middle-Eas Joual of cieific eseach 9 : 57-6 Devi, A, D Padma ad C ubbaao Effec of Polyme oluios o Efflux ime fo wo Exi Pipe ysem Ieaioal Joual of Advaces i Eieei, ciece ad echoloy IJAE epembe- Novembe, -6 Du, J E ad K ajaopal 995 Fluids of diffeeial ype: ciical eview ad hemodyamic aalysis Ieaioal Joual of Eieei ciece 5, Faooq, M, M ahim, Islam ad A M iddiqui eady Poiseuille flow ad hea asfe of couple sess fluids bewee wo paallel iclied plaes wih vaiable viscosiy Joual of he Associaio of Aab Uivesiies fo Basic ad Applied cieces, 9- Faooq, M, M ahim, Islam ad M Aif, eies soluios of lifi ad daiae poblems of a oisohemal modified secod ade fluid usi a veical cylide Joual of Applied Mahemaics Fobes, K ad G C ocki 7 Useady daii flows fom a ecaula ak Physics of Fluids 9, Fobes, Kad G C ocki Useady daii of a fluid fom a cicula ak Applied Mahemaical Modelli, aesia, D 9 Chemical Eieei aboaoy Maual NJI, Newak Joye, D D ad B C Bae he ak daiae poblem evisied: Do hese equaios acually wok? he Caadia Joual of Chemical Eieei 5, 5-57 Memo, K N, A Kha, Islam, N A Zafa, F hah ad A M iddiqui Useady Daiae of Elecically Coduci Powe aw Fluid Applied Mahemaics & Ifomaio cieces 5, 7 65

8 K N Memo e al / JAFM, Vol, No 6, pp 69-66, Memo, K, A iddiqui ad hah 7 Exac oluio of ak Daiae fo Newoia Fluid wih lip Codiio idh Uivesiy eseach Joual-UJ ciece eies 9 Papaasasiou, C 99 Applied fluid mechaics Peice all ajaopal, K ad A Gupa, 9 O a class of exac soluios o he equaios of moio of a secod ade fluid Ieaioal Joual of Eieei ciece 97, 9- eddy, G V K ad C V ubbaao Compaiso of Efflux imes bewee cylidical ad spheical ak houh a exi pipe, Ieaioal Joual of Eieei & Applied cieces IJEA, 6-6 iddiqui, A M ad P N Kaloi 96 Ceai ivese soluios of a o-newoia fluid Ieaioal Joual of No-iea Mechaics 6, 59-7 ubbaao, C V Compaiso of efflux ime bewee cylidical ad coical aks houh a exi pipe Ieaioal Joual of Applied ciece ad Eieei 9, ubbaao, C V, Madhavi, D A Naidu ad P Ki Use of Polyme oluios fo Da educio i Gaviy Dive Flow ysems Ieaioal Joual of Applied ciece ad Eieei, ubbaao, C V, P ao, G M J aju ad V K Pasad low daii of lae spheical ak ude aviy Chemical Eieei, Elixi Ieaioal Joual5 6- ubbaao, C V, P Yadav ad P Ki Da educio by sufaca soluios i aviy dive flow sysems Iaia Joual of Chemisy ad Chemical Eieei IJCCE, 9- Va Doe, D B ad E C oche, 999 Efflux ime fom aks wih exi pipes ad fiis Ieaioal Joual of Eieei Educaio 5, 6- Wa, C Y 966 O a class of exac soluios of he Navie-okes equaios Joual of Applied Mechaics, Wa, C Y 99 Exac soluios of he useady Navie-okes equaios Applied Mechaics eviews, 69- Wa, C Y 99 Exac soluios of he seadysae Navie-okes equaios Aual eview of Fluid Mechaics, Wilkes, J O 6 Fluid Mechaics fo Chemical Eiees wih Micofluidics ad CFD Peaso Educaio 66

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