Analytical solution of tank drainage for electrically conducting power law fluid

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1 Pepis NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v Aalyical soluio of ak daiae fo elecically coduci powe law fluid K. N. Memo,*, A. M. Siddiqui, Syed Feoz Shah, S. Islam 4 Depame of Basic Sciece, Meha Uivesiy of eieei echoloy jamshoo, Pakisa Pesylvaia Sae Uivesiy, Yok Campus, Edecombe 74, USA Depame of Mahemaics ad Saisics, QUES, Nawabshah, Pakisa 4 Mahemaics, Abdul Wali Kha Uivesiy Mada, KP Pakisa. saeedislam@awkum.edu.pk ABSAC his pape ivesiaes he ak daiae poblem of a isohemal, useady, icompessible elecically coduci Powe law fluid. Aalyic soluio have bee obaied fom ovei coiuiy ad momeum equaios subjec o appopiae bouday codiios by usi Peubaio mehod. he Powe law fluid model soluio wihou MD is eieved fom his poposed model o subsiuio. Declaaio o behalf of velociy pofile, volume flux, aveae velociy, coecio of ime wih espec o leh of he ak ad equieme of ime fo whole daiae of fluid ae acquied. Special effecs of umeous emei paamee s o velociy pofile vz ad deph of he fluid i he ak ae aphically peseed. Keywods: ak daiae, Powe law MD fluid, Aalyical soluio. INODUCION I cue yeas, o-newoia fluids have icease siifica cosideaio o accou of hei umeous bioloical, idusial ad echoloical applicaios. ee few cases of o- Newoia fluids such as ooh pase, dilli mud, eases, pais, blood, polyme mels, clay coais ec. I is a expasive class of fluids so; hee is o ay sile model ha ca hadle all he popeies of such fluids as is doe by he Newoia fluids descibed by he well-kow Navie-Sokes equaio. I his ead, seveal cosiuive equaios have bee poposed o pedic he physical sucue ad behavio of such ypes of fluids fo diffee maeials [-]. 8 by he auhos. Disibued ude a Ceaive Commos CC BY licese.

2 Pepis NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v Pesely he class of o-newoia fluids, he powe law model have bee boadly coceaed o accou of umeical effolessess ad fa eachi mode applicaios. Amid he las fou decades, ciical advaceme has bee ackowleded i he impoveme of diaosic aaeme ad umeical calculaios of powe law liquid seam issues [-5]. he daiae of a fluid houh pipe of a ak ude he acio of aviy is a old, howeve complicaed poblem. he ak may be daied by a aach pipe o may be daied houh evehaded hole oifice siuaio. he pipe possibly could be hoizeal o veical o may coai a complee pipi sysem wih hoizeal exesio ad veical dop wih fiis ad valve, ec. Usually ak has a shape of cylideical coai a veical wall howeve boom may be coical hemisheical o by fla o mih be addiioal shape. hee is someimes ies i daii he ak should be oally dy i which siuaio he boom shape eeds o be accoued fo ad occasioally o. Classificaios of aviy daii fluid s ae used exesively houhou idusies, a small umbe of such classificaios ae: daii codesae io soae, wae disibuio, wase wae maaeme ad dams, eieval of chemicals fom ak fam. he eeaed model will accuaely epese ak daii behavio fo all aks wih a simila seup. Ed effecs, accuacy of ime measueme, accuacy of heih measuemes ad ficio losses will be ake io cosideaio [6]. o day sciece due o pacical coceaio, he sudy of ak daiae flow has eceived siifica cosideaio. Numeous aalyss have podeed he beak dow hese pes of flows sice hei fomulaio. he powe fluid s model have bee uilized fo ak daiae flow by [] o ivesiae ad solve he poblem exacly. Fo simple viscous fluid, he heoy depici he efflux ime cocei a ak has bee efflux ime of a ak has bee ifeed by Cosby [7] ad by Bid, Sewa, ad ihfoo [8], ad addiioally exeded o sysems wih he isalled fiis by aesia [9]. I is foude fac ha, whe fluid is daied by mea of hole fom he ak, he equaio of oicelli s is uilized o defie he dischae velociy field ad flow ae ha is ive [-], hese ypes of he issues fuhe evisied i []. Fo he ubule flow a he exi pipe, he elaio amos he heih of he fluid o he boom of he exi pipe ad he efflux ime is calculaed by []. Fuhe a sho oe o mechaics of he slow daii fo lae ak is wie by [4]. wo dimesioal ad wo laye fo ecaula ak daii

3 Pepis NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v useady flows is ive i [5] ad hee dimesioal fo wo fluid i he cicula ak is sysem by [6]. Efflux ime ad compaiso of cylidical wih diffeeial is specified i [7-] ad slow daii ude he acio of aviy fo lae spheical ak is sudied i [], hey have compae he mahemaical ad expeimeal values ad esablish o be i ood aaeme wih he model. Usae of polyme soluio s fo da educio ude he acio of aviy is paicula i [-]. I his aicle, we cosideed he poblem of ak daiae fo Powe law MD fluid. Aalyical soluios of he cosequeial diffeeial equaios focus o bouday codiios, ae foud by usi peubaio mehod ad he subsiuio peubaio paamee, we eieve he esuls fo Powe law fluid wihou MD []. Also elaioships fo velociy pofile, flow ae, aveae-velociy, deph of fluid i he ak a ay ime ad ime equieme of ime fo o complee daiae ae cosideed. As pe he bes of ou isih, he soluio of he poblem has o bee accoued fo i he lieaue. his pape is sucued by meas of follows: Secio umbe povides basic ovei equaio s fo Powe law MD fluid. Secio umbe deals wih fomulaio ad he soluio of poblem. Secio umbe 4 deals wih flow ae, aveae velociy, elaioships how does he leh of fluid chae s wih espec o ime ad equieme of ime fo o complee daiae. esuls ad discussio ae specified i secio umbe 5, fially coclusios ae deliveed i secio umbe 6. Basic Equaios Esseial ovei equaios fo icompessible Powe law MD fluid flow, diseadi hemal effecs ae: V, DV D p b J B,

4 Pepis NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v he symbol epeses cosa desiy, p sad fo he dyamic pessue, V be he velociy D veco, b epese o he body foce, sads fo he exa sess eso ad he opeao D deoes he maeial deivaive. As a esul oez foce pe ui volume be J B [,, B v ], whee is he elecical coduciviy, B [,, B ] be he uifom maeic field, hee B be he applied maeic field ad J be he cue desiy J, which is z J E V B, 4 B J. 5 ee E is he elecic field which is o cosideed i his sudy ad be he maeic pemeabiliy [4-7]. he exa sess eso descibi a Powe law fluid [4-5] is made by: ad A, 6 eff A : A eff, 7 hee epese cosisecy coefficie, is he powe-law idex ad A be he s ivli Eickse eso, epeseed as: ak daiae A V V. hik abou a ak of cylidical shape havi a icompessible Powe law MD fluid. e suppose he adius of he ak is, diamee of he ak be D ad be he iiial deph of he fluid i he ak. he fluid which is pese i he ak, which is daied dow houh by cylidical pipe havi leh ad adius be. Pomoe all he moe lei be he deph of fluid i he ak a a all he ime. Ou saey is o deemie he velociy, pessue, volume flux, aveae velociy, elaioship how does he ime flucuae wih leh ad he ime equied fo fiish daiae. ee we ake

Pepis www.pepis.o NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v cylidical coodiae s,, z wih -axis omal o cylidical pipe ad z -axis alo he cee of he pipe i veical diecio.

5 Pepis NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v cylidical coodiae s,, z wih -axis omal o cylidical pipe ad z -axis alo he cee of he pipe i veical diecio. As he flow is oly io he diecio of z, he ad compoe s of velociy field ae equal o zeo, V [ v, v, vz ],, vz,. 8 Fiue : Geomey of he ak daiae flow dow houh pipe. Uilizi velociy field 8, he equaio of coiuiy is idisiuishably fulfilled ad he momeum equaio dimeshio owad compoe of momeum: p, 9 compoe of momeum: p, z compoe of momeum: v z p vz z v z B vz. Accodi o defiiio of maiude eeds ha he esul be a posiive umbe. hus we sellec v of si ha poduces z i equaio depeds o whehe he deivaive v z is posiive o eaive. I he cue example as iceases, he velociy deceases - he velociy is a is maximum a he cee of he pipe. he deivaive v z vz vz is eaive, ad heefoe.

6 Pepis NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v Fom equaios 9 - we ca see ha he equaio of moio is ow quie simple, yieldi ha he pessue is oly fucio of z ad ad he equaio o be solved fo v z, is vz p vz B vz. z Equaio is a paial diffeeial equaio fo p ad v z. he velociy emais ealy cosa wih ime i he pipe flow due o slow daii such ha we may elec ime v deivaive z. Also flow be i he pipe is due o boh hydosaic pessue ad aviy, a he pipe eace ad exi he pessue is, a z, p p, a z, p p, so ha p z he equaio of moio hus educes o he elaed bouday codiios ae a, vz B vz. vz d, a, v 6 z 4 5 Peubaio soluio: We ake B o be a small paamee ad velociy pofile v z, ca be saed as a powe seies ive by, v z, v v v... 7 By uizi equaio 7 io he equaio 4, 5 ad 6 ad equai coefficies of like powe s of, we acquie he followi se of poblems alo wih hei associaed bouday

7 codiio s: zeoh ode poblem:, : v 8 wih elaed bouday codiios,, a d dv 9. a v Fis ode poblem:, : v d dv d dv d d houh beloi codiios,, a d dv. a v Velociy pofile: Zeoh ode soluio: he soluio of equaio 8 by mea of bouday codiios fom equaios 9 ad is. v 4 Fis-ode soluio: eplaci he zeoh ode soluio fom equaio 4, io equaio ad subjec o codiios fom equaio ad is specified by v 5 hus he soluio wih peubaio echique coec upo fis ode is, z v 6 ee impoa oe ha if we selec o he peubaio paamee i equaio 6, we e Pepis NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v

8 he soluio fo same poblem wih Powe law fluid wihou MD [] ad fo sei, ad we e he soluio fo he Newoia wihouh MD fluid [8]. 4 Flow ae, aveae velociy ad ime equied fo o complee daiae he flow ae " Q pe ui widh is specified houh he fomula., d v Q z 7 Usi velociy pofile 6 i equaio 7, oe ca calculae he flow ae. 4 Q 8 We deemie he aveae velociy, V by uilisi he fomula. Q V 9 Afe subsiui he value of flow ae io equaio 9, so he aveae velociy will be.. V Mass balace ove he eie ak is. Q d d Subsiui flow ae fom equaio 8 io equaio ad he sepaai vaiables o boh sides of equaio oe obais l 7 Pepis NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v

9 ad he ime equied fo complee daiae is obaied by aki i l 7 Fiue : Effec of o velociy pofile,.,,, 5., /.78,.5 cm B cm cm cm poise whe Fiue : Effec of o velociy pofile, Pepis NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v

10 Pepis NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v whe.5 poise,.78 / cm,., 5cm, cm,., cm. Fiue 6: Effec of o velociy pofile, Fiue 4: Effec of o velociy pofile, whe.5 poise,.78 / cm,. 5cm, cm,., B.5. whe.5 poise, 5cm,., cm, cm,., B.5. Fiue 5: Effec of o velociy pofile, whe.5 poise,.78 / cm,. cm, cm,., B.5. Fiue 7: Effec of o velociy pofile, whe.5poise,.78 / cm 5cm, cm,., B,..5. whe.78 / cm, 5cm, cm, cm,., B.5,.. Fiue 8: Effec of o velociy pofile,

11 Pepis NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v.5poise,.78 / cm, B.5 cm,., Fiue 9: Effec of o velociy pofile, whe.78 / cm, 5cm, cm, cm,., B.5,.5. Fiue : Effec of o ime w.. o deph, whe.6 poise,.8 / cm,., cm, cm, 5, B.5, 5 Fiue : Effec of o flow ae, whe.5 poise,.78 / cm, cm,.,., B.5. Fiue : Effec of o ime w.. o deph, whe.6 poise,.8 / cm,., cm, cm, 5, B.5,. Fiue : Effec of o flow ae, whe

12 Pepis NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v 5 esuls ad discussio I he ovehead secios we coemplaed ak daiae poblem uilizi a icompessible Powe law MD fluid, Aalyical soluio s fo he oliea diffeeial equaio is acquies by usi peubaio mehod. he vaiaio of velociy pofile v z, flow ae Q ad ime equied fo o complee daiae has bee ivesiaed o diffee paamees. he effecs of he elecical coduciviy, applied maeic field B, dyamic viscosiy, deph, leh of pipe, pipe adius, desiy ad fo Powe law idex o velociy pofile ae obseved houh fiues - 9 as well as effec of he deph ad Powe law idex o flow ae ae show i fiues - ad effec of he adius of ak as well as Powe law idex o o ime equied o complee daiae is examied i fiue. I fiues 9 i is deeced ha he maiude of velociy iceases as he expasio of elecical coduciviy, applied maeic field B, deph, pipe adius ad desiy ad deceases fo he icease of leh of pipe, dyamic viscosiy ad Powe law idex. Fom fiue 9 we ca summaized ha as he fluid is becomi hie he maiude of velociy iceases. I fiues - fo he icease we deeced ha flow ae iceases ad decease fo iceasi. Fiues ae ploed fo he ime equied fo o complee daiae wih espec o deph, we poi ou ha wih icease of adius of ak ad powe law idex he i will ake a ime fo compleely dai fom he ak. I is evide fom fiue ha fluid desceds moe quickly as he value of deceases. 6 Coclusios We have peseed esuls fo useady, icompessible, isohemal ak daiae flow fo he Powe law MD fluid ad obaied exac soluios fo velociy pofile, flow ae, aveae velociy, elaio of deph of he ak ad ime equied fo complee daiae. ee i is oed ha fo he peubaio paamee, soluio of he poblem educes o Powe law fluid [] ad fo subsiui ad, we ecove he soluio cocei o Newoia wihouh MD fluid [8]. A elaioship, how does he ime shif wih leh is ifeed. I is oiced ha as he fluid is ei o be hicke, velociy of he fluid deceases ad hie fo aki velociy vice vesa.

13 Pepis NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v efeeces [] N. S. Deshpade, M. Baiou, Vibaioal flow of o-newoia fluids, Chemical Eieei Sciece, 56, [] M. Kemiha, X. Fak, S. Poci,. Z. i, Oii of he eaive wake behid a bubble isi i o-newoia fluids, Chemical Eieei Sciece, 6, [] Memo, K. N., S. Islam, A. M. Siddiqui, S. A. Kha, N. A. Zafa ad M. Akam, if ad daiae of Elecically Coduci Powe aw Fluid o a Veical Cylide, Ieaioal Joual Applied Mahemaics ad Ifomaio Scieces, Appl. Mah. If. Sci. 8, No., [4] A. M. Siddiqui, M. Akam, K. N. Memo, S. Islam ad Khalid kha, Wihdawal ad daiae of hi film flow o a veical Cylide, Scieific eseach ad Essays, 7, [5] J. N. Kapu, Flows of Powe-aw Fluid Pas a Fla Plae wih uifom sucio ad bewee wo paallel plaes wih uifom sucio ad ijecio, Joual of he physical sociey of Japa, 8, [6] Joe eoaed, S.. Mackli, Jeife Ouyomi, ak daiae modeli, Oklahoma Sae Uivesiy, School of Chemical Eieei, Ui opeaio laboaoy /5/9 [7] E. J. Cosby, Expeimes i aspo Pheomea, Wiley, New Yok, 96. [8]. B. Bid, W. E. Sewa, ad E. N. ihfoo, aspo Pheomea, Wiley, New Yok, 96. [9] D. aesia, Chemical Eieei aboaoy Maual, NJI, Newak, 984. [] De Neves, N., Fluid Mechaics fo Chemical Eiees, d ed., McGaw- ill, New Yok, NY. p.64, 99. [] Bid,.B., W.E. Sewa ad E.N. ihfoo, aspo Pheomea, d ed., Joh Wiley & Sos, New Yok, NY, pp. 99, 6 8, 8 9,. [] Doald D. Joye ad Bade C. Bae, he ak Daiae Poblem evisied: Do hese Equaios Acually Wok?he Caadia Joual of Chemical Eieei, Volume 8, Ocobe

14 Pepis NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v [] Wilkes, J.O. Fluid Mechaics fo Chemical Eiees, Peice-all P, Uppe Saddle ive, NJ,. p.68, 999. [4] David b. Va doe, Efflux ime fom aks wih Exi Pipes ad Fiis, I. J. E Ed. Vol. 5, No., pp. 6±, 999. Josue Njock ibiia, Mechaics of he slow daii of a lae ak ude aviy, Am. J. Phys. 7 ~!, Novembe,. [5] awece K. Fobes, Useady daii flows fom a ecaula ak, Physics of Fluids 9, 84, 7. [6] awece K. Fobes, Gaeme C. ocki, Useady daii of a fluid fom a cicula ak, Applied Mahemaical Modelli ,. [7] Ch. V. Subbaao, Compaiso of Efflux ime bewee Cylidical ad Coical aks houh a Exi Pipe Ieaioal Joual of Applied Sciece ad Eieei. 9, : -4,. [8] A. Uma Devi, P.V. Gopal Sih, G.V.S.K. eddy, S.J. Dhawal ad C..V. Subbaao, A eview o Efflux ime, Middle-Eas Joual of Scieific eseach 9 : 57-6,. [9] G.V.S.K.eddy, Ch.V.Subbaao, Compaiso of efflux imes bewee cylidical ad spheical ak houh a exi pipe, Ieaioal Joual of Eieei & Applied Scieces IJEAS Vol., Issue 6-68,. [] A. Uma Devi, D. V. Padma & C. V. Subbaao, Effec of Polyme Soluios o Efflux ime fo wo Exi Pipe Sysem, Vol. No. Sepembe-Novembe, [] Ch.V.Subbaao, P.Sivasa ao, G.M.J.aju ad V.S..K.Pasad, Slow daii of lae spheical ak ude aviy, Elixi Chem. E. 5,46-48,. [] Ch. V. Subbaaoa, Madhavi, D. Appala Naidu, ad P. Ki, Use of polyme soluios fo da educio i aviy dive flow sysems, Ieaioal Joual of Applied Sciece ad Eieei., : 59-69,. [] Subbaao, Chiavui Vekaa, Da educio by sufaca soluios i aviy dive flow sysems, Ia. J. Chem. Chem. E. Vol., No.,

15 Pepis NO PEE-EVIEWED Posed: 5 Febuay 8 doi:.944/pepis8..v [4] Mohyuddi M, Goz. esoace behavio of viscoelasic fluid i Poiseuille flow i he pesece of a asvesal maeic field. Ieaioal Joual fo umeical mehods i fluids, 49, p [5] ossow VJ. O flow of elecically coduci fluids ove a fla plae i he pesece of a asvese maeic field, NASA, epo o. 58, 489, 958. [6] Pop I., Kumai M., Nah G. Cojuae MD flow pas a fla plae, Aca Mech. 6: [7] Abel S., Veea P., ajaopal K., Payi VK. No-Newoia maeohydodyamic flow ove a sechi suface wih hea ad mass asfe, I. J. No-iea Mech. 9: [8]. C. Papaasasiou, Applied Fluid Mechaics, P Peice all 994.

Exact Solution of Unsteady Tank Drainage for Ellis Fluid

Exact Solution of Unsteady Tank Drainage for Ellis Fluid Joual of Applied Fluid Mechaics, Vol, No 6, pp 69-66, Available olie a wwwjafmoliee, IN 75-57, EIN 75-65 DOI: 95/jafm69 Exac oluio of Useady ak Daiae fo Ellis Fluid K N Memo,, F hah ad A M iddiqui Depame