Stokes Flow of an Incompressible Couple Stress Fluid past a Porous Spheroidal Shell

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1 Soe Flow of a Iopeble ouple Se Flu pa a oou Spheoal Shell T. Sa Lah aha T.K.V Iyega ba The pee pape eal wh he poble of he Soe flow of a ouple e flu pa a poou pheoal hell og of a pa of o foal pheo S a S whee S wh S. The ego wh S flle wh ouple e flu he aula ego bewee S a S aue o be poou a he ae ouple e flu a ha wh S flow wh a ufo veloy he fee ego.e oue S. The poble foulae ug he V.K. Soe equao ebg he flow oue he hell a well a he flow e he hell whle a aalogue of he laal ay law he heoy of poou ea ue wh he hell ego. Ue Soea appoxao he oluo he ough aalyally a he expeo fo he flow fel vaable ae obae e of Legee fuo aoae Legee fuo aal polae pheoal wave fuo a agula polae pheoal wave fuo. The ee ag o he hell ae eae a he ag expeee by he boy obae. Nueal uy ueae o uy he effe of he peeably of he eu ouple e paaee a he geoe paaee o he ag a he eul ae peee ug gaph. I fou ha fo a fxe S a he eey of he oue pheo eae he ag eeae. Iex Te Spheoal hell ouple e flu ag Seale. I. INTOUTION ouple e flu heoy popoe by V.K. Soe [] he ple pola flu heoy ha how all he poa feaue a effe of ouple ee flu aue ue o he ehaal eao ha ou e a efog ouu. g feaue of h oel ha eul equao ha ae la o he Nave Soe equao hee by falag a opao wh he eul fo he laal o pola ae. Seveal flow pa axye boe ealg wh ouple e flu have bee ue by Lahaa ao Iyega [] a Iyega a Svaahaya []. ll hee poble eal wh flow ag he oex of pevou boe le phee pheo a appoxae phee. To he exe he auho have uveye o uh leaue fou o he flow of ouple e flu pa poou boe exep ha by aaa Muhy e.al [] who ue he flow of a opeble ouple e flu flow pa a poou phee. Spheo beg a oe geealze fo ha a phee he pee vegao eal wh a opeble ouple e flu pa a poou pheoal hell. The ue ealg wh poou pheoal hell ae of ue he uy of pla phyology. Th ape ha bee well bough ou by Joh oe a Ke ow who have ue he heoy of hell o uy oe ape of he phy of fu gowh [5]. T.S.L aha wh Vhu Iue of Tehology havaa e-al: aha.l@ gal.o. T.K.V Iyega o who oepoee o be ae wh he Naoal Iue of Tehology Waagal e-al: yega_w@yahoo.o.. he laal Nave Soe equao he ouple e flu flow equao alo ae o lea aue a eve of hghe oe ha he Nave Soe equao. Thu oly vey few poble a have exa oluo ubje o pefe bouay a egulay oo. Whe we oe he poble of flow pa boe le phee o pheo alo poble o f a exa oluo. Hee eeahe e o olve a plfe veo of he flu flow equao by pog oe aupo bae o uo. Oe of hee aupo ha gve by Soe: whe he flow low a he flu hghly vou vou foe peoae he eal foe eae o he boy. Th aupo help egleg he olea eal e he oeu equao a hee by ag he poble oe aheaally aable. I h pape we uy he Soe flow of a opeble ouple e flu pa a poou pheoal hell og of wo ofoal pheo whee hee ouple e flu fllg he ego e he e pheo a he ego oue he oue pheo. The aula ego bewee he wo ofoal pheo aue o be poou aue a he flow govee by ay law hee. The flow vaable peag o he e ego Ғ oue ego Ғ a he poou ego Ғ ae obae aalyally. The expeo fo ea fuo a peue ae obae e of Legee fuo aoae Legee fuo aal polae pheoal wave fuo a agula polae pheoal wave fuo. The ee ag o he pheo ae evaluae a a expeo fo he ag obae. Though he expeo fo ee ae oplae he expeo fo ag ae a elghfully ple fo. Nueal uy ueae o u he vaao of ag wh epe o he aeal paaee geoe paaee a he peeably paaee. The ag ee o eae wh he eae he peeably paaee a eeae wh he eae he eey of he oue pheo a well a a eae he ouple e paaee. ouple e flu oel gve by V.K. Soe bae o he peupo ha he flue eu a ua ouple ee. Hee we have he o ye e eo j a he ouple e eo j gve by j pδ j vqδ j j ε j { ω ρ } δ ω ω j j j j whee q eoe he flu veloy veo ω ul q ω he j p eo he ae of efoao eo p he flu j peue a ρ he boy ouple veo. The quae a ae he voy oeffe a ae he ouple e voy oeffe. Thee aeal oa ae oae by he equale

2 The paaee a haae eaue of he polay of he flu whh zeo he ae of o pola flu. The ouple e flu equao ae gve by ρ ρv q q ρ ρ f ul ρ vτ ul vm 5 whee ρ he ey of he flu τ he ye pa of he foe e a a M he ouple e a a f ae he boy foe pe u a a boy ouple pe u a epevely II. FOMULTION oe wo ofoal polae pheo S a S wh fo a whee u. Le O be he po of. Ioue he ylal pola ooae ye θz wh epe o O a og a O exee o ehe e a Z ax. Le u oe he low aoay flow of a opeble ouple e flu pa he pheo S wh a ufo flow wh veloy U he eo of he z-ax fa away fo he boy. Le he ego Ғ bewee S a S be poou. Le he ego Ғ wh he pheo S be flle wh he ae ouple e flu a oue S. We exae he flow geeae wh he aupo ha he flow he poou ego haaeze by ay law. Se he flow low we aue ha he flow ax ye a he ae ay ea plae a hu he flow vaable ae epee of he azuh agle φ. We hall oue he polae pheoal ooae e e e a bae veo a h h h a he ξφ wh ξ φ oepog ale fao hough he efo ξ z oh 6 Le q p eoe he veloy a peue he ego Ғ a le q p be he veloy a peue he poou ego Ғ. I vew of he lowe of he flow we ae ξ eξ v ξ e p ξ q u 7 p 8 Igog he boy foe a boy ouple f a l epevely he fel equao he ba equao goveg he Soea flow a be we he fo v q fo 9 ga p ul ul q ul ul ul ul q fo a q gap whee he peeably oa. I vew of he ouy equao we oue he ea fuo ψ hough ψ ; ψ h h u h h v fo ξ Ug 7 a fo ulq ψ e φ h ululululq h ψ eξ h ψ e hh h ξ whh he Soe ea fuo opeao gve by h h h 5 hh ξ hh ξ hh Ug he expeo fo ul ul q ul ul ul ul q he ba equao ebg he flow ego Ғ a Ғ ae gve by p h ψ ψ 6 ξ hh p h 7 ψ ψ hh ξ ξ lag p fo 6 a 7 we have 6 ψ 8 whee 9 Thu he flow vaable he ego Ғ a Ғ ae opleely eeable fo he ye of paal ffeeal equao 8. The flu peue p a be obae ug equao 6 a 7. The flow he poou ego Ғ govee by v q q gap whh ple ha he peue p a hao fuo gve by he equao p The eeao of he eleva flow fel vaable ψ a p ubjee o he followg bouay a egulay oo. ouy of he oal veloy opoe o he efae: v v u u o S u u o S Tageal veloy opoe vah o he efae: v o S v o S ouy of peue o he efae: p p o S p p o S 5 o S ulq ulq o S 6 The veloe ae egula o he ax a fa away fo S he flow a ufo ea whh ea a fy

3 U 7 ψ Ohe fo of bouay oo a alo be ae. Howeve we ae ug he pee bouay oo oly a a al al fo a oplae e of equao wh a oplae geoey. Se we ae ealg wh a polae ooae ye we have h h h 8 9 whee o ohξ ; III. SOLUTION FO TH FLOW IN TH GION Ғ The equao goveg ψ ψ The oluo of equao a be obae by upepog he oluo of he equao ψ a ψ vew of he lea ouave opeao a. Soluo of equao : The oluo of equao a be we he fo ψ ψ ψ 5 whee a ψ U 6 ψ G 7 whee he evave of wh epe o. The fuo ψ equao 6 epee he ea fuo ue o a ufo ea of ague U paallel o he ax of yey fa away fo he pheoal hell. We oe ha ψ a hee ψ. I vew of h ψ u afy ψ 8 I a be vefe ha he expeo f 9 whee he evave of Legee fuo of eo wh epe o afe f. I vew of h we hall poe he eo o he fuo hough ψ o ha ψ. G Now opeag o he equao 7 a equag he eul wh he gh ha e of equao we ge { G } G G Followg [] we oe ha govee by he ffeeal equao G G G g whee g The equao a ae val fo a he e volvg o be elee he gh ha e of equao 5 a o be epee a o oba g. Ug he eho of vaao of paaee we oe ha G g g fo whee epee he value pefyg he oueo pheo pa whh he flow beg ue. Thu he flow ego Ғ gve by >. ψ u e o. I vew of h we have o ae. Hee he appopae expeo fo G gve by G g g fo 5 g volve oe e { } of abay oa he fuo G volve wo e of abay oa { } a { } ψ.. Ug h equao 5 we ge Soluo of equao : To olve he equao vz. ψ we ue he eho of epaao of vaable a ae he oluo he fo ψ S 6 Subug h he equao we oe ha a S epevely afy he ffeeal equao Λ 7 a S S Λ S 8 whee Λ a epaao oa. [6]. Thee ae pheoal wave ffeeal equao of aal a agula ype epevely. To eue egulay of oluo a fy a he flow ego we have o hooe he oluo of equao 7 a 8 he fo a whee S π K 9 5

4 5 he aoae Legee fuo of he f. The oeffe he above expao ae oa epeg o he paaee a he uffx ha he value 5 o 6 epeg upo he o o eve value of. We have heefoe he oluo 5 ψ S whee ae oa. Hee he ea fuo fo he ego Ғ gve by ψ U We a ee ha G ψ S S a S ψ whh ae eoe fo fuue ue. IV. eue buo Ғ The equao 6 a 7 fo a ug equao 7 ae p ψ ψ 56 a p ψ ψ 57 Ug he expeo equao 5 a a 57 o egao we ge p 58 Thu ψ a p gve equao 5 a 58 epevely ae he ea fuo a peue buo fo he ego Ғ. Thee volve he hee e of oa a a be ee fo equao 5 a 58. { } V. SOLUTION FO TH FLOW IN TH GION Ғ The equao fo ψ gve by equao ψ 59 Soluo of equao 59 a be obae by upepog he oluo of ψ a. The poeue o oba ψ he oluo a be ae ou exaly o la le a he ae of Ғ wh he ffeee ha hee boue << a og a a pa of he ax of yey ae he flow fel. ayg ou he poeue exaly a he ae of Ғ we ge he ea fuo uable fo Ғ a ψ G whee S 6 g g G 6 wh g Hee aga 6 val fo wh he ueag ha he e o be elee whe we ae a o be epee a. oe a h age oe. opag he expeo fo g ha we ge hee a he expeo fo g of equao we obae he ae of Ғ we oe ha hee we have a epevely plae of a hee equao. Th eleo of he appopae Legee fuo gue by he boue aue of he flow ege Ғ a he uboue aue of he flow ege Ғ. Fuhe he fuo gve equao 6 a aal pheoal wave fuo gve by K π 6 valuag ψ ψ a ug he expeo equao 6 we oe ha peue buo Ғ gve by p 6 Thu a ea fuo p ψ a peue buo fo he flow ege Ғ ae gve by equao 6 a 6 epevely. Thee volve hee e of oa { } { } { } a a be ee fo equao 66 a 6. VI. SOLUTION FO TH FLOW IN TH GION Ғ We have ee eale ha he flow he poou ego Ғ govee by he equao a whh lea o he equao. The equao ple ha he peue buo p Ғ hao a hee gve by p whee a 65 oue aohe e of abay oa o be eee. The veloy opoe u a v a be eee fo equao a 65. Thu all we have egh e of uow { } { } a oa { } { } { } { } { }

5 hee a be eee by ug he egh bouay oo gve by he equao 5 a 6. VII. VLOITY OMONNTS IN TH GIONS Ғ Ғ Ғ The veloy opoe a be obae by ug he expeo fo ψ a ψ gve equao 5 6 a p gve equao 65. Thu he expeo fo he veloy opoe u v ; u v ; u v a be we explly. Ug hee expeo a hoe of p a p he bouay oo gve by equao 5 a 6 we a we he equao ha lea o he eeao ofhe abay oa. VIII. TMINTION OF ITY ONSTNTS I vew of he equao -7 a he ohogoaly popey of Legee fuo a he aoae Legee fuo we have U δ [ ] o U δ 68 [ ] o eeao of he abay oa he poble a a que oplae bu of oue o uuouable. lag he oa we ee ha a have o ulaeouly afy a ye of fe o hoogeeou ye of lea equao. I heaeg o oe ha a he oeffe ae zeo fo o value of we a egegae he above ye of equao a o wo ub ye oag a. The ub ye volvg a ee o be hoogeeou a vew of h fo all pove egal value of. Fally we e up wh he followg o hoogeeou ye. fe a agh fowa bu leghy algeba U δ 7 a 75 whee * 76 * 77 * 78 * 79 a aalyal eeao of o poble we have o eealy eo o a ueal eeao of he oa. Fo h we uae he wo ye 7 a 75 o a o gve a by ye wh 9 a 9 fe eeg hee poble o evaluae ueally he ohe oa. The eal of he apulao ae oe vew of he leghe of he expeo a he fal ye oly epoe hee. IX. TMINTION OF G The ag a be we he fo ξ ξξ π 8 a h plfe o S p π 8 Ug he elao fo Hobo[7] gve by 8 a 8 he ag plfe o

6 } 8 π fe fuhe plfao we ee ha he ag ue o he ufae e gve by 85 whee 86 We efe o a he o eoal ag a h epe upo he eey of he pheo he o polay paaee a aoal aeal oa a he peeably paaee. X. NUMIL ISUSSION [9]. Happel J. a ee H. : Low eyol ube Hyo ya Mau Njhoff publhe The Hague 98. Fg: Vaao of ag wh epe o fo ffee p whe. a. The ag o he pheoal hell ueally evaluae fo eveal paaee value a he vaao of he ag playe hough fgue o. Fo eah value of he peeably paaee p he ag eag a eae. eae ple a eeae he ouple e voy. Hee we oe ha a eae o oao eeae he boy expeee a geae ag. Fo a fxe fo a eae p he ag ee o be lghly eag ee fg a he eae o gfa. eae ae a eae he ze of he oue pheo. The fg how ha a he ze of he oue pheo eae fo a fxe whe he ze of he e pheo fxe he ag eag. Fuhe a he paaee eae fo a fxe he ag eae. Fg: Vaao of ag wh epe o fo ffee whe. p. I fg we ploe he vaao of ag fo fxe value of p a wh epe o vayg a vee value of. Hee alo we oe ha a he ze of he oue pheo eae he ag eae. lo a he ouple e paaee eae he ag gfaly fluee. We e o plo he eale pae fo ffee value of p wh.5 a.. Hee alo he eale pae la o he oe obae by he auho he ae of flow of a opola flu pa a poou pheoal hell [8]. The eale pae fo he oue ego a well a he poou ego lghly ube whee a fo he flu oe ego hee a oeable ubae a he pae ae la o he oe obae by Happel a ee ee page 9 of [9]. Fg:Vaao of ag wh epe o fo ffee whe. p. efeee: []. Soe V.K.: ouple ee flu hy. of Flu Vol 9 pp []. Lahaa ao S.K. a Iyega T.K.V : alyal a opuaoal ue ouple e flu flow U.G.. eeah poje -8-8 S III 985. []. Svaahaya.: Soe flow of a opeble ouple e flu pa a appoxae phee h. he 995. []. aaa Muhy J.V Svaahayulu.N paa. : Ufo flow of a opeble ouple e flu pa a peeable phee ull. al. Mah. So. Vol 99 pp 9-7. [5]. Joh oe Ke ow: hyal pe of Fu gowh. Theoeal aaly of buo of ufae gowh foe fu elao o ag a plg. la hyol Vol 68 pp [6]. baowz M Segu I. : Haboo of Maheaal fuo wh foula gaph a aheaal able ove publao IN NewYo 965 [7]. Hobo.W: The Theoy of Spheal a llpoal Hao helea ublhg opay New Yo 955. [8]. Iyega T.K.V aha T.S.L.: Soe flow of a opeble opola flu pa a poou pheoal hell ouae. Fg: Seale pae fo. a p-.5 Fg5: Seale pae fo.5 a p-.5