Role of Viscosity Ratio in Liquid-Liquid Jets under Radial Electric Field

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1 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 Inda (phn: ; - mal: sdgadkar@tb.ac.n). Rchsh Thakar s Asscat Prfssr at Dpartmnt f Chmcal Engnrng, IIT Bmbay, Mumba Inda (phn: +91 () ; 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

2 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

3 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 ) ) ( 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) () 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

4 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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