Journal of Engineering Science and Technology Review 5 (1) (2012) Research Article

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1 Jest Jounl of Engineeing cience nd echnology eview 5 ( ( 5-56 esech Aticle On Hydomgnetic Modified hemohline Convection - n Enegy eltionshi H.i Mohn JOUNAL OF Engineeing cience nd echnology eview Det. of Mthemtics, Intentionl Cente f Distnce Eduction nd Oen Lening (ICDEOL, Himchl Pdesh Univesity, umme Hill himl-75 (HP, Indi. eceived 7 June ; evised August ; Acceted 5 July Abstct he oblem of modified themohline mgnetoconvection is consideed in the esent e. An ttemt is mde to estblish the eltionshi between vious enegies in Veonis tye configutions. he nlysis mde bings out tht f Veonis tye configution the totl kinetic enegy ssocited with distubnce exceeds the sum of its totl mgnetic nd theml enegies in the mete egime. A simil enegy eltionshi f ten s tye 7 configution is lso estblished. Futhe, these esults e vlid f quite genel ntue of boundy conditions. Keywds: DFIG, Wvelet, ime-fequency Loclition, nsients.. Intoduction hemohline convection me genelly double diffusive convection hs mtued into subject ossessing fundmentl detue fom its countet, nmely single diffusive convection, nd is of diect elevnce in the fields of ocenoghy, stohysics, liminology nd chemicl engineeing etc. F bod nd ecent view of the subject one my be efeed to Bndt nd Fenndo [], Blmfth et l. [], Mlshetty et l. [3] nd Huet et l. []. wo fundmentl configutions hve been studied in the context of themohline instbility oblem, the fist one by ten [5] wheein the temetue gdient is stbiliing nd the concenttion gdient is destbiliing nd the second one by Veonis [6] wheein the gdient is destbiliing nd the concenttion gdient is stbiliing. he min esults deived by ten nd Veonis f thei esective configutions e tht both llow the occuence of sttiony tten of motions oscillty motions of gowing mlitude ovided the destbiliing concenttion gdient the temetue gdient is sufficiently lge. Howeve, sttiony tten of motion is the efeed mode of setting in of instbility in cse of ten s configution whees oscillty motions of gowing mlitude e efeed in Veonis configution. Me comlicted double-diffusive henomenon es if the destbiliing theml/concenttion gdient is oosed by the effect of mgnetic field ottion. Bnejee et l [7] esented modified nlysis of theml nd themohline instbility of liquid lye heted undeside by emhsiing nd utiliing the oint tht line E-mil ddess: hm_mth_hu@ediffmil.com IN: Kvl Institute of echnology. All ights eseved. theeticl exlntion of the henomenon of gvity dominted theml instbility in liquid lye heted undeside (Bend convection should deend not only uon the yliegh numbe which is otionl to the unifm temetue diffeence mintined coss the lye but lso uon othe mete so tht ovision could be mde in the they to ecognie the fct tht eltively hotte lye with its het diffusivity ently incesed/decesed s consequence of n ctul decesed/incesed (deending on the fluid in its secific het t constnt volume must exhibit Bend convection t highe/lowe yliegh numbe thn coole lye unde lmost identicl condition othewise nd futhe this qulittive effect is not quntittively insignificnt. Chndsekh [8] in his investigtion of mgneto hydodynmic simle Bend convection oblem sought unsuccessfully the egime in tems of the metes of the system lone, in which the totl kinetic enegy ssocited with distubnce exceeds the totl mgnetic enegy ssocited with it, since these considetions e of decisive significnce in deciding the vlidity of the incile of exchnge of stbilities. Howeve, the solution f w ( = cons tn t(sin is not cect mthemticlly (nd Chndsekh ws we of it. Bnejee et. l. until 985 did not usue thei investigtion in this diection nd consequently did not see this connection. his g in the litetue on mgnetoconvection hs been comleted by Bnejee et. l. [9] who esented simle mthemticl oof to estblish tht Chndsekh s conjectue is vlid in the egime Q nd futhe this esult is unifmly licble f ny combintion of dynmiclly fee igid boundy when the egion outside the liquid e efectly conducting insulting. Bnejee et l. [9] showed tht in

2 Hi Mohn /Jounl of Engineeing cience nd echnology eview 5 ( ( 5-56 the mete egime the totl kinetic enegy ssocited with distubnce is gete thn the totl mgnetic enegy ssocited with it. Bnejee et l. [] futhe extended these enegy considetions to me genel oblem, nmely, mgnetohydodynmic themohline convection oblem, of ten s tye nd estblished tht in the mete egime, the totl kinetic enegy ssocited with distubnce exceeds the sum of its totl mgnetic nd theml enegies. A simil chcteition theem in mgnetothemohline convection of the Veonis tye ws lso estblished by Bnejee et. l in the subsequent ye. Mohn et l. [] deived chcteition theem in hydomgnetic double diffusive convection nd estblished tht the totl kinetic enegy ssocited with distubnce is gete thn the sum of its totl mgnetic nd concenttion enegies in the mete egime. ecently, Mohn [] extended these esults to the oblems of double-diffusive convection couled with cossdiffusions. he esent nlysis extends these enegies considetions to the hydomgnetic modified themohline convection oblem of Veonis nd ten s tye configutions. he ntue of system of equtions f the esent oblem is clely qulittively diffeent fom those of themohline convection oblem nd the esults e obviously not deivble by the method doted by Bnejee et.l nd Mohn et.l.in this diection on ccount of nontivil couling between θ, φ nd w in the eqution of het conduction. Howeve, close nd citicl look t deivtion of this eqution mkes one feel tht this difficulty cn be tken ce of by n oite tnsfmtion. he im of the esent e is to constuct such tnsfmtion which ovecomes the bove difficulty nd enbles us to deive the desied enegy eltionshi in the esent modified set u.. Mthemticl Fmultion nd Anlysis Following Bnejee et l. [7], the elevnt govening equtions nd boundy conditions of the modified themohline instbility in thei non-dimensionl fm e given by: D D w= θ φ QD D h ( D α θ ˆ α φ = ( α w ˆ α w 3 w D φ = 3 nd D h = Dw ( ( (3 ( ςith w = = θ = φ on both the boundies, Dw= on tngent stess fee boundy eveywhee, Dw = on igid boundy, h = on both the boundies if the egions outside the fluid e efectly conducting, Dh = h t = Dh = h t = if the egions outside the fluid e insulting (5 he menings of symbols fom hysicl oint of view e s follows: is the veticl codinte, d/d is diffeentition long the veticl diection, is sque of hiontl wve υ numbe, = is the theml Pndtl numbe, υ = is the κ η η mgnetic Pndtl numbe, = is the Lewis numbe, κ gαβd = is the theml yleigh numbe, κυ gαβd = is the concenttion yleigh numbe, κυ H d Q = µ is the Chndsekh numbe, w is the ρυ veticl velocity, θ is the temetue, φ is the concenttion, is the comlex gowth te, α is the coefficient of secific het due to vition in temetue nd ˆα is nlogous coefficient due to vition in concenttion, h is the veticl mgnetic field. In ( (5, is el indeendent vible such tht d, D = is diffeentition w..t, is constnt, d > is constnt, > is constnt, > is constnt, nd e ositive constnts f the Veonis' configution nd negtive constnts f ten's β configution, 3 = is the tio of concenttion gdient β to theml gdient, = i i is comlex constnt in genel such tht nd i e el constnts nd s consequence the deendent vibles w( = w ( iw i (, θ ( = θ ( iθ i ( nd φ ( = φ ( iφ i ( e comlex vlued functions(nd thei el nd imginy ts e el vlued. We now ove the following theems: heem : If (, w,θ, φ, h, = i i, is solution of ( ( togethe with boundy conditions (5 with > > nd, then 7 > φ Dw w d Q Dh h d d Poof: Eqution ( uon utiliing (3 cn be witten s ( D ˆ3 ( D α θ α φ = ( α w (6 5

3 Hi Mohn /Jounl of Engineeing cience nd echnology eview 5 ( ( 5-56 Using the tnsfmtions w% = w ( α % θ = θ φ ˆ α3 % φ = φ h% = h Equtions (, (3, ( nd (6 nd the ssocited boundy conditions (5 ssume the following fms: D D w= QD D h θ φ { ( α } D θ = Bw (9 w D φ = D h = Dw with w = = θ = φ on both the boundies, (7 (8 ( ( Dw= on tngent stess fee boundy eveywhee, Dw = on igid boundy, h = on both the boundies if the egions outside the fluid e efectly conducting, Dh = h t = Dh = h t = if the egions outside the fluid e insulting, ( whee = ˆ α3, = ( α ( α ˆ α 3, ( α B= ( α > nd the symbol ~ hs ˆ α3 been omitted f convenience. Multilying ( by h (the comlex conjugte of h, integting the esulting eqution ove the nge of by ts suitble numbe of times, nd mking use of the boundy conditions ( we get (3 M Dh h d h d = w Dh { } whee M ( h ( h =. Equting the el t of (3, we get M Dh h d h d = el t of w Dh d wdh d wdh d / / w d Dh d ( (using chwt inequlity ince, theefe fom (, we get / / Dh d < w d Dh d < Dh d w d (5 Using (5, it follows fom ( tht ( Dh h d < w d (6 ince w ( = = w (, theefe using yleigh-it inequlity [3], we get w d< Dw d (7 It follows fom (6 nd (7 tht ( < < ( Dh h d Dw d Dw w d / s / < ( s φ Dh h d φ d Dw w d d (8 Multilying ( by the comlex conjugte of ( nd integting by ts ove the veticl nge of f n oite numbe of times nd mking use of the boundy conditions ( we get ( φ φ φ ( φ φ D D d D d φ d = ince, w d, theefe, fom (9, we get (9 53

4 Hi Mohn /Jounl of Engineeing cience nd echnology eview 5 ( ( 5-56 ( φ φ φ D D d< w d ( ince φ ( = = φ (, theefe using yleigh-it inequlity [3], we get φ d < Dφ d nd lso φ d D φ d (using chwt inequlity ( It follows fom ( nd ( tht φ < ( d w d φ d < w d < Dw w d φ d< ( 7 Dw w d, ( since the minimum vlue of 3 f > is 7 ( > ( Dw w d Q Dh h d / s φ d ( nd this comletes the oof of the theem. We note tht the left hnd side of ( eesents the totl kinetic enegy ssocited with distubnce while the ight hnd side eesents the sum of its totl mgnetic nd concenttion enegies, nd heem my be stted in the following equivlent fm: At the neutl unstble stte in the hydomgnetic modified themohline convection oblem of the Veonis' tye configution, the totl kinetic enegy ssocited with distubnce is gete thn the sum of its totl mgnetic nd concenttion enegies in the mete egime / s nd this esult is unifmly vlid f ny 7 combintion of dynmiclly fee igid boundies tht e eithe efectly conducting insulting. heem : If (, w, θφ,, h, = i i, is solution of (8 ( togethe with boundy conditions ( with <, <, nd / B then 7 ( > ( Dw w d Q Dh h d / θ d (5 Poof: Putting = nd =, (8 in the esent cse ssume the following fm: / / s s φ < 7 ( d Dw w d Now fom (8 nd (, we get / s Dh h d φ d / s < ( Dw w d 7 (3 D D w= / / θ φ QD D h (6 Poceeding exctly s in heem, (8 in the esent cse cn be witten s Dh h d θ d φ < Dw w d d (7 heefe, if / s, then fom (3, we get 7 Multilying (9 by the comlex conjugte of (9 nd integting by ts ove the veticl nge of f n oite numbe of times nd mking use of the boundy conditions ( we get 5

5 Hi Mohn /Jounl of Engineeing cience nd echnology eview 5 ( ( 5-56 θ θ θ D D d α θ θ D d ( α θ d = B w d ince,, theefe fom (8, we get ( θ θ θ (8 D D d< B w d (9 ince θ ( = = θ (, theefe using yleigh-it inequlity [9], we get θ d < Dθ d nd lso θ d D θ d (using chwt inequlity (3 It follows fom (9 nd (3 tht θ < ( d B w d B θ d < B B θ d< 7 w d < Dw w d Dw w d, ( ( since the minimum vlue of 3 f > is B θ d< 7 ( Dw w d 7. (3 Now fom (7 nd (3, we get Dh h d θ d B < 7 Dw w d. (3 heefe, if B, then fom (3, we get 7 ( > ( Dw w d Q Dh h d θ d (33 nd this comletes the oof of the theem. We note tht the left hnd side of (33 eesents the totl kinetic enegy ssocited with distubnce while the ight hnd side eesents the sum of its totl mgnetic nd theml enegies nd heem my be stted in the following equivlent fm: At the neutl unstble stte in the hydomgnetic modified themohline convection oblem of the ten s tye configution, the totl kinetic enegy ssocited with distubnce is gete thn the sum of its totl mgnetic nd theml enegies in the mete egime B nd this esult is unifmly vlid f 7 ny combintion of dynmiclly fee igid boundies tht e eithe efectly conducting insulting. 3. Conclusions In the esent e, the hydomgnetic modified themohline convection oblem of Veonis nd ten s tye configution is consideed. he nlysis mde bings out the following min conclusions: i. At the neutl unstble stte in the mgnetohydodynmic themohline convection oblem of the Veonis tye configution, the totl kinetic enegy ssocited with distubnce is gete thn the sum of its totl mgnetic nd concenttion enegies in the mete egime nd this esult is unifmly vlid f 7 ny combintion of dynmiclly fee igid boundies tht e eithe efectly conducting insulting. ii. At the neutl unstble stte in the hydomgnetic modified themohline convection oblem of the ten's tye configution, the totl kinetic enegy ssocited with distubnce is gete thn sum of its totl mgnetic nd theml enegies in the mete / egime nd this esult is unifmly 7 vlid f ny combintion of dynmiclly fee igid boundies tht e eithe efectly conducting insulting. 55

6 Hi Mohn /Jounl of Engineeing cience nd echnology eview 5 ( ( 5-56 efeences. A. Bndt nd H.J.. Fenndo, Ameicn Geohysicl Union (996.. N. J. Blmfth,. A. Ghdge, A. Kettum nd.d. Mnde, J. Fluid Mech. 569, 9 (6. 3. M.. Mlshetty nd B.. Bid, Phys. Fluids 3, doi:. 63/.368,.. E.H. Huet nd J.. une, J. Fluid Mech. 6 ( M.E. ten, ellus, ( G. Veonis, J.Ms. es., ( M.B. Bnejee, J.. Gut,.G. hndil, K.C. hm nd D.C. Ktoch,.Mth.Phy.ci. 7 ( Chndsekh, Philos. Mg. 3, 5 ( M.B. Bnejee nd.p Ktyl, J. Mth. Anl. Al (988.. M.B. Bnejee, J.. Gut nd.p. Ktyl, Indin J. Pue Al.Mth. 8(9 ( H. Mohn, P. Kum nd P. Devi, Gnit. 57( (6 9.. H. Mohn, tudi Geo. ech. et Mech. XXXII, 3 (. 3. M.H chult, line Anlysis, Pentice-Hll, Englewood Cliffs, N.J. (