Fundamental Solutions for Micropolar Fluids

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Hector Cunningham
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1 So: Jol of Egg Mh Vol No pp 9-79 ; DO: 7/-7-9- dl Solo fo Mopol ld JN-JN SH * d JENN SHN LEE Shool of Mhl & op Egg Nyg hologl y Nyg Sgpo 979 N fdl olo fo opol fld dd pl fo fo o- d h-dol dy odd Sok d O flo d o po fo d po opl ldg h o-dol opol Sokl h o- d h-dol opol Sok opl h h-dol opol Ol d h h-dol opol O opl h fdl olo do o No flo d o h of ooo loy fld h flo d o h gl fl fo ddg d dyg ol flo pplo h dg off fo old ph o l yld h l lo-yold- opol flo dd d opd h ho opodg o No flo h dg off opol fld g h ho No fld odo h phyl h of h d o po ll-l y dff gfly fo ho ol qp [] dl d ppld go of ol pho fld h od y dlop h of ologl oll hy hog olo d ofld l lg h o h fld d o d h flo god y h N-Sok qo h o odl h h pop of h l y ooly hogho h flo do No o h h fld odld d ggg of pl pog d lol loy Ho h fld qo h h N- Sok qo do o o fo h ool ff of h fld o-o * ho o ho opod hold ddd

2 h hoy of opol fld [] gd pl od ll ol l o o h od of h ol l h oo dd y dpd o-oo o Mopol fld ppo ody opl d h oool ff h hoy of opol fld h ho po fo pdg fld ho ol Ppky l [] fod h l odl fo flo ohl d o hoy of opol fld g pdo of pl l h ho od g h N-Sok qo Mopol fld odl oop fld lqd yl h gd oll g fld lod h d ddy fld d o ologl fld [] of h pol pplo ol fld h d o-no fld h oh plog fdl olo h fdl olo fo Sok flo [] d O flo [] d o po fo ooly d h Sokl d h Ol h fdl olo d o po fo dy Sok flo f dd y h Nol L Loz f k 9 [] h olo o ko y h Sokl lhogh Sok k o h Sokl od y Hok 9 [7] Ooly Hok of h Loz ok Hd h ko o h gh h opd fo Lozl d of Sokl h y ho l h h dy Hok-d Sokl dd y Ldyzhky 9 [] y g o-fo hod of o ppop f og h dy Sokl d Ol fod [9] h lod-fo fdl olo fo glzd dy Sok d O flo od h y -dpd lol d ool oo h dd y Sh & Chg [] opol fld h ooo fdl olo d o po fo h opol Sokl d opol Ol d ho d o po opl h opol Sok opl d opol O opl Sh fdl olo do o No flo d o h of ooo loy fld koo & Mjd [] lzd h gog qo of opol fld d ppld o fo o o h h-dol opol Sokl Old & Mjd [] dd h o-dol opol Ol d opol O opl h pp d fdl Sok d O olo of opol flo h do o h h po fo d po opl pd y do Copodg l fo o-dol flo lo pd Sok d O flo of opol fld d o po fo Cod po fo odd q opl opol fld Who lo of gly h po fo pld h og d h f- loy k o d dy d o h O ppoo h gog qo [] d o h l fld flo ()

3 h h fld dy p ( ) () () h o d h No oool d o gl o ply d o o h qo () h dg of d zo hh fd h l p of h o h p p lol loy d ooo loy qd o dy odd flo p Sppo h f olly gl fo h dy fy of o fo of h fo f dfd y f f d f h -dol opl h h fod l of d h gy h dg of () yld p ( ) () hh h p ho yh p h pol o ol () fo p k h o fo fdg kg h o fo fd p l p Sok d O flo d o po fo h h p fld gdl of hh h fld No o opol () o () h

4 () kg h l of () g ) ( (7) Sg () (7) d pl dffl o qo og oly o ko () h d o ol pl dffl qo of hgh od h () y foz h hgh od pl dffl opo o pod of lo od [] h hod d y Old & Mjd [] olly popod h L (9) h L foh od pl dffl opo d d o Whl h hod of fozo lohp h p fo L o d h dd fozo o foz h dffl opo () h follog : L () Coqly qd h W h f qo d oly fo ko o pd h olo h l of k o zo hfo h fo

5 Coqly Ho o hh g () H h pl dffl opo () fozd o h llo () o () h d h o fozo ld d h phyl o of h p g y () o ol () fo o o k h o fo hh g () h o fo g hh h opol Ol of ( ppd)

6 d d () h j j d oh d h odfd l fo of h od kd pd h l of h pod of l fo d h o o h dg of zo H h po d l o h dg of g zo fd () pod h opol Sokl l () Eqo () g h ooo loy h p of po fo h l of () g h l of h l of hh g o d () d h po h h dg of zo kg h o fo fd Sg () d () h o qo ld o h o fo of h po yld h fo

7 7 lly fd h lol loy g y h opol Ol of ( ppd) d d l () h l h lol loy d ooo loy fld dopl h d h po () of plf o d d l (7) W h h No Ol od h l h opol Ol of () o h opol Sokl of l l () h olo of fo opol fld h o opld h h fo No fld Sok d O flo of opol fld d o po opl Cod po opl odd q opl opol fld d o h O ppoo h gog qo [] lzd (9) p () ()

8 h h po opl h o o Who lo of gly h po opl d o pood h og W g y kg h dg of () hh p p h p fld p h h p h d h gd of p () o zo o o h lol loy fld k h l of () () h Vo d d (9) d o p h l of () h o fo o p () of lo k of () hh () Sg () () ld o () () d () dl fo foz h pl dffl opo () d h phyl o g y () h ho o fo () h o fo of û yld h opol O opl of ( ppd) d d () o pg h h opol O opl of () l o h opol Ol of () p h h fo d y po opl hl h l d o po fo

9 9 h l h opol O opl of () o h opol Sok opl of l (7) W k h dg of () o l h dg of d fd f () h f d o pd h olo k h o fo of () d fd f (9) ho o fo f h o d h l of k h o fo of () d () o fd () o l f o k of h o dy: f ho o fo f W (9) d () h o qo o p of l

10 h o fo of h opol O opl of ( ppd) d d d d d l d l () h () g h opol Sok opl of l l () h h-dol opol Sok opl of lo g h h of Eg [] p fo og g h f Dg o lg old ph opol o flo Cod h flo podd y old ph of d lg h loy opol fld of f p h flo d o h ph y od of po fo d pol dpol oh pld h of h ph h of Sok flo [] H h loy g y

11 O h h ol gh of h pol dpol d h yold d o ll qg h ody odo o h f of h ph yld o lg qo fo h off d O O O ho olo O O h dg o lly fo h po fo h dol dg off D O C Pg o h l fo h ll o flo C D Dg o lg l yld opol o flo Cod h flo podd y l yld of d lg h loy opol fld of f p h flo d o h yld od of o-dol po fo d o-dol pol dpol oh pld h of h yld l l l O qg h ody odo o h f of h yld fd / l / l

12 h dg o lly fo h o-dol po fo h dol dg off C D l / y pg h l fo h ll o-dol o flo od [7] C D l / Colo N fdl olo fo opol fld h dd pl fo h pol of o- d hdol dy odd Sok d O flo of opol fld d o po fo d po opl odd h fdl olo fo Sok d O flo h o-dol opol Sokl g y () d () h o- d h-dol opol Sok opl g y (7) d () h h-dol opol Ol g y () d () d h h-dol opol O opl g y () d () h fdl olo pol d o h of ooo loy fld opol fld h fdl olo g fh fdl olo y dffo h p o h gl po [] y of ll fdl olo g l l : l of fdl olo dd h pp gl po doly Sok flo O flo po fo -D N [] -D [] N po opl -D N [] -D N N h fdl olo fo opol fld d h ldg lok o o olo of ol flo pol y ployg h ody gl hod o h gly hod dod h h fdl olo d o ll h dg off fo lg old ph d l yld ply opol fld lo yold h dg off opol fld g h ho No fld y h fo

13 PPENDX Cod h pl dffl qo h d o y o W pply h o fo o o Lg ; ; g o g d h g kg h o fo of h o qo h g h o fo of ĝ ; g hfo h fdl olo ; ; of h pop of h o fo fd

14 d d d ; g ; h j j d y o o d fol () () () d () pfo h o fo: h d h y o Pl fo po g W old h ; ; ; ; ; ; EEENCES [] Ppky J zzl l d z L fld ho ohl g opol fld hoy So d o : Phyl 7(-) - (999) [] J-J Sh Mol h f f j g pl f Spl d Mo (- ) - ()

15 [] C Eg Moo fld ho : fl Md (Spg-Vlg N Yok ) () [] G G Sok O h ff of h l fo of fld o h oo of pdl o of h Cdg Phloophl Soy 9() - () [] C W O Ü d Sok'h fol d ü d fg d hydodyk k fö Mk (9) - (9) [] H Loz E lg llg o d gg loof jg g d fgld golg Zglg d okljk kd Whpp d 7 (9) ( Dh) ld o Eglh y H k gl ho o h oo of fld h fo d f l dd fo Jol of Egg Mh 9 (99) [7] G J Hok h lf-poplo of oop og hogh lqd Podg of h oyl Soy of Lodo S -Mhl d Phyl S 7() 9- (9) [] O Ldyzhky Mhl pol of h dy fo o opl fld (zgz) (9) ( ) Eglh lo h hl hoy of o opl flo (Godo d h) (9) [9] H k H Loz: Skh of h ok o lo o flo d o oh fld h d h kgod g hh o Jol of Egg Mh (-): - (99) [] J-J Sh d Chg Glzd fdl olo fo dy o flo Phyl E () () [] H koo d S Mjd Dg o lly y ody h Sok flo of opol fld Phy of ld 9() - (97) [] W E Old d S Mjd dl O olo fo h -dol flo of opol fld ol Jol of Egg S () - (9) [] G Łkzz Mopol fld: hoy d pplo (khä oo) (999) [] D Zllg Hdook of dffl qo (d P) (99) [] C Pozkd ody gl d gly hod fo lzd o flo (Cdg y P) (99) [] M oh d Pop Vo opl flo fo lo yold (W P) () [7] H L O h fo oo of ph hogh o fld h Lodo Edgh d Dl Phloophl Mgz d Jol of S () - (9)

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