A study Of Salt-Finger Convection In a Nonlinear Magneto-Fluid Overlying a Porous Layer Affected By Rotation

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1 Innion on o Mchnic & Mchonic Engining IMME-IEN Vo: No: A O -ing Concion In Nonin Mgno-i Oing oo Ac Roion M..A-hi Ac hi o in -ing concion in o o nonin gno-i oing oo c oion. o in h i i gon Ni-o qion n in h oo i i gon c-inn. W ho h h nonini in hi o o no h n c on h ion inii c. Ho h nonini h oon c on h oii c. In Ni-o qion c-inn gn onoi nonin gno-i oii. I. INROUCION h inii ho h c coni in n h n cogni o o nn ionc in n i o i nic. h i in o on h on o h inii in i ho o n [ ]. n o ih hin o n incoi ico i ning on ic inin conn. h c n ing in conc ih h i o. In hi in n c h cin ciic gin c o inii cn in. h inii o o i h o o n c o Coioi oc h n i Chnh [ ] o ion n oii c. H ho h h nc o h oc h h c o inhiiing h on o h concion. Ni [5] coni h on o -ing concion in oo.non. [6] coni h hohin inii n -ing in oo i n o h on o. n [7] h i o coni ch o n h M..A-hi Mhic n Cog o cinc i Unii i i Ai.q777@hoo.co hooing ho o o h in ii qion. n [7] n Ni [8] c in oing h qion o oo. In c o oion in oo i h c inc oo h c o h Ni-o icoi hi in h oii c (inn o) gg inn [9] h Ni- o icoi i i. Chn & Chn [] coni h i- o hn h o i h n o o n h oion o h o i oin ing hooing ho. in & Ogn [] o in h inion o c ho in o h gnion o io ign. [] nion o Chh onoi o inig n ign o iing o o icing ini concing inn co o h h on gnic in inii. hi [] i h c o c-nion in o concing i ih n io gnic i n oin hn h c i o n non-o. H o h ing Chh c ho n h oin o in o h o Chn & Chn []. ghn [] i h h concion in i oing oo n coni h o o o h o o n c nion in on h o on o. In [5] h o ih h o coniing h io h o h i n inig h c o h iion o n i n oo i oi. Ahin [6] i n concion in hoion oo concing i in h nc o gnic i n coioi oc. A-hi & hi [7] i h ing concion in hoion oo o i c oion n ic in gnic i on oh. h oion i oin ing gn onoi hn h h n h concnion c o o. In hi o A-hi & hi o in nonin gno-i. Nic oion oin ing gn onoi. Conqn i i on h h o o o inii ion IMME-IEN Ag IEN I E N

2 Innion on o Mchnic & Mchonic Engining IMME-IEN Vo: No: inii hn h h n h concnion c o o n oii hn h h n h concnion c o o. II. MAEMAICA REAMEN W coni i oing oo o h h o o h oo och h oo o h i. h n inc n h o i h on o h i i n h o on o h oo i i h h ci n no h i n oo i ci. W o h h i i ih n incoi h n cic concing ico i coniing o hich o in i n gon Ni-o qion. Ho h o i occi oo i h i oing in i n gon c-inn. oh c o conn ic nonin gnic i n c oion on ih conn ng oci. Gi g c in h ngi icion o. Concion i in h ning on h i ni n ing n icoi. h Oc-oinq oiion i h ni o i i conn h c in h o oc h h ni i in ooion o n concnion i... () Wh no h Kin o h i i h concnion h ni o i n (conn) h h coicin o o nion o h i n (conn) h ing coicin o o nion o h i. V H n E ci h onoi oci o h i h gnic i h gnic incion h cn ni n h cic i. Hnc iv i () h ion n h gnic i n h gnic incion in nonin h o H () Wh i h gnic ciii ρ i h inn ng ncion.an h M qion c E ch () h h icn cn h n ngc in h con o h M qion i co in iion hn chg i innno i. iing o () () in o () oin c V V c c ρ c (5) h i cic iii. ing c ic c c c g i n V V V hn qion (5) in V V h qion o oion i η ρ. V V. V V g V (7) h i hoic i h nic icoi n i h-inion cin oo. An iing o () () in o h on oc oin c H ρ. ρ. (8) Hnc h qion o oion co V V. V V 8 (9) g V. ρ n o h goning qion o h i V V V V g α ρ β (6) V Ω. ρφ ρ μ ρ c V V () () () c ρφ () V V η () IMME-IEN Ag IEN I E N

3 Innion on o Mchnic & Mchonic Engining IMME-IEN Vo: No: n h goning qion o h oo i V V V g α φ ρ K (5) β V Ω. ρφ ρ μ (6) φ V (7) c ρφ (8) ρ c ρc V V V η ρφ. ρ h 8 (9) ρ h oii 8 o h i n h oo i ci. n h cnig oc o h i n oo i ci V V h onoi n g oci ci V V h coioi ccion o h Ω i n oo i ci h Kin o h i n oo i ci concnion o h i n oo i ci h iii o h i n oo i ci μ μ gnic ii o i n oo ci h h n o h concii o i n oo ci i h inic icoi K i h ii c h h n o c i i ooi n h cci ni o o h i n oo i conn. In c ρ c φρ c φρ c h c i h h cci ni o o h oo. o h i igi n inin conn n conn concnion n i in n inin conn n conn concnion hn h on coniion cn in ζ () on h on n ζ () on h o on h n h no i oci coonn o h i in i n oo i ci ζ n ζ h no i oci coonn o h i in. i n oo i ci. h on coniion on h inc n on h ion h concnion h no n ngni i oci no n ngni conino o h μ μ μ μ μ μ. () Eqion ()-(9) h n qiii oion iing h on coniion ()-( ) on h o V V ρ g ρ g Ω Ω Ω Ω con n () Ω Ω con n Β Β conn Β Β conn n ih h on coniion IMME-IEN Ag IEN I E N

4 Innion on o Mchnic & Mchonic Engining IMME-IEN Vo: No: IMME-IEN Ag IEN I E N () n h inc coniion (5) h qiii i hoic n concnion in h i n oo i ci (6) h. W h ion ooing in ion qnii V (7) V o h goning qion in h i n oo i ci n o h on coniion. A ion h non-inioniion i ing ρ (8) ρ o h i n ing K ρ (9) ρ o h oo i h c ρ n c ρ h h iii o h i h n oo i ci hn h qion ()-() co R R () () () c c () () Wh i h ni co in h -icion n R R n non-inion n no h ico n n h Righ n Righ n Chnh n o n i n gnic n n n gnic n o h i n gin gα R gβ R πρη Ω ρη n h qion(5)-(9)co

5 Innion on o Mchnic & Mchonic Engining IMME-IEN Vo: No: 6 φ G φ R R c c (5) (6) (7) (8) h G ρc ρc n R R (9) n non-inion n no h c n ico n n h Righ n Righ n Chnh n i n gnic n n n gnic n o h oo i n gin K gα K R R gβ K Ω K πρη ρη n h hn hing o o hn hing o o hn concnion o o hn concnion o o. h on coniion ()-() co h n n gin Ω n Ω n γ R γ R R R n. () iniion i on ngc oc n o (high hn h i) o h in ion qni n o h ( ) ci hn i h c o h qion () n (5) oin R R () R R i n o h oigin qion () n (5) in hi c h (c c). h () IMME-IEN Ag IEN I E N

6 Innion on o Mchnic & Mchonic Engining IMME-IEN Vo: No: IMME-IEN Ag IEN I E N R R () R R () An i h c o qion() n(9) ih ing() n(8) oin (5) (6) No h hi coonn o qion()()()(6) (7) (9) n()-(6) (7) R R (8) (9) (5) (5) (5) φ (5) R R φ (5) G (55) φ (56) (57) (58) h i o-inion cin oo n. W h no oion o h o n i Φ Φ o h ncion n.h h goning qion R R (59) (6) (6) (6) (6) (6) R R φ (65) - - φ (66) G (67) φ (68) (69)

7 Innion on o Mchnic & Mchonic Engining IMME-IEN Vo: No: IMME-IEN Ag IEN I E N (7) h n n n noninion n in h i n oo i ci i h goh n n. h on coniion in h in o on (7) on - φ γ γ (7) on (7) III. MEHO O OUION A gn onoi i i o o h qion (59)-(7) ih h n on coniion (7)-(7) n n ino h noion n ci n g h. hn o h α M 8 h i h 8 in hn h qion (59)-(7) cn in in o n oin ini qion o i o inc n i hn o h ign o cn o in h o Y Y Y A h A n Β 8 8 ic. h in ign o c o V V E h ic E n h h oc o. h on coniion c h Mh Mh 8Mh o o E n. IV. REU AN REMARK Uing gn onoi h ign o (59)-(7) ih h on coniion (7)-(7) no o o on oin ini qion o i o in h i n o on oin ini qion o i o in h oo ih n igh on coniion. In hi o i ic h nic hogh o c hn h h n concnion c o o n o. C : h h n h concnion c o o. H n h o h inii h Righ n o h oo i 5 R o in h Righ n o h oo i R cooning o h n o h in o n. In hi c h ign n h h ion inii hn. h nic oin h ii o ho ic in A-hi & hi [7].

8 Innion on o Mchnic & Mchonic Engining IMME-IEN Vo: No: 9 In oh o h nc o non-in gnic i o no h n c on h on o inii hogh ion concion c. C : h h n h concnion c o o. H n h o h inii h Righ n o oo i R o in h Righ n o oo i R cooning o n o in o n. In hi c h ign co n h h oii hn hon in h ooing ig..-5. ho go h ooing : A inc R c hich n h h inc o h oion c h inc o h -ing concion ing o n inc in h inii o h i hon in ig.. A inc R c hich n h h nonini h iiing c o h oii c hon in ig.. h nc o h gnic i h i n i in ig.. h inc o h ii h c h cn o h -ing concion ning h h ii i inc in h i i in ig.. A inc R inc ning h h inc o h h io n h o h i o i ig.5. R =.5 = =.75 = ig.. h ion n n R o in o. 5. n.. R = =5 = ig.. h ion n n R o in o n.. =. =. R = =.5 = R ig.. h ion n n R o in o.. 5 n.. ig.. h ion n o n R o in n IMME-IEN Ag IEN I E N

9 Innion on o Mchnic & Mchonic Engining IMME-IEN Vo: No: 5 =. =. R ig. 5. h ion n n R o o in n. V. REERENCE [] n H. oion ci n n n iqi. R gn cinc iq 6-7 n 9-8 (9). [] n H. oion ci n n n iqi non ch concion n gi nn. Ann. Chi. h. 6- (9). [] Chnh. h inii o o i h o n c o Coioi oc. oc. Ro. oc.(onon) Vo (95). [] Chnh. n E.. h inii o o i h o n c o Coioi oc.ii. oc. Ro. oc.(onon). No. 98- (955). [5] Ni. A. On o hohin concion in oo i oc R (968). [6] non. & ighoo E. hohin inii n ing in oo i h. O i 5 No (97). [7] n W.. Concion Inii in o oo n h.. iion Unii o Minno Minnoi. MN. 97. [8] Ni. A. On o Concion in i oing oo i. i Mch (977). [9] inn H. C. A ccion o h ico oc oing i on n o ic. A. ci. R. A. 7- (97). [] Chn. & Chn C.. On o ing concion in hoion oo ning i.. o H n. -9 (988). [] in K. A. & Ogn R. R. A cic Inion o c Mho Rin o h Gnion o io Eign Nn. 9/ Unii o Ggo n o Mhic in i (99). [] C.. Hognic Inii In h Eh Co h.. hi n o Mhic Unii o Ggo 99. [] hi A.. A c Eign Mho o Mi- Conin. h.. hi Unii o Ggo co [] ghn. c-nion-in concion in i oing oo. Co. h. 7-7 (). [5] ghn. Ec o o iion n oing on concion in i oing oo In.. N. An. Mh. Goch (). [6] Ahin. n concion in hoion oo concing i in h nc o gnic i n coioi oc An M. c. hi U A- Unii. [7] A-hi M.. & hi A-h K. -ing concion in hoion oo o i c oion n ic in gnic i on oh. Anc n Aicion in i Mchnic o. 9-6A.() IMME-IEN Ag IEN I E N