Run-Up Flow of a Maxwell Fluid through a Parallel Plate Channel

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1 Amerin Journl of Compuionl Mhemi, 03, 3, Pulihe Online Deemer 03 (hp:// hp://x.oi.org/0.436/jm un-up Flow of Mxwell Flui hrough Prllel Ple Chnnel Sye Yeull Qri, M. Veer Krihn Deprmen of Mhemi, S. Joeph PG College, Kurnool, Ini Deprmen of Mhemi, yleem Univeriy, Kurnool, Ini Emil: eeive Augu 7, 03; revie Sepemer, 03; epe Sepemer 8, 03 Copyrigh 03 Sye Yeull Qri, M. Veer Krihn. Thi i n open e rile iriue uner he Creive Common Ariuion Liene, whih permi unrerie ue, iriuion, n reprouion in ny meium, provie he originl work i properly ie. ABSTACT We onier he flow of n inompreile viou Mxwell flui eween wo prllel ple, iniilly inue y onn preure grien. The preure grien i wihrwn n he upper ple move wih uniform veloiy while he lower ple oninue o e re. The riing flow i referre o run-up flow. The uney governing equion re olve iniil vlue prolem uing Lple rnform ehnique. The expreion for veloiy, her ree on oh ple n ihrge re oine. The ehvior of he veloiy, her ree n m flux h een iue in eil wih repe o vriion in ifferen governing flow prmeer n i preene hrough grph. Keywor: un-up Flow; Mxwell Flui; Lple Trnform; eynol Numer n Prllel Ple Chnnel. Inrouion In ome ehnologil prolem rele o peroleum inury, Luriion ehnology e., he flui flow experiene phenomenon viz. run-up whih rie ue o uen wihrwl of he preure grien uing he flow while i ounrie innneouly move from re. Uner hi phenomenon, ey flow in he unperure e gin uneine ler. Mny reerh worker hve pi enion o he uy of Mxwell flui. In luriion heory n in mny phyil iuion where we ome ro lip flow, here rie l of prolem referre o run-up n pin-up flow. The growing imporne of he ue of non-newonin flui in moern ehnology n inurie h le vriou reerher o emp ivere flow prolem rele o everl non-newonin flui. One uh flui h h re he enion of reerh worker in flui mehni uring he l four ee i he Mxwell flui. Thi heory h everl inuril n ienifi ppliion well, whih omprie pumping flui uh ynhei flui, polymer hikene oil, liqui ryl, niml loo, ynovil flui preen in ynovil join n he heory of luriion (Nuvinmni e l. [-5], Lin n Hung [6]). Kzki n ivlin [7] iniie he uy of hee flow n ler ivlin [8-0] elorely uie he run-up n pin-up flow of vio-eli flu- i eween rigi prllel ple n in irulr geomerie. mhryulu n ju [] inveige he runup flow of viou inompreile flui in long irulr yliner of porou meril. mkrihn [] iue he run-up n pin-up flow rele o uy viou flui. Ler, M. Devkr n T. K. V. Iyengr [4] exmine he run-up flow of n inompreile ouple re flui eween wo infinie rigi prllel ple. The flow w ume o e iniilly inue y onn preure grien eween wo infinie rigi prllel ple. Afer he ey e w ine, he preure grien w uenly wihrwn n he prllel ple were e o move innneouly wih ifferen veloiie in he ireion of he pplie preure grien. The ime epenene of he reuln flow w inveige. Sugunmm e l. [5] nlyze he r-up flow of n inompreile vio-eli ivlin-eriken flui. The iniil flow i ume ue o he movemen of ounrie. A n inn of ime, he ounrie re uenly rough o re n he flow i minine ue o prerie preure grien. Veer Krihn e l. [6] iue he hll urren effe on uney MHD flow of roing Mxwell flui hrough porou meium in prllel ple hnnel. ji ey n Smiv o [7] nlyze run-up flow of viou inompreile flui hrough rengulr pipe, pipe of equilerl ringu- Open Ae

2 98 S. Y. QADI, M. V. KISHNA lr ro eion, prllel ple hnnel n yliner. They olve hem y uing ADI numeril ehnique. Bh [8] exene he nlyi of he me y oniering vio-eli ivlin-eriken flui eween prllel ple ujee o onn uion. Mllewri [9] iue run-up flow of ivlin-eriken flui wih porou lining. Wih he reen reerhe in non-newonin flui flow ie erlier, we onier he flow of n inompreile Mxwell flui eween wo prllel ple, iniilly inue y onn preure grien. The preure grien i uenly wihrwn while he ple re impulively re imulneouly. The up-flow i referre o run-up flow. In hi preen pper, we re uying hi flow of Mxwell flui.. Formulion n Soluion of he Prolem Conier he flow of n inompreile Mxwell flui eween wo infinie rigi prllel ple y = 0 n y = h long he ireion of x-xi (Figure ). Sine he flow i long he x-ireion, we ke he veloiy q uy,,0,0, whih ifie he oninuiy equion. We onier Crein yem O x, y o h he flui flow ke ple wihin ounry ple y = 0 n y = h. The liner momenum equion governing he flow u(y, ) i given y u u p p u () x x y where he eniy, p i he preure, i he oeffiien of vioiy n i he relxion ime. The ounry oniion re U; y h u 0 () 0; y 0 We onier he run-up flow of he Mxwell flui hrough he prllel ple hnnel. Iniilly he flow ue o prerie preure grien wih ounrie re n he ime > 0, he preure grien i wihrwn n he upper ple move wih uniform veloiy while he lower ple oninue o e re. The equion gov- Figure. Configurion of he prolem. erning he iniil flow i u p y x The orreponing ounry oniion re 0; y h u 0; y 0 0 We inroue he non-imenionl vrile * x * y * u * p * x, y, u, p, U h h U U h (3) (4) Uing non-imenionl vrile, he governing equion re (ropping he erik) where, u u u (5) y u P y Uh i he eynol numer, (6) U h p i he Mxwell flui prmeer, P i he on- x n of preure grien. Uing he ounry oniion (4), he Equion (6) reue o P u yy (7) Applying he Lple rnform o he Equion (5) n uing he Equion (7), we ge, Coh Sinh u y A y B y P y y P Now he rnforme ounry oniion re ; y u 0; y 0 On olving Equion (8) uing (9); we ge u y, y (8) (9) P Sinh y Sinh y Sinh Sinh P Sinh P y y P Sinh (0) On king he invere Lple rnform [3] for he Equion (0); we ge Open Ae

3 S. Y. QADI, M. V. KISHNA 99 P u y y y y, n n e 9Sinh y 3Sinh y n n n e 0Sinh y6sinh y n () The her ree on he upper ple n he lower ple re given y u P L y n y0 n n e Coh n n n e 3 4Coh n u U y y n n ( ) e Coh n n n e 3Coh 4 n P The m flux Q i given y P Q uy y0 n n e Coh Coh n n n 3 4 e Coh Coh n () (3) (4) 3. eul n Diuion The flow governe y he non-imenionl prmeer he eynol numer, he Mxwell flui prmeer. The veloiy, he her ree on he ple n ihrge eween he ple re evlue nlyilly n ompuionlly iue for ifferen vriion in he governing prmeer n. I i inereing o noe h he ehvior of he flow very muh epen on he preure grien, in orne wih he run-up flow, he iniil ey flow i ue o he prerie preure grien, while he perure flow i ue o he uen movemen of he ounry in ene of he preure grien. The flow in ireion of he movemen of he ounry my e oniere ul flow, where he flow ue y he preure grien ume o e gin he ireion of he ounry movemen he reverl flow. Figure -9 repreen he ehviour of he veloiy omponen u for vriion in low n high eynol numer wih Mxwell flui prmeer n for vriou vlue of ime. Tle -3 how ree on oh ounrie n m flux. Figure n 3 epi he vriion of he veloiy u lighly inree for inree in low eynol numer while i experiene enhnemen wih inree in high Figure. The veloiy profile for u on low eynol numer wih P, 0., 0.5. Figure 3. The veloiy profile for u on high eynol numer wih P, 0., 0.5. Figure 4. The veloiy profile for u on he Mxwell flui prmeer wih low eynol numer = 0, P =, 0.. Open Ae

4 300 S. Y. QADI, M. V. KISHNA Figure 5. The veloiy profile for u on he Mxwell flui prmeer wih high eynol numer = 0, P =, 0.. Figure 8. The veloiy profile for u wih preure grien P on low eynol numer 0 wih 0., 0.5. Figure 9. The veloiy profile for u wih preure grien P on low eynol numer 0 wih 0., 0.5. Figure 6. Time evelopmen of he veloiy omponen u for 0.5 wih low eynol numer = 0, P =. Tle. The her ree on he upper ple. I II III IV I II III IV Tle. The her ree on he lower ple. I II III IV Figure 7. Time evelopmen of he veloiy omponen u for 0.5 wih high eynol numer 0, P. eynol numer eing he oher prmeer fixe. The veloiy profile (4 n 5) for u on he Mxwell flui prmeer wih low n high eynol numer = 0 n = 000 repeively. The imilr ehviour i oerve erlier menione. We noie h he Mxwell flui prmeer wih he eynol numer ffe he flow in he enire region. An inereing phenomenon i oerve in Figure 6 n 7, he mgniue of he veloiy enhne wih inree in ime while fixing n. Figure 8 n 9 inie h for fixe, n I II III IV , P inree he veloiy eree for ny y. Thi i in orne wih he f h n inree in P implie eree in preure whih nurlly reul in eree of veloiy. The mgniue of he ree on upper n lower ple enhne wih inreing in, n reue in Open Ae

5 S. Y. QADI, M. V. KISHNA 30 Tle 3. The Dihrge eween he ple. I II III IV I II III IV lower ple inree in upper ple wih inreing (Tle n ). The ihrge eween he ple enhne wih inree in oh n (Tle 3). 4. Conluion The run-up flow of n inompreile Mxwell flui eween wo infinie prllel ple i uie uing Lple rnform ehnique. Anlyil expreion for he flui veloiy fiel re oine in Lple rnform omin. The mgniue of he veloiy enhne wih he inreing in oh n n reue wih he inreing in P. The ree on upper n lower ple enhne wih he inreing in, reue in lower ple n inree in upper ple wih inreing. The ihrge eween he ple enhne wih he inreing in oh n. 5. Aknowlegemen The uhor re hnkful o Prof.. Siv Pr, Deprmen of Mhemi, Sri Krihnevry Univeriy, Annpur, Anhr preh, Ini, n Journl for he uppor o evelop hi oumen. EFEENCES [] N. B. Nuvinmni, P. S. Hiremh n G. Guruvrj, Squeeze Film Luriion of Shor Porou Journl Bering wih Couple Sre Flui, Triology Inernionl, Vol. 34, No., 00, pp [] N. B. Nuvinmni, P. S. Hiremh n G. Guruvrj, Surfe oughne Effe in A Shor Porou Journl Bering wih Couple Sre Flui, Flui Dynmi eiul, Vol. 3, No. 5-6, 00, pp [3] N. B. Nuvinmni, P. S. Hiremh n G. Guruvrj, Effe of Surfe oughne on he Couple Sre Squeeze Film eween Sphere n Fl Ple, Triology Inernionl, Vol. 38, No. 5, 005, pp [4] N. B. Nuvinmni, S. T. Fhim n P. S. Hiremh, Hyro Dynmi Luriion of ough Slier Bering wih Couple Sre Flui, Triology Inernionl, Vol. 36, No., 003, pp [5] N. B. Nuvinmni, S. T. Fhim n P. S. Hiremh, Effe of Surfe oughne on Chrerii of Couple Sre Squeeze Film eween Anioropi Porou engulr Ple, Flui Dynmi eiul, Vol. 3, No. 5, 003, pp [6] J.. Lin n C.. Hung, Comine Effe of Non- Newonin Couple Sree n Flui Ineri on he Squeeze Film Chrerii eween Long Cyliner n n Infinie Ple, Flui Dynmi eiul, Vol. 39, No. 8, 007, pp [7] J. Y. Kzki n. S. ivlin, un-up n Spin-Up in Vio-Eli Flui I, heologi A, Vol. 0, No., 98, pp. -7. [8]. S. ivlin, un-up n Spin-Up in Vio-Eli Flui II, heologi A, Vol., No., 98, pp [9]. S. ivlin, un-up n Spin-Up in Vio-Eli Flui III, heologi A, Vol., No. 3, 98, pp. 3-. [0]. S. ivlin, un-up n Spin-Up in Vio-Eli Flui IV, heologi A, Vol., No. 3, 983, pp [] N. Ch. Phi mhryulu n K. A. ju, un-up in Generlize Porou Meium, Inin Journl of Pure Applie Mhemi, Vol. 5, No. 6, 984, pp [] D. mkrihn, Some Prolem in he Dynmi of Flui wih Prile Supenion, Ph.D. Thei, Kkiy Univeriy, Wrngl, 986. [3] G. Honig n U. Hire, A Meho for he Numeril In-verion of Lple Trnform, Journl of Compuionl Applie Mhemi, Vol. 0, No., 984, pp [4] M. Diwkr n T. K. V. Iyengr, un-up Flow of Couple Sre Flui eween Prllel Ple, Nonliner Anlyi: Moelling n Conrol, Vol. 5, No., 00, pp [5] V. Sugunmm, M. S. Lh n N. Sneep, un-up Flow of ivlin-eriken Flui hrough Porou Meium in Chnnel, Inernionl Journl of Mhemil Arhive, Vol., No., 0, pp [6] M. Veer Krihn, S. V. Suneeh n. Siv Pr, Hll Curren Effe on Uney MHD Flow of oing Mxwell Flui hrough Porou Meium, Journl of Ulr Sieni of Phyil Siene, Vol., No., 00, pp [7]. ey Sheelm, Compuionl Tehnique in Trnien Mgneo Hyro Dynmi Duy Viou n un- Up Flow, Ph.D. Thei, Omni Univeriy, Hyer, 99, pp [8] M. Bh, Vio-Eli Flui Flow n he He Trnfer hrough Porou Meium, Ph.D. Thei, SriKrihnevry Univeriy, Annpur, 994. [9] D. Mllewri, Uney Flow of ivlin-eriken Flui hrough Plnnr Chnnel wih Porou Lining, Ph.D. Thei, Sri Pmvhi Mhil Viwviylym, Tirupi, 00. Open Ae

6 30 S. Y. QADI, M. V. KISHNA Appenix ( ) n n n 4 n π 4 n π 4 n π n n n n n, P P n n 3 n n n 4 Coh P 5 n n n n n n n 6 n n n P Coh 7 8 n n Coh n n n n Coh n n P n n n n 9 n Coh n n Coh n n n n n n 0 n Coh n n Coh n n n n n n P Open Ae

7 S. Y. QADI, M. V. KISHNA 303 P n n n n n Coh n n n Coh n n P n n Coh n n n n n n n n n 3 n Coh n n Coh n n P P n n. Coh 4 n n n n n P n n n n n n n n Coh n Coh n n n n P n n Coh n n n n n P n n n n 3 n n Coh n Coh n n n n P n n Coh 4 n n n n n n n Open Ae