On the Linear Stability of Compound Capillary Jets

Transcription

1 ILASS Americs, th Annul Conference on Liquid Atomiztion nd Spry Systems, Chicgo, IL, My 7 On the Liner Stbility of Compound Cpillry Jets Mksud (Mx) Ismilov, Stephen D Heister School of Aeronutics nd Astronutics, Purdue University, W Lfyette, IN Abstrct Compound cpillry ets re utilized in the mnufcture of coted tblets within the phrmceuticl industry nd present n interesting tomiztion problem Differences in density nd surfce tension between the inner fluid (the medicine) nd the outer fluid (the coting) provide for complex interctions reltive to cpillry instbility The present study ws motivted by ILASS 6 presenttion by Bin nd Mshyek who developed one dimensionl nonliner tretment of the instbility The present work ws initilly studied s homework problem in grdute clss tught by the co-uthor t Purdue during the fll 6 semester An xisymmetric, inviscid liner instbility nlysis hs been developed to compliment the Bin nd Mshyek work nd to provide insight into the droplet sizes formed under vriety of conditions Inner nd outer fluid density nd surfce tension re to be vried prmetriclly to ssess their influence on the droplet sizes formed The effect of the thickness of the outer fluid will lso be ssessed prmetriclly Corresponding uthor, Grdute Reserch Assistnt Professor

2 Introduction A compound et, see Fig, is comprised of two liquid ets, one of which, the inner, is surrounded by the other, the outer This technique is used in the printing nd phrmceuticl industries to obtin fine ink et or medicine wrpped in cpsule The inner liquid is inected through nozzle into quiescent plenum contining the other liquid, sher forces t the periphery of the et form thin covering lyer moving eventully with the sme speed s the internl et There re mny publictions concerned with the stbility chrcteristics nd breking regimes of compound ets The current study ws motivted by the ILASS 6 presenttion by Bin nd Mshyek [], who developed numericl model to simulte the behvior of -D viscous compound et, nd hs been conducted s homework computtionl proect The purpose of the study ws to find dispersion reltion for n inviscid compound et, nd bsed on tht, to investigte how such prmeters s density, tension nd rdius of the outer liquid would ffect the stbility of the et, given sme prmeters of the inner liquid remin unchnged In generl, the et strts to generte droplets t the most unstble wve-numbers nd highest growth rtes Bsed on these numbers the students hd to mke n estimtion for the droplet sizes of the core et The predicted droplet sizes re similr to those plotted in [] for the sme influentil prmeters Another pper of interest is by Snz nd Meseguer [], who lso considered the stbility of n inviscid compound et They used continuity nd xil momentum s governing equtions, involving velocities of inner nd outer liquids in them At lineriztion these velocities re eliminted to get the sought 4-th ordeispersion reltion In our study we strt with n ssumption tht the coordinte system is moving with the bulk speed of compound et, nd use continuity, kinemtic nd dynmic boundry conditions s governing equtions The velocities re represented by velocity potentils After lineriztion -nd ordeispersion reltion is obtined by solving the Bessel s eqution r Vcuum pg, g, g, g Outer liquid p,,, Inner liquid p,,, U Figure Compound et scheme z Assumptions Flow is xisymmetric, incompressible, inviscid, irrottionl Liquid columns re infinitely long, immiscible No body forces present Coordinte system is moving with the bulk speed of both ets, U, see Fig Smll-mplitude wves considered, velocity nd liquid surfce perturbtions re smll Liquids re inected into the quiescent gs with the pressure equl to zero Equl surfce elevtion for both liquids Governing equtions Applying continuity to xisymmetric flow gives: u rv () z r r nd considering tht u nd v, z r continuity reltions for two liquid ets re: () z r r r The unstedy Bernoulli eqution provides one of the two boundry conditions for the interfces Along the linerized interfce rdil position r this condition my be expressed: ˆ p G t, (3) where we neglect body forces G nd higher order term (HOT), ie the dynmic pressure is higher order quntity when one considers coordinte system trveling with the bulk velocity of the et The locl pressure cn be represented s sum of the initil liquid pressure nd disturbnce s: p P pˆ Considering tht dpˆ d p P we cn rewrite eqs(3) for ech liquid et in the form of P p (4) t Consider surfce bout r Subtrcting of eqs(4) in this cse yields P P p p, (5) t t where the initil pressures cn be written s: P nd P P (6) The locl pressures t the interfce between two liquids re relted by p p Kin (7) Here the xisymmetric surfce curvture cn be expressed s sum of two terms,

3 , (8) K in R in in R which re: in R z nd z in 3 R z Neglecting s HOT term reltive to unity nd z expnding in Tylor s series round we rewrite (8) s K in (9) z Combining eqs(5) through (9) we obtin the linerized dynmic boundry condition for the interfce: () t t z Consider surfce bout r For this surfce eqs(4) cn be represented s: g P pnd g Pg pg, () t t where we ssume vcuum conditions,, P, p for this prticulr study Here the g g g g pressures re relted s: P P g nd pg p Kout () Following the derivtion of eq(9) we cn similrly obtin K out (3) z Combining eqs() through (3) yields the dynmic boundry condition for the free surfce (4) t z Kinemtic boundry conditions evluted t r lines cn be written: (5) r t z z Neglecting s higher order terms, we cn z z rewrite (5) for two rdil men levels s: t r r r t, (6) t r r t (7) Lineriztion of Governing Equtions We ssume tht both velocity potentils nd surfce elevtion cn be represented in the form of Fourier wves: F expt ikz, () exp t ikz, () where F F r is n unknown function nd is some initil surfce elevtion Then plugging eq() into () yields n expression for F r : F r F r r r k r F r, r r which is modified Bessel eqution We cn write solution for it for ech liquid in the following form: F r C I kr C K kr, (3) F r C I kr C K kr, (4) where C nd C re constnts to be defined At the limit r perturbtions should vnish, which mens tht C Consider liquid surfce bout r Using eq() we cn write F t ikz, (5) exp F t ikz, (6) exp where from eqs(3) nd (4) it follows tht F C I k, F C I k C K k Plugging eqs(5) nd (6) into kinemtic boundry condition (6) nd using eq() we obtin the first two reltions to determine the unknown constnts: C ki k, (7) C I k C K k (8) k Consider liquid surfce bout r Here we similrly hve F t ikz, (9) where exp F C I k C K k Plugging eq(9) into dynmic boundry condition (4) nd using eq() provides the third reltion C I k C K k k () Solving eqs(7), (8) nd () for the unknown constnts we obtin:

4 C C C kik C Kk k I k I k k k Ik Ik Kk Kk Ik By this, the functions (3) nd (4) nd hence the equtions for the velocity potentils t both liquid surfces (), re fully defined in terms of known vribles Dispersion reltion To get the dispersion reltion for this study we use the dynmic boundry condition t the interfce () Evluting the velocity potentils t r nd fter number of mthemticl mnipultions we cn obtin E E k D D k, () E A BB D where the functions of k, ABDE,,,, re defined s: I k A I k B, I k, I k I k D K k K k, I k I k E K k K k I k Prmetric study In this study the influence of the severl prmeters of the outer liquid on the stbility of the whole compound et is observed These prmeters include density, surfce tension nd the men rdius The inner liquid is ssumed to hve the following prmeters: kg, 3 74 N, 5mm m m The vrition of the density, see Fig, shows tht s the outer liquid gets denser, the growth rte decreses by bout %, while the rnge of disturbnce wve-numbers does not chnge This implies tht the compound et is more stble when the outer et is denser The reson could be tht oscilltion of molecules of the inner liquid close to the interfce is dmped by denser pcked molecules of the outer liquid, thereby diminishing overll disturbnces in the et Vrition of density (kg/m 3 ) ( = 75 mm, = 74 N/m ) Growth rte (/s) = 5 = = Wve number k (/m) Figure The effect of density The increse of the surfce tension of the outer liquid, see Fig3, results in the higher mximum growth rte with shorter rnge of unstble wvenumbers Tht is the increse in the surfce tension cuses the mplifiction of low-frequency disturbnces with dmping of high-frequencies In other words, the surfce tension forces re weker thn the inerti forces of the long wve-length liquid prticles In contrst, when the liquid oscilltes with higher frequency, the mplitude of motion of the prticles is smll, nd the surfce tension forces re ble to lessen this mplitude to lower vlues Growth rte (/s) Vrition of surfce tension (N/m) ( = 75 mm, = N/m 3 ) = 37 = 74 = 5 5 Wve number k (/m) Figure 3 The effect of surfce tension

5 Growth rte (/s) Vrition of outer rdius (mm) ( = 74 N/m, = N/m 3 ) = 55 = 75 = 5 5 Wve number k (/m) E E k D D k E A BB D Tht mens tht if the stbility curve would hve mximum, it would not depend on prticulr mx vlues of nd, but on rtios of densities nd tensions At the sme time, since both nd re present in Bessel s reltions ABDE,,,, then mx will depend on ech prticulr vlue of nd rdius rtio Noting this, we present the results for the droplet size predictions for the chosen inner rdius mm 5 Figure 4 The effect of outer rdius The vrition of the outer rdius, see Fig4, shows cler trend towrds enhnced stbility s the rdius of the outer liquid grows This cn be explined by the fct tht the hevier the outer liquid gets, the hrder it is for the inner liquid to disturb it ccording to conservtion of momentum We cn lso see tht the rnge of the unstble wve-numbers widens towrds the high-frequencies s the outer rdius decreses Which tells us, tht the et gets sensitive to even wek disturbnces with smll mplitudes Droplet size pproximte prediction For ech unstble wve-number on the stbility curve there exists corresponding droplet size, s droplet cn be potentilly formed s result of seprtion of liquid segment from the bulk of the liquid Here we consider only the droplets tht cn be generted t most unstble wve-numbers We ssume tht t the moment of seprtion the droplets cn be treted s liquid columns with the length equl to corresponding most unstble wve-length nd the rdius equl to After seprtion the droplets re ssumed to form sphericl bodies Thus, for the pproximte prediction of the droplet size we hve 4 3 mx, 3 where mx Then the corresponding droplet kmx rdius is s 3 5 rd kmx We cn rewrite eq() fctoring out nd / Droplet size by density vrition (Fixed prmeter: / =5) 3 4 / / =5 / = / =5 Figure 5 Density vrition with fixed rdius rtio / Droplet size by tension vrition (Fixed prmeter: / =5) / =5 / = / =4 5 5 / Figure 6 Tension vrition with fixed rdius rtio

6 Here we consider severl cses where one of the three prmeters,, hs been vried t different vlues of the second prmeter, while the third prmeter hs been held constnt, totlly six different combintions The plots of droplet sizes by vrying density nd surfce tension t fixed rdius rtio, see Figs5 nd 6, confirm the trends observed on Figs nd 3 where the increse in density nd surfce tension of outer liquid led to the lower vlues of the most unstble wve-numbers nd hence to biggeroplet sizes / Droplet size by outer rdius vrition (Fixed prmeter: / =) / =5 / = / = / Figure 7 Outer rdius vrition with fixed density rtio / Droplet size by tension vrition (Fixed prmeter: / =) / / = / =5 / = Figure 8 Surfce tension vrition with fixed density rtio The plots of vrying rdius nd tension rtio t fixed density rtio on Figs 7 nd 8 repet the stbility behvior on Fig 3 nd 4 Here we cn see gin tht the increse in surfce tension nd rdius of the outer et results in lower k mx, nd consequently in bigger droplet sizes The vrition of rdius nd density rtio t fixed tension rtio, see Figs9 nd results in biggeroplets s rdius nd tension rtio re incresed, s on Figs3 nd 4 / Droplet size by outer rdius vrition (Fixed prmeter: / =) / =5 / = / = / Figure 9 Outer rdius vrition with fixed surfce tension rtio / Droplet size by density vrition (Fixed prmeter: / =) 3 4 / / = / =5 / = Figure Density vrition with fixed surfce tension rtio Overll, we cn observe tht with the set of prmetric rtios considered here the droplet sizes lie in the rnge of 45 of the outer rdius of compound et We cn lso note from Figs 8 nd tht when solely surfce tension oensity rtio is vried with rdius rtio below 5, the size of the droplets chnges very little This cn be of benefit when both

7 et rdii cn be well controlled nd the liquids of the compound et re vried frequently, to produce droplets with stble sizes Conclusions This study is concerned with the stbility of the compound et with regrd to vrition of severl prmeters of the outer liquid while these of the inner liquid remin unchnged The prmeters vried were density, surfce tension nd the rdius The plots due to dispersion reltion hve shown some distinguishble trends which re the following With denser outer liquid the compound et is more stble, nd the stbility region does not chnge If the outer liquid hs higher surfce tension the lower wve-numbers re mplified while the higher ones re dmped, nd we cn observe tht the mximum growth rte is higher In cse of incresed outer rdius we see tht growth-rte, s well s the rnge of unstble wve-numbers, decrese The stbility chrcteristics obtined provide us with tool for n pproximte prediction of droplet sizes, s droplets re pinched off from the bulk of the liquid The plots of droplet sizes with continuous vrition of rdius, density nd surfce tension rtio show n greement with trends observed previously where these prmeters where chosen discretely Superscipts -inner nd in interfce between two liquids -outer liquid References Bin, X, nd Mshyek, F, Nineteenth ILASS Americs Annul Conference on Liquid Atomiztion nd Spry Systems, Toronto, Cnd, My 6 Snz, A, nd Meseguer, J, Journl of Fluid Mechnics 59:55-58(985) Nomenclture rdius of n undisturbed liquid surfce d droplet rdius I Bessel function of the st kind of order I Bessel function of the st kind of order K Bessel function of the nd kind of order K Bessel function of the nd kind of order K k p ˆp P R r d rz, uv, xisymmetric surfce curvture wve number locl pressure pressure perturbtion pressure before ny disturbnce rdius of curvture droplet rdius from the inner liquid rdil nd xil coordintes xil, rdil velocity perturbtions wve-length fluid density undisturbed surfce velocity perturbtion potentil Subscripts g mbient gs -inner nd -outer liquid in interfce between two liquids out outer surfce mx most destbilizing disturbnce