Intrnatnal Jurnal f Chmcal and Blgcal Engnrng 6 01 Rl f Vscsty Rat n Lqud-Lqud Jts undr Radal Elctrc Fld Sddharth Gadkar and Rchsh Thakar Abstract Th ffct f vscsty rat (λ, dfnd as vscsty f surrundng mdum/vscsty f flud t) n stablty f axsymmtrc (m=0) and asymmtrc (m=1) mds f prturbatn n a lqud-lqud t n prsnc f radal lctrc fld (E 0 ), s studd usng lnar stablty analyss. Th vscsty rat s shwn t hav a dampng ffct n bth th mds f prturbatn. Hwvr th ffct was fund mr prnuncd fr th m=1 md as cmpard t m=1 md. Invstgatng th ffct f bth E 0 and λ smultanusly, an pratng dagram s gnratd, whch clarly shws th rgns f dmnanc f th tw mds fr a rang f lctrc fld and vscsty rat valus. Kywrds lqud-lqud t, axsymmtrc prturbatn, asymmtrc prturbatn, radal lctrc fld E I. INTRODUCTION LECTRIFIED lqud ts r thrads hav bn wdly studd n flud mchancs. Hwvr th sam cannt b sad abut mmrsd lqud ts, whr a t f lqud s submrgd n anthr mmscbl lqud f fnt vscsty. In such a cas, th vscsty rat f th t and th mdum flud (λ = µ mdum /µ t) plays an mprtant rl n dtrmnng th stablty f th systm. Als, systms nvlvng mmrsd ts whn subctd t lctrc flds ar nw attractng ncrasng attntn. An mprtant ccurrnc s th lctrdsprsn f a cnductng lqud t submrgd n an mmscbl dlctrc lqud and subctd t lctrc fld [1] [3]. A stady cn- t can b ralzd fr a rang f systm paramtrs, whch can undrg axsymmtrc r asymmtrc nstablts [4] [7]. Th tchnqu s nw usd t gnrat mulsns wth narrw dstrbutns f drplt szs cntrllabl n th rang frm mcrmtrs t tns f nanmtrs [4], [8]. Thr s xtnsv ltratur avalabl n bth xprmntal and thrtcal nvstgatn f lqud t brakup, wth r wthut th prsnc f lctrc fld and svral rvw artcls hav dscussd th tpc n ampl dtal [9] [11]. Althugh cnsdrabl wrk has bn dn n th ffct f lctrc fld n t nstablts, th ffct f surrundng mdum has nt bn adquatly addrssd. Th studs n th ltratur fr a t undr radal lctrc fld hav thr cnsdrd th nvscd t lmt (λ = ) [1] [14] r lkd at a vscus t (λ = 0) n an nvscd mdum [15] [5]. Sddharth Gadkar s a Phd studnt f th IITB-Mnash Rsarch Acadmy, IIT Bmbay, Mumba 400076 Inda (phn: +91-9769066819; - mal: sdgadkar@tb.ac.n). Rchsh Thakar s Asscat Prfssr at Dpartmnt f Chmcal Engnrng, IIT Bmbay, Mumba 400076 Inda (phn: +91 () 576 741; mal: rchsh@ch.tb.ac.n Thr ar fw studs whch hav xplctly cnsdrd vscsty f bth t and surrundng, hwvr, lmt thr dscussn t partcular valus f vscsty rat [6] [31] and hav nt cnsdrd th ffct f arbtrary vscsty rat n th dffrnt mds f prturbatns. In th prsnt wrk, w cnsdr a chargd lqud t ssung nt anthr mmscbl lqud and subctd t radal lctrc fld. Rlatv mtn btwn th tw fluds s cnsdrd zr. W cncntrat n th cas f hgh Ohnsrg numbr (Oh; a dmnsnlss paramtr rprsntng th rat f vscus and ntrfacal tnsn frcs), and us lnar stablty analyss t study th ffct f vscsty rat n th axsymmtrc and asymmtrc nstablty f a vscus t (a prfct cnductr) submrgd n anthr vscus flud ( prfct dlctrc) subctd t radal lctrc fld. Th hgh Oh lmt whch has bn assumd n ths wrk can b asly satsfd fr hghly vscus lctrfd ts. Als t s vald n th study f stablty f nutrally buyant lqud brdgs mmrsd n an utr bath f anthr mmscbl lqud n th prsnc f lctrc fld [3] [34]. Ths typ f flw s als ncuntrd n plymr phas sparatn, whr lqud drplts f n f th phass nuclat ut and grw and ar strtchd n xtnsnal flws. Phas sparatn & mrphlgy undr lctrc fld wuld thn dpnd upn th stablty f such thrads [35]. Th thry dvlpd n th currnt wrk, can thus b usful t study all th abv ntrstng systms. II. FORMULATION OF THE PROBLEM Cnsdr an nfntly lng cylndrcal t f radus a f an ncmprssbl lqud wth vscsty µ, mmrsd n an mmscbl flud f vscsty µ. Th subscrpt dnts nsd flud t whras subscrpt stands fr utsd surrundng mdum. Th flud t s a chargd cnductr wth dlctrc cnstant ε, charactrzd by zr fld nsd, surfac ptntal ψ s and charg σ s whras th utsd mdum s a prfct dlctrc wth dlctrc cnstant ε. Th t s subctd t radal lctrc fld f strngth E 0. A. Gvrnng Equatns Th gvrnng quatns f mtn fr th systm ar gvn by ɶ ɶ = 0 (1) v vɶ tɶ ɶ ɶɶ ɶɶ ɶ ɶ ɶ ɶ ρ + v v = p + µ v + ρc E () 10
Intrnatnal Jurnal f Chmcal and Blgcal Engnrng 6 01 whr s st as fr th nnr flud and fr th utsd flud. E s th lctrc fld; v s th vlcty fld, p th prssur, ρ c s th fr charg dnsty and ρ s th flud dnsty n th bulk. Th tld rprsnts dmnsnal quantts. In th absnc f any fr charg.. ρ c = 0, th ptntal (φ) s dscrbd by ɶ ɶ = 0 (3) and E φ = φ Th abv gvrnng quatns ar nn-dmnsnalzd usng th fllwng scalng: th dstanc s scald by a, th tm by µ aγ, th vlcts ar scald by γ/µ and th strsss and th prssur by γ/a, whr γ rprsnts th ntrfacal surfac tnsn btwn th t and th surrundng flud. Th γ ε ε and scalng fr ptntal and lctrc fld ar, a / ( 0) γ / ( aε ε ) rspctvly, 0 whr ε 0 s th prmttvty f fr spac. Usng th abv scalng, w gt, = 0 (4) v 1 v + v v = p + c v (5) ( Oh ) t = 0 (6) φ 1/ whr Oh (Ohnsrg numbr) = µ / ( ρ γ ) and c = µ / µ r 1 fr th nnr t r utr mdum rspctvly. Th vscsty rat µ/µ s rprsntd by λ. In th prsnt wrk, w spcfcally lk at th cas f vry hgh Oh, Stks flw cndtns, whch as dscrbd n th ntrductn ar cmmn n many ndustral prcsss and blgcal systms. Ths cndtns ar satsfd fr flw systms wth hghly vscus flud t and/r fr vry small damtr cylndrcal ts. Equatn 5 s thus rducd t, B. Bundary Cndtns 0 = p + c v (7) Th lctrstatc bundary cndtn at th ntrfac f th cnductr t and th dlctrc surrundng s gvn by cnstant ptntalɶ φ = ɶ φ, whr th nn-dmnsnal surfac s ptntal s gvn by ( 0 ) φ = φ = ψ = ɶ φ / ɶɶ γ / ε (8) s s a Th hydrdynamc bundary cndtns ar th cntnuty f vlcty and th frc balanc at th ntrfac. W us th dfntns f th unt nrmal and th unt tangnts t th ntrfac and th vlcty vctr v = v r r + v θ θ + v z z t a wrt th bundary cndtns as n ( v ( r, θ, z, t) v ( r, θ, z, t)) = 0 (9) t ( (,,, ) (,,, )) 0 1 v r θ z t v r θ z t = (10) t ( (,,, ) (,,, )) 0 v r θ z t v r θ z t = (11) F( θ, t) n v ( r, θ, z, t) = 0 t (1) n ( τ ( r, θ, z, t) + τ ( r, θ, z, t) τ ( r, θ, z, t) τ ( r, θ, z, t)) n + H ( θ, t) = 0 (13) ( (,,, ) t (,,, ) (,,, ) 1 τ r θ z t + τ r θ z t τ r θ z t τ ( r, θ, z, t)) n = 0 ( (,,, ) t (,,, ) (,,, ) τ r θ z t + τ r θ z t τ r θ z t τ ( r, θ, z, t)) n = 0 (14) (15) whr, n, s th unt nrmal and t 1 and t ar th mutually rthgnal unt tangnt vctrs rspctvly. C. Lnar Stablty Analyss (LSA) In LSA a typcal varabl f s xprssd as, ' f = fm + δ f (16) ' whr f s th bas stat (stady stat valu) and m f s th prturbatn varabl δ bng a small paramtr. Th analyss s cnductd t O(δ). Th prturbd quantty ' f s ' ( kz+ mθ f f ( r) ds dk dm, = k and m ar th n n - dmnsnal axal a n d a z m u t h a l wavnumbrs and s s th dmnsnlss grwth rat. Th prturbd shap f th ntrfac s gvn by F( θ, t) 1 δ D ( kz+ mθ = + (17) whr 1 s th nn-dmnsnal cylndr radus and D s a cnstant. Th curvatur at th prturbd ntrfac s gvn by ( kz+ mθ H = 1 δ D(1 m k ) (18) whr th man curvatur f th cylndr s gvn by 1. Smlarly all th thr quantts, such as th prssur p, vlcty cmpnnts (vr, vθ, vz ) and th ptntal φ fr bth nsd and utsd flud ar dcmpsd nt a bas part and a prturbatn part. Ths quantts whn substtutd back nt th gvrnng quatns prvd gn functns fr th dffrnt prturbatns varabls. Th cmplt Egn functns fr th ptntal ar drctly 103
Intrnatnal Jurnal f Chmcal and Blgcal Engnrng 6 01 btand by usng ptntal bundary cndtns, φ = ψ (19) s E D φ ψ δ ln r + 0 ( ) ( kz+ mθ = s E0 Km kr (0) Km( k) Th bundary cndtns ar appld at th unknwn ntrfac F(θ,t) and th valu f a typcal varabl f s btand frm th unprturbd ntrfac as f δ f t ( r= 1 + D) = f( r= 1) + D r r= 1 ( ) kz + m θ + st Substtutng th gn functns n th bundary cndtns, n can assmbl all th quatns n a matrx frm as MX=0 whr matrx M wuld b a functn f s, k, m, E and λ, and X wuld b a clumn matrx mad up f all th cnstants. Th matrx quatn MX= 0 has a nn-trval slutn nly whn th Dt[M] = 0. Slvng th dtrmnant f M and quatng t t zr gvs th dsprsn rlatn, azmuthal wavnumbrs alng wth thr pratng paramtrs. Th dsprsn rlatn s farly lng and cmplcatd and hnc s nt shwn hr. Th dffrnt lmnts f th matrx X ar prvdd n th Appndx. A. Valdatn III. RESULTS AND DISCUSSION T valdat ur prblm frmulatn and slutn prcdur, w frst cmpar ur rsults wth rlvant xprssns that hav bn prvusly rprtd n th ltratur fr spcfc valus f vscsty rats, λ. asymmtrc m = 1 md was drvd. As prvusly bsrvd [39] [41], th asymmtrc md was fund t b stabl (-v grwth rat) fr all valus f k, at zr lctrc fld. Fr m = 1 md, th grwth rat xprssn fr a t n a nnvscus mdum (λ 0) s, s = kγ k + k + k c + k c + kc 3 ( ( (5 ) (1 ) ) 3 4 3 ( k + k (9 + k ) c 1 kc ( 4 + k + 4 k ) c ) whr c = I 1 (k)/i 0 (k). (4) ) Wth Radal Elctrc Fld Th xprssns f grwth rat fr axsymmtrc and asymmtrc prturbatns n a vscus cnductng t, n an mmscbl vscus dlctrc surrundng, subctd t radal lctrc flds ar drvd. Fr th lmtng cas f λ 0, a lw k analyss was prfrmd fr bth m = 0 and m = 1 md. Th xprssns btand ar, 1 [ s γ ε ER ], m = 0 (5) 6 4 [1 + ε R ln( )], = 1 s E k m 3k (6) Equatns (5) and (6) agr wth th xprssns drvd by Savll [16] undr smlar cndtns. Radal lctrc fld s knwn t hav a dual ffct n axsymmtrc prturbatns, stablzng th lng wavs whl dstablzng th shrt ns. 1) Wthut Elctrc Fld Frstly, rsults f th mdl wthut th prsnc f any lctrc fld ar prsntd. Prvd blw ar th xprssns f grwth rat fr axsymmtrc (m = 0) md fr th spcal cas f vscsty rat λ = 0, 1 and. Ths thr vscsty rats crrspnd t a nn- vscus vacuum surrundng, smlar vscsty fluds and a nn-vscus t rspctvly. λ = 0, s= λ =, s= [ k 1] / K ( k) 0 (1 + k k ) K 1( k ) [1 k ] / K ( k) 0 (1 + k k ) K 1( k ) [1 k ] 1 0 1 (1) () k λ = 1, s= ( I ( k) K ( k) I ( k) K ( k)) (3) Equatns (1), () and (3) agr wth that drvd prvusly undr sam cndtns by Raylgh [36], Tmtka [37] and Stn and Brnnr [38] rspctvly. Smlar t axsymmtrc md, th grwth rat xprssn fr th Fg. 1 Effct f radal lctrc fld n m = 0 md at λ=0.5 Fg. Effct f radal lctrc fld n m = 1 md at λ=0.5 On th thr hand, asymmtrc prturbatns hav bn shwn t bcm mr unstabl wth ncrasng radal fld fr all wavlngths. Th rsults btand n th prsnt wrk agr wth prvus nvstgatns as shwn n fgurs 1 and. 104
Intrnatnal Jurnal f Chmcal and Blgcal Engnrng 6 01 Fg. 3 Effct f λ n m = 0 md at E 0 = 3 Wth th scalng usd n th prsnt wrk, fr th radal lctrc fld rntatn th nrmal and tangntal lctrc strsss d nt dpnd upn th dlctrc cnstant rat, β = ε /ε B. Effct f Vscsty Rat n Prsnc f Radal Elctrc Fld Th ffct f vscsty rat λ n lqud-lqud ts whn subctd t radal lctrc fld s nw dscussd. Th analyss s rstrctd t m = 0 and m = 1 md f prturbatn. Fgurs 3 and 4 shw th grwth rat vs wavnumbr plts at dffrnt λ fr m = 0 and m = 1 md rspctvly whras fg. 5 shws th varatn f maxmum grwth rat (s m) wth λ at E 0 = 3. Fgurs 3 and 4 suggst stablzatn f bth axsymmtrc and asymmtrc nstablty wth ncras n λ. Th maxmum grwth rat, s m fr bth m = 0 and m = 1 md dcrass wth λ (Fg. 5). Fg. 5 Effct f λ n s m fr m = 0 and m = 1 md at E 0 = 3 Fg. 6 Opratng dagram shwng dmans f pr-dmnanc f m = 0 and m = 1 mds fr radal lctrc fld Fg. 6 shws that th m = 1 md can nly b ralzd n th lwr λ lmt. Als, at λ valus whr m = 1 md dmnats, a mnmum thrshld lctrc ptntal must b prvdd t vrcm th axsymmtrc m = 0 md. Wth ncrasng λ ths thrshld lctrc fld als ncrass, hwvr, ths rul s nly vald up t a crtan crtcal λ abv whch th m = 0 md s always dmnant. Fg. 4 Effct f λ n m = 1 md at E 0 = 3 Thus t s sn that lctrc fld and vscsty rat hav ppsng actns n th grwth rats f th tw mds f nstablts. Radal lctrc fld n n hand dstablzs whras vscsty rat n th thr stablzs ths prturbatns. Addtnally, th xtnt wth whch bth ths paramtrs act s dffrnt fr th tw mds. Thus t s vry mprtant t study th ffct f lctrc fld and vscsty rat smultanusly. T ths nd, an pratng dagram shwng dmans f prdmnanc f th tw mds fr any gvn valu f E 0 and λ s prsntd n fg. 6. IV. CONCLUSION Th currnt study prsnts th lnar stablty analyss n a flud t mmrsd n anthr mmscbl flud and subctd t radal lctrc fld. Th analyss rducs t th prvusly rprtd rsults fr axsymmtrc prturbatns n th apprprat lmts f th vscsty rat and xtnds t nclud asymmtrc prturbatns alng wth th ffct f changng vscsty rat and appld lctrc fld. Whl th ffcts f lctrc fld n flud ts ar alrady knwn, t s fund that vn th vscsty rat f th fluds was crtcally mprtant n dcdng th mst dmnant md f prturbatn. Incrasng λ has a tndncy t damp bth axsymmtrc (m=0) and asymmtrc (m=1) mds f nstablts, hwvr th ffct s mr prnuncd fr m = 1 md as cmpard t m = 0 md. Thus as λ gs up, th thrshld lctrc fld rqurd t xprss m = 1 md als rss. An pratng dagram t prdct th pr-dmnant md at any gvn valu f lctrc fld and vscsty rat s prsntd. Ths dagram can b f grat hlp n crrctly prdctng th pratng cndtns rqurd t xprss any dsrd nstablty fr a partcular applcatn. 105
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