Magnetohydrodynamic Stability of Self-Gravitating Fluid Layer Embedded into Different Fluid

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1 Australan Jurnal f Basc and Appld Scncs, 3(4): , 009 ISSN Magnthydrdynamc Stablty f Slf-Gravtatng Flud Layr Embddd nt Dffrnt Flud Ahmd E. Radwan and AbdlTawab A.A. Shalaby Dpartmnt f Mathmatcs, Faculty f Scnc, An-Shams Unvrsty, Car, Egypt Numrcal Wathr prdctn Cntr, Rsarch Dpartmnt, Egyptan Mtrlgcal Authrty, Car, Egypt Abstract: Th magnthydrdynamc stablty f slf-gravtatng flud layr mbddd nt an nfnt flud f dffrnt dnsty has bn dvlpd. Such a mdl has crucal applcatns n svral dmans f scnc. Th prblm s frmulatd, th partal dffrntal quatns ar slvd, th bundary cndtns ar appld and th dsprsn rlatn s drvd. Th stablty crtrn stablshd fr all knds f th prturbatns. Th slf-gravtatng frc s dstablzng as th nt dmnsnlss wav numbr s lss than 0.64 whl t s stablzng fr all th rst. Th lctrmagntc frc has strng stablzng nflunc and that nflunc ncrass and supprssd th slf gravtatng dstablzng ffct and stablzng sts n. Ky wrds: INTRODUCTION Th stablty f a sm-nfnt layr undr natural frcs has bn dscussd by Raylgh (945). Chandraskhar (98) dd dscuss th stablty f tw sm-nfnt fluds nstablty undr th flud prssur gradnt and th xtrnal gravtatng frcs as th fluds ar ntally at rst (Raylgh-Taylr) r nt (Klvn- Hlmhltz nstablty). Sn, aftr wards Chandraskhar (98) mad svral xtnsns fr such studs and thrs. S als Kant and Malk (985), Ogansyan (96), Radwan (987), (990) and (007) and Crak (966). Th lattr authr lad th thrtcal fundatn fr th shar prturbatn stablty, s als Bnjamn (959) and Mls (96). Th rspns f th slf-gravtatng nstablty f a flud cylndr fr axsymmtrc prturbatn s bng studd fr frst tm by Chandraskhar and Frm (953). Latr n Chandraskhar (98) (Nbl prz wnnr 986) dvlpd and mad mscllanus xtnsns. In vry chaptr f Chandraskhar s bk (98) thr has bn n attmpt t gv xhaustv rfrncs nr ds th authr clam t hav rad r vn fund vry rlvant papr. Hr w study th magnthydrdynamc stablty f slf-gravtatng layr mbddd nt dffrnt flud, fr all mds f prturbatns. Basc Stat: W hav a flud layr f thcknss h mbddd nt a dffrnt nfnt flud. Th lattr flud f dnsty and th flud layr f dnsty ar assumd t b ncmprssbl-nn-vscus and prfctly cnductng and prvadd by th magntc flds H, H,0,0 () Th fluds ar actd upn by th prssur gradnt, slf-gravtatng and magnthydrdynamc frcs. Th basc quatns whch ar rqurd fr dscussng th stablty f such knd f prblms ar th cmbnatn f th quatns cncrnng th lctrmagntc thry, rdnary hydrdynamc quatns and th Nwtnan slf-gravtatng quatns. Ths quatns ar gvn as fllws s s s s s s s s s u u H H V P t () Crrspndng Authr: Ahmd E. Radwan, Dpartmnt f Mathmatcs, Faculty f Scnc, An-Shams Unvrsty, Car, Egypt 46

2 Aust. J. Basc & Appl. Sc., 3(4): , 009 s s u 0 H 0 s s s s s s s s u H H u H dvu u dvh t s s V 4 G, s, (3,4) (5) (6) Equatns () ar th magntgravtatnal vctr quatns f mtn, quatns (3) ar th cntnuty quatns, quatns (4) ar th cnsrvatn f flux, quatns (5) ar th valuatn quatns f magntc flds and quatns (6) ar Nwtnan s slf-gravtatng quatns. Hr, up, ar th flud mass dnsty, vlcty vctr and prssur, and H ar th magntc fld prmablty and ntnsty and V and G ar th slf-gravtatng ptntal and cnstant, rspctvly. Prturbatn Tchnqu: Fr small dparturs frm th unprturbd stat, vry physcal quantty Qx, y, z; t valu n th unprturbd Q zbsds an nfntsmal ncrmnt Q x, y, z; t, vz. culd b xprssd as ts,, ;,, Q xyzt Q z tq xyz Hr Q x, y, z; t stands fr u, P, H, V and th z-dstanc f th flud layr, wth t s, th ampltud f prturbatn at nstant tm t, gvn by xpt whr at t 0 s th ntal ampltud and s th scllatn frquncy. Du t th prturbatn, th z-dstanc f th prturbd wav alng th flud layr-nfnt flud ntrfac s bng (7) (8) z z z z h, (9) wth z xp[ kxly t ] (0) s th lvatn f th surfac wav masurd frm th unprturbd pstn whr k and l ar th wavnumbrs n th x and y drctns. Frm th vw pnt f th xpansns (7) - (0), th magntgravtatnal lnarzd quatns dscrbng th prturbd stat f th flud layr ar gvn by t curl u u u P H H 0 H 0 u u, 0 H H u u t V 0 () (,3) Snc w us hr th nrmal md analyss, basd n th lnar prturbatn tchnqu, w hav t sk a slutn f prturbatn quatns whs xyt,, dpndnc n th frm (4) (5) *,, ; kx ly t Q x y z t Q z (6) Cnsquntly, quatn (4) ylds H kh u (7) 46

3 Aust. J. Basc & Appl. Sc., 3(4): , 009 Takng th crculatn f quatn (), w btan t curlu curlgradp curlcurl H H upn usng th dntts HH H curlh HH H H H H P 0, u (8) (9) (0) quatn (8), rducs t A 0, A k H () and s u y ux ux u u z y ux 0, 0, 0 x y z x x y () Cmbnng quatn () wth th cntnuty quatn (): ku lu x y u z z (3) w gt dux( z) l k ux ( z) 0 dz (4) Th nn-sngular slutn f quatn (4), s gvn by u c csh, z mz m l k x (5) and cnsquntly u csh y c l k z mz u snh z m k c mz (6) (7) and H kh u (8) Als, th nn-sngular slutn f quatn (5) s gvn by V csh c mz (9) whr c andc ar cnstants f ntgratn t b dntfd. Fr th flud f dnsty surrundng th flud layr th prturbatn quatns ar gvn by quatns ()--(5) but wth suprscrpt nstad f. Th nn-sngular slutns f ths systm f prturbd quatns, as z, ar gvn by tkxly mz ux c3 u l k u y x u m k u z x (30) (3) (3) 463

4 Aust. J. Basc & Appl. Sc., 3(4): , 009 tkxly mz V c4 H kh u (33) (34) whr c3 and c4 ar cnstants f ntgratn t b dtrmnd. In rdr t btan th prssurs f th flud layr and th surrundng flud, lt us rturn t th systm f quatns ()--(5). Equatn () ylds u kh H t wth P H H V whr V P H H s th ttal MHD prssur whch s th sum f th kntc and magntc prssurs. Th z-cmpnnt f quatn (35), s bng u z A z frm whch, w btan A csh c mz k whr (35) (36) (37) (38) H k A (39) s th Alfvn wav frquncy dfnd n trms f H. In smlar stps, fr th flud f dnsty surrundng th flud layr, culd b btand n th frm k mz A c3 (40) wth P H H V (4) As w hav sn n th prvus dtals w gt all th varabls dscrbng th prturbatn stat f th flud layr and th surrundng flud f dnsty. Th cnstants f ntgratns c, c, c3 and c4 appar and asscatd wth ths varabls culd b dtrmnd by usng th bundary cndtns avalabl t th prsnt mdl. (I)Bundary Cndtns: Th slutn f th basc quatns ()--(5) fr th flud layr (5)--(9) and (37)--(39) and fr th surrundng flud gvn by (30)--(34) and (40)--(4) must satsfy th fllwng bundary cndtns. () Slf-gravtatng cndtns Th slf-gravtatng ptntal V and ts drvatv must b cntnuus acrss th fluds ntrfac (9) at z h. Ths cndtns rad V V V z V z z z at z h V V V V z z z z z z (4, 43) frm whch w btan 464

5 Aust. J. Basc & Appl. Sc., 3(4): , G c3 m xp mh (44) and 4 G c3 csh mh m (45) (II) Knmatc Cndtns: Th nrmal cmpnnt f th flud layr vlcty must b cntnuus wth th vlcty f th surrundng flud and at th sam tm must b cp wth th vlcty f th prturbd fluds ntrfac (9) at z h. Ths cndtns gv z uz uz at z h t (46) frm whch, w gt k c msnh mh (47) and k c3 xpmh m (48) (III) Magntdynamc Cndtn: Th nrmal cmpnnt f th magntc fld must b cntnuus acrss th flud layr-surrundng flud ntrfac (9) at z h. Ths cndtn rads H H at z h z z (49) r altrnatvly Hu Hu z z (50) By usng (7), (3) wth (30) and th rsults (47) and (48) fr th cndtn (50), w s that th cndtn (49) s dntcally satsfd at z h. Fnally, w hav t apply sm cmpatblty cndtn acrss th ntrfac (9) at z h. Ths cndtn stats that th nrmal cmpnnt f th ttal strss n th flud layr must b cmpatbl wth that f th surrundng flud acrss th ntrfac (9) at z h. Ths cndtn rads P H H P H H whch may b rwrttn, n vw f (36) and (4), n th frm (5) V V (5) By substtutng fr V, V, and, th fllwng dsprsn rlatn s btand Hk Hk cth mh cth mh 4G cth mhxp mh (53) whr, m l k (54) 465

6 Aust. J. Basc & Appl. Sc., 3(4): , 009 DISCUSSIONS Equatn (53) s dsrd dsprsn rlatn f magntgravtatnal flud layr mbddd nt magntgravtatnal nfnt flud, whr bth f th layr and surrundng flud ar actd by th flud prssur gradnt, slf-gravtatng and magntdynamc frcs. It s vald fr all knds f prturbatns. Equatn (53) rlats th scllatn frquncy wth th dnsts f th flud layr and f th surrundng flud and thr rat, th wav numbrs k and l n x-and y-drctns and als th rsultant wav numbr m k l and wth th paramtrs, and H f th prblm. Th analytcal nvstgatn and vrfd numrcally f th rlatn (53) as 0, H 0, shws that th flud layr s slf-gravtatng unstabl as th dmnsnlss wav numbr mh 0.64 whl t s stabl fr all wavs wth wav numbr mh 0.64 whr th margnal stablty stats ccurrd at mh As 0andG 0, th dscussns f th rlatn (53) ndcat that th lctrmagntc frc has strng stablzng ffct. Th lattr ffct ncrass th slf-gravtatng stabl stats and smultanusly dcrass ths f nstablty and thn stablty sts n. In dscussng th rlatn (53) n ts gnral frm, w s that th ffcts f th prssur gradnt and lctrmagntc frcs hav mdfd th stablty crtrn and mprvd th nstablty f th prsnt mdl. REFERENCES Bnjamn, T. and J. Flud, 959. Mch., 6: 6. Chandraskhar, S. and E. Frm, 953. Astrphys. J., 6: 8. Chandraskhar, S., 955. Vstas n Astrnmy. Chandraskhar, S. Nbl prz wnnr, 986. Hydrdynamc and hydrmagntc stablty (Dvr publ., N.Y, USA). Crak, A. and J. Flud, 966. Mch., 6: 369. Kant, R. and S. Malk, 985. Phys. Fluds, 8: Mls, J. and J. Flud Mch., 3: 433. Ogansyan, R., 96. Svt Astrnmy, 4: 634. Radwan, A. and S. Elazab, 987. Smn Stvn, 6: 93. Radwan, A., 990. Arch. Mch. Pland, 4: 3. Radwan, A.J., 005. Appl. Maths. And Cmput., 60: 3. Radwan, A., 007. Phys. Scr., 76: 50. Raylgh, J.W., Th Thry f Sund. (Dvr Publ., N. Y., USA). 466