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1 Global Joural of Scic Froir Rsarch Mahaics & Dcisio Scics Volu Issu Vrsio. Typ : Doubl lid Pr Rviwd Iraioal Rsarch Joural Publishr: Global Jourals Ic. US Oli ISSN: 9-66 & i ISSN: Oscillaory Fr Covciv Flow Through Porous Mdiu i a Roaig Vrical Porous Chal y K.D.Sigh & lphosa Mahw Hiachal adsh Uivrsiy Shila bsrac - horical aalysis of h ffcs of prabiliy ad h ijcio/sucio o a oscillaory fr covciv flow of a viscous icoprssibl fluid hrough a highly porous diu boudd bw wo ifii vrical porous plas is prsd. Th ir sys roas abou h axis oral o h plas of h plas wih uifor agular vlociy Ω. For sall ad larg roaios h dpdc of h sady ad usady rsula vlociis ad hir phas diffrcs o various parars ar discussd i dail. Kywords : Oscillaory roaig porous chal Porous diu Fr covcio. GJSFR-F Classicaio : rica Mahaical Sociy subjc classificaio: 76 W oscillaoryfrcovciv flow hroughporousdiuiaroaig vricalporous chal Sricly as pr h copliac ad rgulaios of :. K.D.Sigh & lphosa Mahw.This is a rsarch/rviw papr disribud udr h rs of h Craiv Coos ribuio-nocorcial. Upord Lics hp://craivcoos.org/licss/by-c/./ priig all o corcial us disribuio ad rproducio i ay diu providd h origial work is proprly cid.
![i h filds of agriculural girig o sudy h udrgroud war rsourcs i prolu chology o sudy h ov of aural gas oil ad war hrough h oil chals/rsrvoirs[].](/docs-images/96/129219065/images/2-4.jpg "sris of ivsigaios hav b ad by diffr scholars whr h porous diu is ihr boudd by horizoal vrical surfacs or paralll porous plas.")
2 Rf.. CollisR.E Flo Fluids hrough porous Marials Rihold Nw York 96.. DWisR.J.M Flow hrough Porous Mdia. cadic ss Nw York969. Schidggr.E Th physics of flow hrough porous dia. Uivrsiy of Toroo ss Toroo 97. Oscillaory Fr Covciv Flow Through Porous Mdiu i a Roaig Vrical Porous Chal K.D.Sigh & lphosa Mahw bsrac - horical aalysis of h ffcs of prabiliy ad h ijcio/sucio o a oscillaory fr covciv flow of a viscous icoprssibl fluid hrough a highly porous diu boudd bw wo ifii vrical porous plas is prsd. Th ir sys roas abou h axis oral o h plas of h plas wih uifor agular vlociy. For sall ad larg roaios h dpdc of h sady ad usady rsula vlociis ad hir phas diffrcs o various parars ar discussd i dail. Kywords : Oscillaory roaig porous chal Porous diu Fr covcio. I. INTRODUCTION Fr covcio flows i a roaig porous diu ar of gra irs i a ubr of idusrial applicaios such as fibr ad graular isulaio gohral syss c. uoyacy is also of iporac i a viro whr diffrc of praurs ca giv ris o coplicad flow pars. I rc yars h probls of fr covcio hav aracd h aio of a larg ubr of scholars du o is divrs applicaios. Th flow of fluids hrough highly porous diu boudd by vrical porous plas fid urous girig ad gophysical applicaios viz. i h filds of agriculural girig o sudy h udrgroud war rsourcs i prolu chology o sudy h ov of aural gas oil ad war hrough h oil chals/rsrvoirs[]. sris of ivsigaios hav b ad by diffr scholars whr h porous diu is ihr boudd by horizoal vrical surfacs or paralll porous plas. Rapis [8] aalyzd h usady flow hrough a porous diu boudd by a ifii porous pla subjcd o a cosa sucio ad variabl praur. Rapis ad Prdikis [9] furhr sudid h probl of fr covciv flow hrough a porous diu boudd by a vrical porous pla wih cosa sucio wh h fr sra vlociy oscillas i i abou a cosa a valu. par fro h abov wo disioal sudis a ubr of hr disioal flows hrough porous diu hav also b sudid. Sigh al. [6] aalyzd h ffcs of priodic prabiliy o h hr disioal flow hrough highly porous diu boudd by a ifii porous surfac. Sigh al. [5] also ivsigad h ffc of prabiliy variaio o h ha rasfr ad hr disioal flow hrough a highly porous diu boudd by a ifii porous pla wih cosa sucio. Sigh ad Vra [] sudid furhr h flow of a viscous icoprssibl fluid hrough porous diu wh h fr sra vlociy oscillas i i abou a o-zro cosa a. uhor : Dpar of Mahaics ICDEOL Hiachal adsh Uivrsiy Shila 7 5Idia. E-ail : gail.co. uhor : Dpar of Mahaics S.Thoas Collg Thrissur Idia. E-ail : alphos_5@yahoo.co.uk 5 Global Joural of Scic Froir Rsarch F Volu XII Issu III V rsio I Global Jourals Ic. US
3 5 Global Joural of Scic Froir Rsarch F Volu XII Issu III V rsio I I h rc yars a ubr of sudis hav appard i h liraur ivolvig roaio o a grar or lssr x viz. Vidyaidhu ad Niga [9] Gupa [] Jaa ad Daa [5] Sigh [7]. Ijcio/sucio ffcs hav also b sudid xsivly for horizoal porous pla i roaig fra of rfrcs by Gaapahy [] Mazudr [7] Mazudr al. [6] Soudalgkar ad Pop [8] Sigh [] for diffr physical siuaio. Th flows of fluids hrough porous diu boudd by roaig porous chals fid ay idusrial applicaios paricularly i h filds of crifugaio filraio ad purificaio procsss. I viw of hs applicaios Sigh ad Shara [] sudid h ffc of h prabiliy of h porous diu o h hr disioal Cou flow ad ha rasfr. I h prs papr a ap has b ad o sudy h ffcs of h prabiliy of h porous diu ad ijcio/sucio hrough h porous paralll vrical plas o h fr covciv flow hrough a highly porous diu. Th ir sys roas abou a axis prpdicular o h plas of h plas. II. MTHEMTICL NLYSIS Cosidr a oscillaory fr covciv flow of a viscous icoprssibl fluid hrough a highly porous diu boudd bw wo ifii vrical porous plas disac d apar. cosa ijcio vlociy w is applid a h saioary pla z ad h sa cosa sucio vlociy w is applid a h pla z d which is oscillaig i is ow pla wih a vlociy U abou a o-zro cosa a vlociy U. Th origi is assud o b a h pla z ad h chal is orid vrically upward alog h x -axis. Th chal roas as a rigid body wih uifor agular vlociy abou h z -axis. Sic h plas ar ifii i x all h physical uaiis xcp h prssur dpd oly o z ad. Doig h vlociy copos u v w i h x y z dircios rspcivly ad praur by T h flow i h roaig sys is govrd by h followig uaios: v u w w v w z u z px / u zz v g T Td u / z p T y vzz u v / / k wtz Tzz C p whr is h kiaic viscosiy is h i is h dsiy ad p is h odifid prssur T is h praur Cp is h spcific ha a cosa prssur k is h hral coduciviy g is h acclraio du o graviy h coffici of volu xpasio ad is h prabiliy of h diu. Th boudary codiios for h probl ar u = v = T d cos T T T a z = u = U = U + cos v = T Td a z = d 5 Rf. 5. Jaa R.N. ad Daa N. 977 Cou flow ad ha rasfr i a roaig sys ca Mch Sigh K.D. oscillaory hydroagic Cou flow i a roaig sys J. ppl. Mah ad Mch Global Jourals Ic. US
4 Nos whr is h frucy of oscillaios ad is a vry sall posiiv cosa. y iroducig h followig o-disioal uaiis z / d u u /U v v /U d / h roaio parar d / h frucy parar w d / h ijcio/sucio parar / d h prabiliy parar T Td g T Td Gr T Td U w h Grashof C p ubr h adl ubr ad supprssig h sars h uaios k o bco whr u iv. U Gr i U U / 6 7 Th boudary codiios 5 ca also b wri i coplx oaios as U i i i i a. I ordr o solv h sys of uaios 6 ad 7 subjc o h boudary codiios 8 w assu i i o 9 i i o. Subsiuig 9 ad io 6 ad 7 ad coparig h haroic ad oharoic rs w g l l Gr Gr Gr ' '' ' '' i 5 ' '' i 6 whr l i i ad i. Th corrspodig rasford boudary codiios rduc o a 8 5 Global Joural of Scic Froir Rsarch F Volu XII Issu III V rsio I Global Jourals Ic. US
5 . a a 7 Th soluios of uaios o 6 udr h boudary codiios 7 ar whr i i i i l l 5 6 ] [ l Gr ] [ Gr ] [ Gr ] [ Gr Global Jourals Ic. US 5 Global Joural of Scic Froir Rsarch Volu XII Issu rsio I V III F Nos
6 5 [ Gr ] Nos III. RESULTS ND DISCUSSION Now for h rsula vlociis ad h shar srsss of h sady ad usady flow w wri u iv ad i i iv u. 5 Th soluio 8 corrspods o h sady par which givs u as h priary ad v as h scodary vlociy copos. Th apliud ad h phas diffrc du o hs priary ad scodary vlociis for h sady flow ar giv by R a v u u v 6 / Th rsula vlociy R for h sady par is prsd i Fig..a b for sall ad larg valus of roaios rspcivly of h vrical porous chal. Th wo valus of h adl ubr as.7 ad 7. ar chos o rprs air ad war rspcivly. I Fig..a b h curv I corrspods o h flow hrough a ordiary diu. I is vry clar fro Fig..a ha R icrass wih h Grashof ubr Gr h roaio of h chal sucio vlociy ad h prabiliy parar. I h cas of adl ubr R is icrasig ar h oscillaig pla. Siilarly for larg roaios show i Fig.b. h apliud R icrass wih Gr h fr covcio currs ad h prabiliy parar ad R also oscillas wih h icras of h roaio of h chal. I is irsig o o ha icras of adl ubr lads o a icras of R ar h oscillaig pla bu o a dcras ar h saioary pla. Howvr h ffcs of h sucio/ijcio a h plas ar rvrsd i.. h apliud R icrass ar h saioary pla ad dcrass hrafr. Th phas diffrc for h sady flow is show graphically i Fig.a b for sall ad larg roaios rspcivly. Fig..a shows ha h phas agl is dcrasig 55 Global Joural of Scic Froir Rsarch F Volu XII Issu III V rsio I Global Jourals Ic. US
7 ar h oscillaig pla wih h icras of Gr or or ad bu icrass wih h prabiliy parar. Siilarly for larg roaios show i Fig.b. h phas diffrc dcrass wih roaio ad adl ubr. u h icras of prabiliy parar Grashof ubr Gr ad h sucio/ijcio a h plas lads o a icras of. Th apliud ad h phas diffrc of shar srsss a h saioary pla for h sady flow ca b obaid as ad a / 7 r x y or oy ox Nos 56 Global Joural of Scic Froir Rsarch F Volu XII Issu III V rsio I 8 i oy whr. ox / Hr ox ad oy ar rspcivly h shar srsss a h saioary pla du o h priary ad scodary vlociy copos. Th urical valus of h apliud r of h sady shar srss ad h phas diffrc of h shar srsss a h saioary pla for h Gr r r Tabl : Valus of r ad r for various Gr ad. sady flow ar prsd i Tabl -. Th prabiliy parar h Grashof ubrgr ad h roaio parar lad o a icras of for boh h cass r Global Jourals Ic. US
8 Nos of sall or larg roaios. I is also obsrvd ha r dcrass wih ad for sall ad larg roaios. Siilarly h valus for h sady phas diffrc icrass wih h sucio parar ad h prabiliy parar for boh h cass of sall or larg roaios. u h ffc is rvrs i h cas of adl ubr. Th icras of lads o a icras i r for sall roaios. u h ffc will b rvrs i h cas of larg roaios. Th soluios 9 ad oghr giv h usady par of h flow. Th usady priary ad scodary vlociy copos u ad v rspcivly for h flucuaig flow ca b obaid as v u R al R al cos I I si r 9 R al R al si I I cos Th rsula vlociy or apliud ad h phas diffrc of h usady flow ar giv by R u a v u v / For h usady par h rsula vlociy or h apliud R ar prsd i Fig..a b. for h wo cass of roaio sall ad larg. I Fig..a b h curv I corrspods o h flow hrough a ordiary diu. I is obsrvd fro figur.a for sall roaios ha R icrass wih adl ubr fr covcio curr Gr h sucio/ijcio parar ad prabiliy parar bu dcrass wih h roaio parar ad h frucy of oscillaios. Fig..b for larg roaios clarly shows ha h apliud R icrass wih all h parar Gr xcp ha wih h roaio parar R dcrass ar h oscillaig plas. Th phas diffrc for h usady par is show i Figur. a b. I Fig..a b h curv I corrspods o h flow hrough a ordiary diu. Figur.a for sall roaios shows ha h phas diffrc icrass wih h adl ubr ad h frucy of oscillaios bu dcrass wih h Grashof ubr Gr h sucio parar h prabiliy parar. d wih h fasr roaio of h chal icrass ar h saioary pla. I is also vid fro Figur.b ha icras of or Gr or or lads o a dcras i bu h icras of h roaio parar frucy of oscillaios boh lad o a icras i. For h usady par of h flow h apliud ad h phas diffrc of shar srsss a h saioary pla = ca b obaid as 57 Global Joural of Scic Froir Rsarch F Volu XII Issu III V rsio I which givs u / iv x i y / a 9 r x y r y / x Th apliud r of h usady shar srss ar show graphically i Figur 5.a b rspcivly for sall ad larg roaios. Fig.5.a b h curv I corrspods o h flow hrough a ordiary diu. I is irsig o o ha h shar srss icrass Global Jourals Ic. US
9 58 Global Joural of Scic Froir Rsarch F Volu XII Issu III V rsio I sharply for sall oscillaios of h frucy ad hrafr dcrass abruply for largr frucy of oscillaios. This figur shows clarly ha h shar srss icrass wih icrasiggr or or. Howvr h ffcs of adl ubr ad h prabiliy parar ar rvrsd. For largr roaio h variaios of shar srss r ar prsd i Figur 5. b. This figur shows ha h apliud r icrass wih h fr covcio currgr h adl ubr h sucio parar h roaio parar ad prabiliy parar. Th phas diffrc r of h usady shar srss is show graphically i Figur 6.a b rspcivly for sall ad larg roaios. I is irsig o o fro hs figurs ha gos o icrasig wih icrasig frucy of oscillaios for boh sall r ad larg roaios. Th phas diffrc r dcrass for boh sall ad larg roaios wih h icras of Grashof ubr Gr ad sucio parar. Howvr for sall roaios icrass for all valus of frucy of oscillaios ad for larg roaios r r dcrass vry ar h oscillaig pla. Th ffcs of adl ubr ad h prabiliy parar lad o a icras i roaios. REFERENCES RÉFÉRENCES REFERENCIS r r vry whr for larg or sall. CollisR.E Flo Fluids hrough porous Marials Rihold Nw York 96.. DWisR.J.M Flow hrough Porous Mdia. cadic ss Nw York969. Gaapahy R. 99 o o oscillaory Cou flow i a roaig sys SME J. ppl. Mch Gupa.S. 97a Eka layr o a porous pla Phys. Fluids Jaa R.N. ad Daa N. 977 Cou flow ad ha rasfr i a roaig sys ca Mch Mazudr.S. Gupa.S. ad Dua N. 976 Flow ad ha rasfr i h hydroagic Eka layr o a porous pla wih hall ffcs I. J. Ha Mass Trasfr Mazudr.S. 99 xac soluio of oscillaory Cou flow i a roaig sys SME J. ppl. Mch Rapis.. 98 Usady fr covcio hrough porous diu I. J. Egg. sci Rapis.. ad Prdikis C.P. 985Oscillaory flow hrough a porous diu by h prsc of fr covciv flow I. J. Egg. sci Schidggr.E Th physics of flow hrough porous dia. Uivrsiy of Toroo ss Toroo 97.. Sigh K.D. oscillaory hydroagic Cou flow i a roaig sys J. ppl. Mah ad Mch Sigh K.D. 999 Thr disioal Cou flow wih raspiraio coolig Z. agw. Mah. Phys Sigh K.D. ad Vra G.N. 995 Thr disioal oscillaory flow hrough a porous diu ZMM. Z. agw. Mah. Mch Sigh K.D ad Shara Raksh. Thr disioal Cou flow hrough a porous diu wih ha rasfr Idia J. pur appl. Mah Sigh K.D. ad Shara Raksh ad Chad Kh. Thr disioal flucuaig flow ad ha rasfr hrough porous diu wih variabl prabiliy ZMM. Z. agw. Mah. Mch Nos Global Jourals Ic. US
10 Nos 6. Sigh K.D. Chad Kh ad Vra G.N. 995 Ha rasfr i a hr disioal flow hrough a porous diu wih priodic prabiliy ZMM. Z. agw. Mah.Mch Sigh K.D. Gorla M.G. ad Has Raj. 5 priodic soluio of oscillaory Cou flow hrough porous diu i roaig sys Idia J. pur appl. Mah Soudalgkar V. M. ad Pop I. 97 O hydroagic flow i a roaig fluid pas a ifii porous pla J. ppl. Mah. d Mch Vidyaidhu V. ad Niga S.D. 967 Scodary flow i a roaig chal J. Mah d Phys. Sci Gr I a II 59 R R b III Gr I Global Joural of Scic Froir Rsarch F Volu XII Issu III V rsio I II III Fig. a b : Rsula vlociy R for sall ad larg roaios du o u ad v Global Jourals Ic. US
11 a G I II III Nos 6 Global Joural of Scic Froir Rsarch F Volu XII Issu III V rsio I b Gr I II III Fig. a b : Phas agl for sall ad larg roaios du o u ad v Global Jourals Ic. US
12 a Gr I II Nos III R 6 R b Gr I II III Global Joural of Scic Froir Rsarch F Volu XII Issu III V rsio I Fig. a b : Rsula vlociy R for sall ad larg roaios du o u ad v Global Jourals Ic. US
13 a Gr I II III Nos 6 Global Joural of Scic Froir Rsarch F Volu XII Issu III V rsio I b Gr I II III IV V Fig. a b : Phas agl for sall ad larg roaios du o u ad v Global Jourals Ic. US
14 a Nos r Gr I r II III b Gr I II III Global Joural of Scic Froir Rsarch F Volu XII Issu III V rsio I Fig. 5 ab : Th apliud r of usady shar srsss for sall ad larg roaios a. Global Jourals Ic. US
15 Gr a I II III Nos r 6 Global Joural of Scic Froir Rsarch F Volu XII Issu III V rsio I r Gr I II III b Fig. 6 a b : Th phas diffrc of usady shar srsss for sall r ad larg roaios a. Global Jourals Ic. US
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