EXACT SOLUTIONS FOR THE FLOW OF A GENERALIZED OLDROYD-B FLUID INDUCED BY A SUDDENLY MOVED PLATE BETWEEN TWO SIDE WALLS PERPENDICULAR TO THE PLATE

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1 THE PUBISHIG HOUSE PROCEEDIGS OF THE ROMAIA ACADEMY Si A OF THE ROMAIA ACADEMY Vol /. 3 EXACT SOUTIOS FOR THE FOW OF A GEERAIZED ODROYD-B FUID IDUCED BY A SUDDEY MOVED PATE BETWEE TWO SIDE WAS PERPEDICUAR TO THE PATE Coi FETECĂU * Moo KHA ** Coi FETECĂU *** Hio QI **** * D of Thoil Mhi Thil Uii of Ii Ii 75 Roi ** D of Mhi Qi-i-Az Uii Il 44 Pi *** D of Mhi Thil Uii of Ii Ii 75 Roi **** Shool of Mhi Sii Shog Uii Wihi Wihi 649 Chi Cooig ho: f@hoo. Th flow of ioil gliz Olo-B fli i l o l w wo i wll il o h l i i of Foi i l fo. Th loi fil wi i of h gliz G fio i w h woi olio h ooig o- woi oiio. Th olio ooig o h gliz Mxwll fli wll h olio fo oi Mxwll Olo-B fli foig h oio oi liiig of h gl olio. I h of i wll l wh h i w wll o ifii h olio ooig o h oio o ifii l o l o. Fill h ff of fiol o h loi fil wll h ifl of h i wll o h fli oio i oligh ghil illio. K wo: Gliz Olo-B fli Si wll Sl o l Ex olio.. ITRODUCTIO Th li flow of g of fli h oli lii foo o i fo li iologil fli o foh o l i of h lil lil io woi ol. Th fo h woi hio if ilf i i of w: o-woi ioi h hiig o h hiig -lxio o-li ig lo of ol iff il. o ol h oo o i h o hii of h fli h ig lifi fli of iffil igl. Aog h ol h h lo h Olo-B ol i l o li o iol xil oooio. I h h o i iig h o of o oli lii ig iw o of h o fl ol fo iig h o of -l of h fli. Rl h fiol ll h o h i h iio of iolii i oig o ll ool o hl ioli oi [ 5]. Eill h hologil oii io wih fiol ii l io ol i iio of h hio of h ol olio l. I il i h how h h iio of fiol ii Mxwll ol i xll g wih h li ioli i h gl iio gl [ 6]. Th li of h liio i i log i ilig fl i fiol w iffio fiol Hiloi i oh oi i hi. Th oi fh wo o h oiiol fiol ii ol. Th ig oi of h fiol ii ol of o- woi fli i ll lil iffil io whih i oifi lig h i ii of ig o o-ll Ri-ioill fiol iffil oo. Thi glizio llow o fi il o-ig o igl o ii [7]. Dig h l

2 4 Coi Fă Moo Kh Coi Fă Hio Qi lo of gig h fli h lih. H w hll f ol o ho gig gliz Olo-B fli GOF [8-6] h f hi. Th i of hi o i o lih x olio fo h loi fil ooig o h flow of ioil GOF w wo i wll il o l o li h ifl of h fiol of h i wll o h oio. Th oio of h fli i o h l whih i l zo i ilil i oio wih o loi V. Th olio h h oi igl i fo i of gliz G fio lih of Foi i l fo. Th w h woi olio h ooig o-woi oiio. Th iil olio fo gliz Mxwll fli wll ho fo oi Mxwll Olo-B fli oi liiig of gl olio. Fho i h of h i wll ll io olio o ho ooig o h oio o ifii l h wll ow lil olio fo h fi ol of So ig lo o liiig.. STATEMET OF THE PROBEM oi ioil Olo-B fli oig h o ifii l il o h -xi w wo i wll i i h l z z [7]. A i h ifii l gi o li i i l wih h o loi V. D o h h h fli o h l i gll o i loi ig of h fo V V i wh i i h i o log h x-iio. Fo hi flow h oi of ioiili i oill ifi. Aig h h x- S wll h loi V i fio of z ol i i o how h i h of o fo gi i h flow iio h goig io fo hi flow i [7] ν > z z wh ν i h ii ioi of h fli h lxio io i. If h fiol ll oh i i h oii liohi of h fli ol h goig io h fo f. wih [] [6] α D ν D > z z 3 wh h fiol iffiio oo D fi [7] f τ D f τ < 4 Γ τ Γ ig h G fio. I hol o h h goig io 3 i o fo iiol oi of iw ol if h w il o h h iio of α α il. I o h ho h i of io h oii io of GOF. How fo ilii w ig h oio lhogh h il o h iff igifiio io E. 3. Th oi iiil o oiio ooig o hi ol [7] > z z 5

3 3 Ex olio fo h flow of gliz Olo-B fli 5 z V > < z < >. 6 Moo h l oiio ifii > 7 h o lo ifi. I h followig h fiol iffil io 3 wih h iiil o oiio 5-7 will ol of Foi i l fo. 3. THE SOUTIO OF THE PROBEM Mlilig E. 3 / π i i z wh π / igig h l wih o z fo o ifii il o ig io o h iiil o oiio 5-7 w fi h lo E. A fo Aix α D ν D νv > π Γ wh ig h ol Foi i fo of [8] h o if h iiil oiio fo > Alig h l fo o E. 8 ig h l fo fol fo il fiol ii [7] w fi h h ig fio of i gi A 8 α νv α π [ ν α ] wh α ν. I o o oi o oi h o llio of i oo igl w l h i i l fo ho [8-6]. How fo il io of h fil l w fil wi h l fo fo E. i h il fo [ α ν ν α α ] ν α ν α ν I iw of h lii A3 fo ix h o fo of h l fo wi ol i h fo α l α l l α l. α α [ ν/ ] [ ν/ ] ν α.![ α ] l!! Ioig io iig h l of h Foi i fol [8] lig h h i i l fo o h oi l ig E. A4 A6 h o A7 fo Aix w fi h loi h fo

4 6 Coi Fă Moo Kh Coi Fă Hio Qi 4 -ν - 8νV π G α ν i z l l G α! α! l! ν i 3 wh l α l [9. 4 5] α j j j Gα. 4 Γ j Γ α j j Io o lio j... j i h Pohh oloil 4V i z - i V z z - f ν f ν ν ν h loi fil ooig o woi fli foig h oio. I h followig fo l w h hg h oigi of h ooi h il of h * hl. Col ig z z h oig o h oio E. 3 5 h o il fo fo i [7 E. 3] fo h woi loi 5 l z 4νV o µ! α z z µ i π h µ!! l l - ν µ - ν ν Gα µ G α µ V z h z -µ V o µ o µ z µ h µ -µ µ fµ ν f µ ν ν ν wh µ π /h f i h ol o fio of G SPECIA CASES. Tig io E. 6 w fi h loi fil 4νV o µ z z z i π h µ ν µ ν Gα α µ ooig o gliz Mxwll fli foig h oio.. B ow lig α io 8 h iil olio 8

5 5 Ex olio fo h flow of gliz Olo-B fli 7 4νV o µ z z z i π h µ ν µ ν G µ ooig o h lil Mxwll ol i oi. 3. Tig h lii of h gl olio 6 fo α h loi fil 9 z 4νV o µ α µ z z i π h µ ν µ ν G µ ooig o oi Olo-B fli i o. 4. Fill i iw of E. A7 i i woh oiig o h h woi olio oi fo o of E. 8 9 o fo. Fho i lo oi liiig of E. 3 6 fo. 5. IMITIG CASE h FOW OVER A IFIITE PATE I h of h i wll l wh h io o lii h loi fil ooig o h oio o ifii l So' fi ol i o []. Th olio ooig o 6 fo i o ν l νv! α i π l!! l ν ν Gα G α α ν i G νv ν π wh V f / i h lil olio ooig o h fi ol of So. A h of o lli w how h h ig of gi E. iil o ho ooig o h iil olio oi i [] iff hi. Pig o h lii io w oi h iil olio fo Mxwll fli wih fiol ii l νv ν ν i Gα α. 3 π Fho ow lig α io 3 h olio fo oi Mxwll fli νv ν ν i G 4 π

6 8 Coi Fă Moo Kh Coi Fă Hio Qi 6 i oi. A fo hi l olio i oll iff of h olio oi i [] []. How of ghil illio w how h o olio 4 i il o h iil olio oi i [] iff hi. 6. COCUSIOS AD UMERICA RESUTS Th i o of hi o i o oi h loi fil fo h flow of ioil gliz Olo-B fli i l o l w wo i wll il o h l o oligh h ifl of h fiol o h fli oio. Th x olio oi ig Foi i l fo igl i fo i of h gliz G fio. Th ooig olio fo fiol Mxwll fli wll ho fo oi Mxwll Olo -B fli lo oi liiig of o gl olio. Fho li h io olio fo h li h olio of h woi olio h ooig o-woi oiio. I h of i wll l wh h ll olio h h oi o h olio ooig o h oio o ifii l o l. I o o l o l hil gig h oi l h ig of giig h loi ofil h il of h hl h w gi fo iff l of h fiol α. Th ff of h wo i l fo Fig. ooi. I h gio h l fo i h loi of h fli i iig fio wih o α ig o wih o. Th ifl of h i wll o h fli oio i li i Fig.. Th loi of h fli i ll i h of h i wll h iff i w o x i i i. Fill i i woh oiig o h h gl olio 3 6 ooig o ioil gliz Olo-B fli o h woi olio if α. Thi l loo xio of Joh [ Sio.] oig Olo-B fli α.3 Fig. Pofil of h loi fil gi E. 6 3 fo V h iff l of α.

7 7 Ex olio fo h flow of gliz Olo-B fli Fig. Pofil of h loi fil gi E. E. fo V h ν. APPEDIX Γ Γ / ig V V V D A } { D D Γ A!!! l l l l A3. R > R f f 4 i π A4 < > R > R G A5 } { } { if } { A6. li < Γ G A7

8 Coi Fă Moo Kh Coi Fă Hio Qi 8 ACKOWEDGEMETS Th ho Coi F Coi F owlg h o fo h Mii of Eio Rh CCSIS hogh PM II-I G o. 6/8-9-7 Co CCSIS ID_593. REFERECES. BAGEY R.. TORVIK P. J. O h fiol ll ol of ioli hio J. Rhol FRIEDERICH C. Rlxio io fio of h Mxwll ol wih fiol ii Rhol. A SOG D. Y. JIAG T. Q. S o h oii io wih fiol ii fo h ioli fli-oifi Jff ol i liio Rhol. A HIFER R. Aliio of Fiol Cll i Phi Wol Siifi P Sigo. 5. ROSSIKHI Y. A. SHITIKOVA M. V. Ali of i hio of ioli o who hologil ol oi fiol ii of wo iff o ZAMP Z Agw. Mh. Ph HEIBIG A. PAADE. I. O h ili of oji fiol ii ioli fli ol J. Mh. Ph PODUBY I. Fiol Diffil Eio Ai P S Digo TOG D. IU Y. Ex olio fo h oiol flow of o-woi fli i l i I. J. Eg. Si TOG D. WAG R. YAG H. Ex olio fo h flow of o-woi fli wih fiol ii i l i Si. Chi S. G: Phi Mhi Aoo KHA M. HAYAT T. ASGHAR S. Ex olio fo MHD flow of gliz Olo-B fli wih oifi D' lw I. J. Eg. Si KHA M. MAQBOO K. HAYAT T. Ifl of Hll o h flow of gliz Olo-B fli i oo A Mh QI H. XU M. So' fi ol fo ioli fli wih h gliz Olo-B ol A. Mh. Si VIERU D. FETECAU Coi FETECAU C. Flow of gliz Olo-B fli o ol lig l Al. Mh. Co FETECAU Coi FETECAU C. KHA M. VIERU D. D of oil ox i gliz Olo-B fli Al. Mh. Co FETECAU C. MAHMOOD A. FETECAU Coi VIERU D. So x olio fo h hlil flow of gliz Olo-B fli i il li Co. Mh. Al FETECAU C. FETECAU Coi KAMRA M. VIERU D. Ex olio fo h flow of gliz Olo-B fli i ol lig l w wo i wll il o h l J. o-woi Fli Mh FETECAU C. HAYAT T. KHA M. FETECAU Coi U flow of Olo-B fli i h ili oio of l w wo i wll il o h l A Mh SEDDO I.. Foi Tfo MGw Hill Boo Co I. w Yo Tooo oo OREZO C. F. HARTEY T. T. Gliz Fio fo Fiol Cll ASA/TP FETECAU C. FETECAU Coi Th fi ol of So fo Olo-B fli I. J. o-i Mh FETECAU C. FETECAU Coi U oio of Mxwll fli o logiil oiol oillio of ifii il li Po. Ro. A. S. A Mh. Ph. Th. Si. If. Si JOSEPH D.D. Fli Di of Violi ii Sig Vlg w Yo 99. Ri M 5 9

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