Break-up and no break-up in a family of models for the evolution of viscoelastic jets

Transcription

1 Z. angew. Math. Phy /3/84-8 c 3 Birkhäer Verlag, Bael Zeitchrift für angewandte Mathematik nd Phyik ZAMP Break-p and no break-p in a family of model for the evoltion of vicoelatic jet Marco A. Fontelo Abtract. The goal of thi paper i to tdy the condition nder which break-p of a thin vicoelatic jet i inhibited or, oppoitely, to provide familie of elf-imilar break-p oltion emerging in the evoltion. The flid i a polymeric oltion and the contittive relation we conider i of the Johnon-Segalman type which contain one parameter. The model nder conideration are the limit of the well know one-dimenional model when the vicoity of the olvent goe to zero or to infinity and when the Deborah nmber i very large. Some conideration on the tationary and travelling wave oltion of the model are alo preented. Mathematic Sbject Claification. 35Q3, 35K55, 35R35, 76M55. Keyword. Vicoelatic jet, One-dimenional model, break-p, econd kind imilarity, beadon-tring trctre.. Introdction The goal of thi paper i triple. Firt, we find condition nder which the oltion of a model for the evoltion of axiymmetric jet of Non-Newtonian flid polymeric oltion to be precie experiment break-p delay de to their vicoelatic natre. Second, we dic the exitence of elf-imilar break-p oltion for the one dimenional model of jet of polymeric oltion in the limit of very high and very low olvent vicoitie. Third, we dedce ome oltion of the model that exhibit pecial featre: tationary and travelling wave oltion. It i a inglar and well docmented fact cf. [] that the addition of polymer to a Newtonian flid in form of oltion i ometime able to delay ignificantly the break-p proce which occr in Newtonian flid de to Rayleigh intabilitie. Even mall trace of polymer can trn a break-p proce which happen in a fraction of econd for the Newtonian flid into a proce of econd and even minte for the oltion. Dring the evoltion, the jet develop a bead-on-tring trctre where a more or le randomly paced eqence of drop i connected by very thin filament. The trctre remain almot tationary for a long period of time of the order of the relaxation time of the polymer after which break-p

2 Vol Break-p and no break-p in a family of model 85 take place. Another intereting phenomenon reported in [] i the drop motion of the bead along the tring in the bead-on-tring configration and the evental coalecence of drop that take place. Stdying the aociated free-bondary problem coniting in olving Navier- Stoke eqation with the appropriate contittive relation and bondary condition in an evolving domain i an almot impoible tak from the analytical point of view. We will e intead the o-called one dimenional model which ha repeatedly proved to be accrate in the analyi of the evoltion of very thin jet cf. [6] and reference therein. The one-dimenional model for the evoltion of a flid jet i dedced baed on the idea that in a fficiently thin jet, the characteritic length of the pertrbation along the axi i mch larger than the characteritic diameter of the jet cro-ection. Under thi hypothei which i experimentally right p to the break-p time, the two component of the velocity field can be written a v z r, z, t = vz,t, v r r, z, t = r vz,t, z where vz,t atifie the eqation v t + v v z = σ κ ρ z + h h v 3ν z z + σ zz σ rr,. with hz,t being a fnction which give the ditance of any point of the bondary κ of the jet to the axi, z i the derivative of the mean crvatre of the bondary given by κ z = h zz,. z + h z 3 h + h z or, eqivalently, by κ z = h h zz h + h z + h z 3 h + h z h K,.3 z and σ zz,σ rr being fnction to be defined below which repreent the Non-Newtonian contribtion. The parameter σ, ρ and ν are the rface tenion coefficient, denity and kinematic vicoity of the flid repectively. Notice that. i, in fact, a one-dimenional Navier-Stoke eqation. Below we hall redce the nmber of parameter and introdce a itable family of adimenional parameter. The fnction hz,t atifie the following hyperbolic partial differential eqation: h t + v h z + h v =..4 z The ytem.,.4 ha a long hitory cf. [6], [9], bt it i only recently that it ha been ed to tdy jet break-p cf. [4], [5]. For Newtonian flid

3 86 M. A. Fontelo ZAMP σ rr = σ zz = the exitence of a elf-imilar break-p oltion ha been hown nmerically cf. [5]. If the flid i invicid, then the exitence of a biparametric family of elf-imilar oltion ha been hown nmerically in [6] and proved mathematically in [7] where ome other inglarity formation mechanim have been fond and decribed ee alo [8]. In a very vico flid, there i a niqe generic elf-imilar oltion ee [7]. At thi point, we hold dic what the term σ zz, σ rr in. aociated to the Non-Newtonian contribtion are and how do they originate. Non-Newtonian flid and, in particlar, polymeric oltion, are mathematically characterized by the fact that their contittive relation i not a imple algebraic and linear relation between tre and train bt, intead, a nonlinear firt order differential eqation. More preciely, the relation between tre and train in a Newtonian flid i S =νd, while in a Non-Newtonian flid of the Johnon-Segalman type cf. [3], [4] i S = S + σ, with S repreenting the tre tenor that one wold have if the flid contain only olvent and no polymer in oltion, i.e., S =ν D, and σ being the polymer contribtion to the tre tenor. The σ term i amed to atify the following contittive relation: σ i defined a σ + λ σ + a {Dσ + σd} =ν p D..5 σ σ t + v σ v T σ+σ v, with analogo definition for D. The parameter λ in.5 i called the relaxation time and can be of the order of econd and even minte, while ν p i the polymer contribtion to the kinematic vicoity phyically, the m ν + ν p wold be the vicoity of the oltion in the limit of zero train. The parameter a i a real adimenional parameter. In mot flid the o-called hear-thinning flid, cf. [] a < and we will take thi a hypothei. It i worth noticing that the limit a = correpond to the Oldroyd-B model which can be dedced from moleclar dynamic. We are going to conider a, a range which contain the Oldroyd-B model and the neighbor one which are hear-thinning. Oldroyd-B model ha been bject of extenive tdy in [3]. Under the one-dimenional approximation, the tenor σ i approximated by the tenor

4 Vol Break-p and no break-p in a family of model 87 σzz z,t, σ rr z,t while D i approximated by vz,t z vz,t. z Thi yield the following eqation for the component of σ : σzz d σzz +λ + a v σ rr dt σ rr z with d dt := t + v z σzz σ rr = ν p v z v z,.6. Eqation.6, together with eqation.,.4 contitte a ytem of for partial differential eqation. By introdcing R, ρr 3 /σ, σ/ ρr and σ/r a characteritic length, time, velocity and prere repectively and defining the nondimenional parameter µ = ν ρ, µ p = ν p ρ σr and D = λ/ ρr 3 /σ, ytem.,.4,.6 tranform into v t + v v z = κ z + h h v 3µ z z + σ zz σ rr,.7 h t + v h z + h v =, z.8 dσzz σ zz +D a v dt z σ zz dσrr σ rr +D + a v dt z σ rr σr =µ p v z,.9 = µ p v z.. The parameter D called Deborah nmber i the ratio between the relaxation time λ and a characteritic time ρr 3 /σ. It can be very large, ince the characteritic time i typically of order 3 in a jet of radi. mm. a the one conidered in [] for intance while λ can be 3 or 4 order of magnitde larger. The large vale of D will be ed to dedce an aymptotic limit of.7-. which will be the one analyzed in thi paper. The big qetion i wether ytem.7-. contain oltion ch that hz,t become zero at ome point z at finite time or not. A we remarked above, the problem i alo open for Newtonian jet althogh nmerical evidence indicate the exitence of elf-imilar oltion with thi kind of behavior cf. [5]. For Non-Newtonian jet the limit µ ha been conidered analytically in [4], [5] with κ in. replaced by h and nmerically in [3]. A far a we know, the limit µ ha never been conidered even thogh thi limit i a natral one when the olvent i water, for intance, with µ 3

5 88 M. A. Fontelo ZAMP in a jet of radi. mm.. Finally, the vale of µ p depend on the polymer contribtion to the vicoity at zero train. In the flid reported in [] it rn between and 3 cp. which wold yield µ p 3. The combination η µp D which will appear a an important one in the paper i of order. Following other athor who have tdied the break-p problem both for Newtonian and Non-Newtonian jet cf. [7], [4], we firt conider that the olvent in the oltion i very vico µ, o that a qaitationary Stoke eqation i conidered intead of the claical Navier-Stoke eqation. The exitence of a niqe generic elf-imilar break-p oltion in a Newtonian jet wa etablihed in [7]. A imple dominated balance argment how that ch a oltion cannot model the whole break-p proce and inertial effect hold be taken into accont in any cae for time fficiently cloe to the break-p time if break-p actally happen in the oltion of Navier-Stoke. Depite thi limitation, it trn ot that thi elf-imilar oltion repreent an intermediate aymptotic tate and mark a tendency to break-p. The experimental work in [] how that thi intermediate aymptotic actally happen and take place in a elf-imilar manner ee alo the reference that [6] make to thi article. Mathematically, the limit µ lead to the ytem.8-. together to the following one-dimenional qaitationary Stoke eqation: = κ z + h z h v 3µ z + σ zz σ rr.. We impoe that both vz,t and hz,t are periodic and derive an eqation correponding to the aymptotic limit D. We how that for a> no break-p i poible. For a there exit elf-imilar oltion analogo to the oltion fond in [7] developing finite time break-p. In the limit of oltion with very low olvent vicoity µ we arrive to the ytem.8-. together to the following Eler eqation: v t + v v z = κ z + h h σ zz σ rr.. z We how the exitence of elf-imilar oltion analogo to thoe fond in [7] when a and the no exitence of thoe oltion for a>. Finally, we will conider tationary and travelling oltion of.7-. in the aymptotic limit D and dic how they model bead-on-tring trctre and drop migration repectively. Thi paper i organized a follow: In Section we introdce the Lagrangian formlation of ytem.7-. which trn it into a ingle integrodifferential eqation. Thi formlation will be ed in the ret of the paper de to the fact that in the limit conidered µ and µ the ytem tranform into a parabolic and an aymptotically elliptic eqation. In Section 3 we dic the break-p delay phenomenon when µ. Break-p oltion in thi limit are fond in Section 4. Section 5 i devoted to the analyi of elf-imilar oltion when µ. Section 6 deal with the tationary oltion which exhibit bead-

6 Vol Break-p and no break-p in a family of model 89 on-tring trctre and the motion of drop along the tring. Finally, Section 7 preent the conclion.. Formlation in Lagrangian coordinate In thi Section we hall decribe a new repreentation of the one-dimenional ytem.7-. for the evoltion of the free bondary of a thin flid tbe, in term of a ingle Partial Differential Eqation. Thi formlation will be efl in the analyi performed in the ret of the paper. We define the Lagrangian evoltion of the poition of a point zζ,t by mean of the eqation: dzζ,t dt = vzζ,t,t,. zζ, = ζ,. where from now on d dt tand by the Lagrangian derivative i.e. for fixed ζ. Notice that for a given fnction fz,t there hold: df dt = f t + v f z,.3 where z, t are the partial derivative for fixed t and z repectively. Differentiating.,. with repect to ζ, we readily derive the eqation: d z = vzζ,t,t z dt ζ z ζ,.4 z ζ, ζ =..5 Uing.3 we can rewrite.4 a: z + v z = v z t ζ z ζ z ζ..6 We then define Gζ,t=h zζ,t,t z ζ ζ,t. Taking into accont.6 and.4 we dedce that: dg =. dt.7 Then: h zζ,t,t= h ζ,. z ζ ζ,t.8 Uing.,.3 into.7 and ing formla.3 it follow that: dv dt = h z ζ h K + h z ζ µ h z v ζ ζ ζ ζ + h σ zz σ rr..9

7 9 M. A. Fontelo ZAMP Let introdce a new patial variable: d = h ζ,dζ,. with the initial condition = at ζ =. Then, taking into accont.8, we can rewrite.9 a: dv dt = z K + µz v + z σ zz σ rr.. We now eliminate v from. ing. and obtain the eqation: z z t = K + µ z z t + z σ zz σ rr.. Notice that.8 and. imply that: z = h..3, t Th, the fnction olve the eqation: t = K + t µ + σ zz σ rr..4 A nice featre of eqation.4 i the fact that every term in it ha a phyical meaning. The term at the left-hand ide i the inertial term in Navier-Stoke. The firt term at the right hand ide i the contribtion of rface tenion. The econd member at the right hand ide i the econd order derivative of a term coming from the olvent vicoity pl a term aociated with the Non-Newtonian contribtion. Thi clear eparation of term will be very important when dicing the relative importance of different phyical effect dring the evoltion. The ytem.6 i, in Lagrangian coordinate, σzz d σzz +D + a t σzz t = µ σ rr dt σ rr σ p rr,.5 which relt in the following linear ordinary differential eqation for σ zz and σ rr : σzz σ zz +D a t t σ zz σrr σ rr +D + a t t σ rr Notice that.6,.7 can be rewritten a e t D e t D t a e t D σzz a e t D σzz t = µ p t,.6 = µ p t..7 = η t a+, = η t a,

8 Vol Break-p and no break-p in a family of model 9 where we have introdced the adimenional parameter η µ p D. If we ame that the initial tree are zero natral from the phyical point of view, then we can integrate the eqation to conclde σ zz = η a e t D σ rr = η a e t D t e τ D t e τ D Integration by part yield t a+ dτ = a ηa e t D t a dτ = a η a e t D σ zz = a η + a ηa a e t η D + a D a e t D σ rr = a η + a η a a e t η D + a D a e t D and we can then compte σ zz σ rr = a η a a t e τ D t t e τ D t a a e t D a dτ, t e τ D a t dτ. dτ, a e τ D a dτ, + η t a D e t D e τ a, t D a, τ a, t a dτ..8, τ In fact, we can write σ zz σ rr = k, t a, t k, t a, t, with k, t = η a e t η t D + ad e t D e τ D a dτ,.9, τ k, t = η a e t η t D + a D e t D e τ D a, τdτ.. Finally, we can expre, in Lagrangian coordinate, K = = ,

9 9 M. A. Fontelo ZAMP where we have ed.3. The ytem.7-. i then eqivalent to the eqation.4 with σzz σrr and K btitted by.8,.. The model correponding to µ relt from eliminating the inertial contribtion in.4 and i therefore repreented by the following partial differential eqation t µ + K + σ zz σ rr =,. which can be integrated twice in to give t µ + K + σ zz σ rr =λt,.3 where we have ed periodicity of, t, K and σ zz σ rr. The fact that the oltion mt be periodic in phyical pace implie that mt be ch that, td = z, td = z +Period z =Cont. Period Period i.e.,, td = t, td =. t Period Period Thi implie, from.3, [ λt K σ zz σ rr ] d =, Period which lead to the following formla for λt: Period λt= [K + σ zz σ rr ] d. Period d From now on we will ame, withot lo of generality, that the period i [, ] and that, tdt =. Notice that, tk, td = d where we have ed that = 3 d, = ince and are identical at

10 Vol Break-p and no break-p in a family of model 93 the border of the period. Therefore [ ] 3 + k + λt = 4 5, t a+ k, t a+ d..4 d The problem.3 read then µ t = λt k + 4 5, t a+ + k, t a+,.5 with λt given by.3. Eqation.5 i a nonlinear parabolic partial differential eqation and the qetion i wether it contain oltion that develop blow-p in finite time or not. The model for µ i impler and relt in the eqation t = K + k, t a k, t a..6 A we have remarked above, the tdy in thi cae will foc on the exitence of elf-imilar oltion. If we ame D o large that η<< Dη and time t of lower order of magnitde than D, then we can write with σ zz σ rr and W, t defined a + η a = k a k a +W, t,.7 k = a η a,k = a ηa,.8 W, t = a η a t D e t D e τ D e t D a a a a, t a, τ a, t a, τ dτ,.9 being O η D. In the following ection we are going to drop all the O η D term. There are two reaon for doing thi. The firt i that according to [] breakp happen at time of the order of magnitde of the relaxation time of the polymer, i.e., when the recaled time t i of order D which i jt the time at which W, t i comparable with the other term of the eqation. Second, the Johnon- Segalman contittive eqation and, in particlar, the Oldroyd model, have been criticized a being poor when polymer experiment large extenion, i.e. when

11 94 M. A. Fontelo ZAMP train are very large, and that happen cloe to break-p experimentally when t i OD. Hence, the range of validity of the model force to conider them only when W, t i mall and, in firt approximation, we can neglect it. Thi amont to aying that the fnction k, t and k, t defined in.9,. will be alway replaced by k and k defined in No break-p in the limit µ We conider the following Nonlinear parabolic eqation, which i a generalization of eqation conidered in [4], [5] and take into accont crvatre effect that are certainly important at ome moment of the evoltion: t where 3 = λt k a+ + k a+, 3. λt = k a+ k a+ d. 3. d The parabolic natre of eqation 3. garantee the exitence of oltion for initial data, in itable fnctional pace like C +α for intance at leat for hort enogh time. We will ame in what follow that, ha at leat two bonded derivative. We can prove the following theorem: Theorem. If a> then the oltion to 3. with initial data, = and λt given by 3. never blow p. Let introdce the following notation: We define m t a a boltion for, t ; i.e., m t, t with m = inf,, m t will be the vale of where the infimm of i reached, M t will be a peroltion for, t; i.e., M t, t with M = p,. The proof of theorem will be pported in the next lemma. Lemma. lower bond on The oltion to 3. with initial data, = and λt given by 3. atifie the following lower etimate: with ν =+ a m + a M., t m e νt, 3.3

12 Vol Break-p and no break-p in a family of model 95 Proof. We take m t to atify m,t = λt m 3 m k m t a+ m λt + k m t a+ m. From eqation 3. it follow immediately that m t i a boltion for, t. Notice that k a+ d p k a+ d which implie pk d d a d = pk, d m,t pk m 3 m k m t a+ m pk m 3 m pk a+ m. 3.4 A m t < thi i neceary ince, td =, we can obtain the following ineqality from 3.4: o that m,t pk + + pk m, m t m e t p k++p kdτ m e νt, 3.5 and thi prove ineqality 3.3. We pa now to the proof of the theorem. Eqation 3. can be written in the form t 5 = λt k a+ + k a+ +A 5 3, with A We define v, t =, t + k a, t k a, t. Or goal will be to how that v, t cannot blow-p at finite time. Thi implie that, t cannot blow-p at finite time notice that the term, t

13 96 M. A. Fontelo ZAMP k a, t i alway bonded by ineqality 3.3. We are going to dedce a differential ineqality for v, t at it maximm o that we can compare with a itable peroltion. Thi comparion will only take place in the interval of time for which the maximm of v i reached at a point where i large enogh. Otide thee interval both v and are bonded. Firt, it i imple to verify that Ax = + 4 x 5 x x +< if x,, 3 and hence A 5 3 < 3.7 when < 5. We can compte v = +a k 3 a +a+k a +k a k a. If v = then k = 3a+ + k, a+ +a k 3a +a+k and one can write p k 3a+ + p k p k + p k 3a+ + a+ +a k 3a +a+k a inf k 3a, provided that a+ < a inf k 3a. Let ame that { [ ] } a >max a inf k, ω. Then p k + p k 3a+ a inf k 3a K if v = and >ω, with K depending on p, inf, p, bt not on the contittive contant. A imple coneqence i that K which implie < 5 whenever >K 3. Therefore, at the maximm of v {, where} v =, we will alway have, provided that i larger than ω max ω, K 3, that ineqality 3.7 i atified. Let mltiply 3. by R, t +a k 3 a +a+k a. Notice that R, t i poitive if >ω and clearly alo if >ω. Then v t R, t 5 R, tλt v if >ω

14 Vol Break-p and no break-p in a family of model 97 If we define J, +k a k a then it i poible to compte o that v =J +k a k a, v =J +J + k a k a + k a a + k a + a, = v J k a + k a k a a + k a + a J = v J + k a k a + k a a + k a + a. J J If v = and >ω then we have hown above that K and J > a inf k a. It i imple to how that J a+3pk +pk a 3. We can conclde then v + a + 3pk + pk K a + p k a J p k a +K p k + a + p k a v + η K a, J J J if >ω and with K i depending on p i =,, and inf bt not on the contittive contant. Therefore, keeping in mind that R, t = J, t,, 5 3 v t v ηk a +R, tλt v J η a 5 +R, tλt v, 3.9 for >ω at the maximm of v where v =. By definition of v, t we have λt = v d d p v. 3. Let v M t be a oltion of dv M t = η a 5 t,t+rt,tλt vm t t,t dt with v M = p v, t and t the vale of at which the premm of v, t i reached at time t. t i a time for which v reache it maximm at a point for which,t >ω. Withot lo of generality, we can ame

15 98 M. A. Fontelo ZAMP t =. We will how that after thi time no blow p for can happen by proving that v M t, which i a peroltion for v, t by 3.9, doe not blow-p. Uing ineqality 3. we dedce dv M t η a 5 t,t. dt In the range of a that we conider a 5 <, o that we can e the etimate3.3 with the vale of ν given in the lemma and conclde dv M t dt η a 5 m e 5 aνt, which implie that v M t i bonded and, therefore, blow-p of, t cannot happen. Remark. The previo theorem doen t exclde in principle a different inglarity mechanim like hock formation finite time blow p for. In fact, one can e parabolic theory to how that thi i not poible while i bonded and the previo theorem garantee the exitence of global oltion. 4. Self-imilar oltion in the limit µ The qetion now i abot the exitence of oltion that develop finite time inglaritie, i.e., oltion ch that, t a and t t. Thee oltion, tranlated into the langage of hz,t and vz,t wold be ch that hz,t become zero at ome point in finite time and, therefore, repreent break-p oltion. When, t become very large in the neighborhood of ome point, one can neglect lower order term in 3. and approximate the eqation by the following one: 3µ t = λt 3 C a+, 4. with C being a contant that we can take, withot lo of generality, a one. Notice that we have alo neglected the firt and econd order derivative of with repect to. Thi i an implicit amption that near break-p the breakp profile are o flat that 5 << and 5 << 3. It will be imple to verify that all the elf-imilar oltion fond in thi ection do in fact verify the amption. 3 + a+ d and +ε ε d diverge when t t If both +ε ε bt the integral are bonded in the interval [, ε] and [ + ε, ], one can write λt +ε ε 3 + a+ d +ε + o ε d [ +ε ε ] d. We will eek for aymptotically elf-imilar oltion, i.e. oltion which can be

16 Vol Break-p and no break-p in a family of model 99 decribed in the form [, t = g t t α t t β ] +ot t δ t t α [ g ξ+ot t δ ], 4. for ome δ>. The contant α and β have to be choen appropriately and poitive in order to have blow-p and gξ ha to be ch that gξ ξ α β a ξ. The fnction, t i bonded otide and we can afely approximate λt 3 + a+ d [ ] + o d. d It i now evident from 4. that the exitence or not of break-p oltion and the break-p profile in the limit conidered will be the relt of a balance between the rface tenion 3 and the Non-Newtonian contribtion a+. We can alo eliminate the parameter 3µ by a itable recaling in time and arrive to the integrodifferential eqation 3 + a+ d t = 3 a d If a 4, ] then Non-Newtonian term are dominant with repect to rface tenion and the break-p oltion will be, at leading order, oltion of the following eqation: t = a+ d d a If a, 4] then rface tenion term are dominant with repect to Non- Newtonian term and the break-p oltion will be, at leading order, oltion of the following eqation t = 3 d d 3, 4.5 which, a we will ee, are imply the well known Papageorgio elf-imilar oltion. We can then tate and prove the following theorem: Theorem. i If a 4, ] then there exit oltion of 4.4 that blow-p in a elf-imilar and geninely Non-Newtonian way. The imilarity oltion i econd kind in the ene of Barenblatt. ii If a, 4] then there exit oltion of 4.5 that blow-p with the ame profile a a jet of Newtonian flid.

17 M. A. Fontelo ZAMP Proof. We are going to eek for ymmetric oltion of the elf-imilar type. A we wrote in 4., thee are oltion of the form, t = t t αg t t β t t αgξ, 4.6 with gξ ch that gξ ξ α β a ξ.ifha the form given in 4.6 then λt a+ d = ga+ ξdξ d t t αa. g ξdξ We fix the vale of α and maybe β in ch a way that all the high order term in 4.3 are of the ame order of magnitde in t t. More preciely, introdcing 4.6 into 4.4 one find the eqation t t α αg + βξg ξ =t t a+α ga+ ξdξ g g a+. g ξdξ Impoing α = a+α α = a then one eliminate the time dependent factor and obtain the following differential eqation for g : a g + βξg ξ = ga+ ξdξ g g a+, 4.7 g ξdξ which ha to be olved together with the condition gξ ξ aβ a ξ. Notice that the parameter β i, in principle, free. In fact, we will how that it ha to be choen in ch a way that the aymptotic behavior condition i atified. Thi characterize a econd kind imilarity cenario in the ene of Barenblatt, cf. []. Let Λ = ga+ ξdξ g ξdξ. Or eqation read then a g + βξg ξ =Λg g a We leave Λ a a free parameter and eek g a a ymmetric and reglar oltion to 4.8 o that a g + βξg ξ dξ = a g+βξg ξ dξ = 4.9 thi i eqivalent to impoing the vale of Λ given above. We have to chooe Λ and β and a fnction gξ o that 4.8 and 4.9 are atified. In fact, if we impoe reglarity of g at the origin we dedce a relation between Λ and β a the following comptation how: if we take g A+Bξ a ξ, we arrive to condition on A and B which can be redced to

18 Vol Break-p and no break-p in a family of model a =ΛA Aa, a +β = ΛA Aa a += a +Aa and then from which it follow Λ= a +Aa = A β a A= a A a a += a +Aa a, a a + β a a β a a a =, β a β a a a Let g = g/a o that g =. Then a g + βξg ξ =ΛAg A a g a+, from which it follow a g + βξg ξ = β a g β a a ga+ λg µg a+. We can compte a g + βξg ξ dξ =A a g+βξg ξ dξ =A. λg µg a+ dξ. 4. Impoing condition 4.9 i eqivalent to impoing that the right hand ide of 4. i zero. Bt notice now that o that d log ξ = g ξ =Ke βdg λg µg a+ a g, βdg λg µg a+ a g, and we can write g βdg βdg λg dξ =Ke µg a+ a g λg µg a+ a g. 4. Therefore from 4. and 4. we conclde that the elf-imilar coefficient β ha to be ch that the following integral eqation i atified: g e βd λ µ a+ a λg µg a λg µg a dg =. 4. a

19 M. A. Fontelo ZAMP a Figre. Self-imilar exponent β a a fnction of a We have olved eqation 4. nmerically and fond the vale of the elf-imilar exponent β a a fnction of a repreented in Figre. One can imply verify that the aymptotic condition gξ ξ aβ a ξ i atified. Let remark that more oltion of 4. do exit, bt the one repreented in the figre are the only which are generic the other reqire the initial condition to be very flat arond the origin. ii The cae a< 4 correpond to the Newtonian inglarity. In thi cae, the eqation 4. redce to β g + β 3 g β+ β g β + dg=. g The elf-imilar exponent β i obtained a root of the previo eqation. Thi root can be fond nmerically and yield β = The correponding final profile arond breakp in phyical pace can be ob-

20 Vol Break-p and no break-p in a family of model 3 tained from z = hz, t,t =, t d = aβ, aβ, t = 4aβ, 4aβ which lead to hz,t =A z with appropriate contant A. Thi yield a final profile A z in the limit a + 4 Newtonian and A z in the limit a. It can be hown that the exponent 4aβ decreae with a increaing. The Non-Newtonian effect in thi range of variation of a i to create a harper breakp profile than the Newtonian one. 5. Self-imilar oltion in the limit µ If µ = then cloe to a break-p point one can write from.6 t = K +Ca. 5. Again, we can take withot lo of generality C = and we can ame a priori that the break-p profile i o flat that K. Thi lead to the following eqation for large and a> 4 : t and the following if a 4 : t We can prove now the following Theorem: = a, 5. =. 5.3 Theorem 3. i If a, then the eqation 5. ha no oltion that blow-p in finite time. ii If a 4, ] then there exit oltion of 5. that blow-p in a elf-imilar and geninely Non-Newtonian way. The imilarity oltion form a biparametric family. iii If a, 4] then there exit oltion of 5.3 that blow-p with the ame profile a a jet of Newtonian flid. The imilarity oltion form a biparametric family. Proof. i We can write eqation 5. in the following eqivalent manner: { t = v, v t =a a. 5.4

21 4 M. A. Fontelo ZAMP The ytem 5.4 i of hyperbolic type and admit the correponding Riemann invariant which, in thi cae, are ee [4] for intance: w = v a ω a dω = v a a, a w = v a ω a dω = v a a. a The Riemann invariant are contant along the correponding characteritic. If both w and w are bonded from above and below at time t =, then w + w = a a a and w w =v will be bonded along the characteritic. Thi implie that both and v are bonded along characteritic and, therefore, cannot blow-p at finite time. ii In the cae a 4, ] the ytem 5.4 i of elliptic type. Analogoly to [7], [8], we introdce the hodograph tranform coniting in conidering, intead of, t and v, t the fnction t, v and, v. In thi way, the nonlinear ytem 5.4 i eqivalent to the linear ytem: { t = v, =a a 5.5 t v. Sytem 5.5 can be rewritten a the following ingle elliptic partial differential eqation for t, v and, v: t a a t vv =, 5.6 a a vv =. 5.7 In thi way, the form of, t at a given t = t may be reprodced from the level crve t, v =t and the vale of, v on it. In the ame way a it wa done in [7], [8], we can contrct large familie of oltion developing inglaritie in finite time. We are intereted here in blowing-p oltion, i.e. oltion for which become very large and grow to infinity in finite time. If we introdce the new variable w = a and v = a a y then 5.6, 5.7 read a w + a w a tw + t vv =, w w a w a w + vv =, and we can introdce polar coordinate r = w + y, θ = arctan y w to obtain t r + t r θ + t a r r tan θ t a r =, 5.8 θ r + r θ + tan θ a r r + a r =. 5.9 θ Let ame that the blow-p time i t = and we conider the evoltion at negative time the change of variable t t t leave the eqation invariant. w

22 Vol Break-p and no break-p in a family of model 5 The qetion now i abot the exitence of oltion ch that t when w eqivalently goe to infinity. We howed in [7] that the oltion contrcted by eparation of variable in , when tranlated from the hodograph plane w, y to the phyical plane, are nothing bt elf-imilar oltion. The oltion are in thi cae of the form where Y θ atifie Y θ+ a and a Xθ = δ r, θ = r δ Xθ, 5. tr, θ = r δ+ a Yθ, 5. tan θy θ+δ δ+ a Yθ=, 5. Y θ co a θ. In the interval π, π there are two linearly independent oltion to 5. and any oltion i linear combination of thee two: Y θ =C Y θ+c Y θ C Y θ+cy θ. The contant C may be aborbed in the definition of tr, θ and r, θ after recaling. The parameter C on the other hand generate different oltion Y θ for different vale. Thi parameter, together with δ define a biparametric family of oltion. Thee oltion, when tranlated to phyical pace, are of elf-imilar type a we howed in [7]. In fact, from 5., 5. one obtain H y w Xθ δ = δ+ Y a δ, δ+ θ t a o that = H Hξ ; that i, ing y = r in θ and formla y w 5. it follow δ δ+ t a w = y Hξ = t in θ δ+ a Y and that implie δ+ a θ δ+, t = t a a M a ξ Hξ t δ+ a Mξ, which i a elf-imilar oltion of 5. with a final blow-p profile of the form with Γ +, Γ being two poitive contant., = Γ ± δa, 5.3

23 6 M. A. Fontelo ZAMP Formla 5.3 force δ, a in order to have, inglar bt integrable. The correponding final profile arond breakp in phyical pace can be obtained from z =, d = ± δa +Γ ± δa +, hz,, = = δa,, Γ± which lead to hz,t =A ± z +δa with appropriate contant A ±. iii We refer the reader to [7] or to the point iii with a = The tability of filament and travelling wave oltion A we have een, when a>, the model tdied don t lead to breakp. We conjectre that the oltion evolve toward an tationary oltion of.4. In fact, thee tationary oltion have been decribed in [8] and contain ome of the main featre of the bead-on-tring trctre. For the ake of completene we preent thee oltion in the lagrangian formalim. Aming jt for implicity that k =k =η and integrating the tationary verion, i.e. withot time derivative, of.4 we arrive to the following differential eqation: d d = 5 λ η a+ + η a+ + 3 d d , d d 6. where λ i ome real poitive contant. Let introdce the recaling λ and λ 3. It can be eaily hown that thi recaling in the phyical plane i a imple dilatation of the form z,h λz,h. Eqation 6. i, in the new variable, d d = 5 ηλ 4a a+ + ηλ +a a The oltion we are going to conider correpond to bonded vale of bt with λ mall. Hence, we neglect, for a>, the term ηλ+a a+ in 6. and the relting eqation can be written in the following firt order atonomo ytem form: = v, 6.3 v = 5 where δ = ηλ 4a. δ a v v g, v,6.4

24 Vol Break-p and no break-p in a family of model 7 We analyze now the orbit in the v, phae plane correponding to the dynamical ytem 6.3, 6.4. If v = then δ a + =. For δ mall enogh and a > there will be two eqilibrim point, i i=, with <. Notice that δ a a δ. In fact, a v tend to zero and keeping only firt order term, one ha g, v 9 +δ a , and g, g i, i a i, where g <, g >. Thi implie that P, i a foc and P, an ntable node. The orbit arond the foc are periodic and rn in the anticlockwie ene. From 6.3 one obtain max d along orbit = min v, which i the period aociated to every oltion. In the phyical z,h plane thee oltion are tationary periodic oltion. A an example, we have taken a =, δ=.395, ==, v = =. The correponding orbit γ i periodic a hown in Figre crve Γ and Γ are given implicitly by g, v = and the trajectorie in the phae plane cro them vertically. In order to repreent thi oltion in the phyical pace, we add the differential eqation z = to ytem 6.3, 6.4, olve the third order dynamical ytem relting and then plot the parametric crve z, ±. The relt i given in Figre 3. In fact, we can move the vale of = o that the periodic orbit called γ in Figre toche the node,. Thi node i reached only a ±. The correponding oltion i a bmp a the one howed in Figre 3 with two filament extending to z ±. It can be hown that tationary oltion do in fact exit alo for the Newtonian model. What make Non-Newtonian jet a the one howed here table i the fact that in the filament we have, and d d + δ a = 3 +a δ a A >, o that we can approximate at the filament = + ũ and eqation.4 by 3µ ũ tt = ũ t +Aũ. 6.5 The diperion relation for 6.5 i σ +Ak + 3µ σk = which yield tability for all k. Thi i not the cae in Newtonian flid where δ = and A <.

25 8 M. A. Fontelo ZAMP,, V Figre. Phae-plane analyi of If µ = then eqation 6.5 i imply the wave eqation. Thi fact gget looking for travelling wave oltion of eqation.4. Again, aming for implicity k = k = η, and introdcing a oltion of the form, t = U ct Uξ, one obtain an eqation which, after recaling U λ U and λ 3, read d U dξ = U 5 c λ 3 U 3 + λµ cu + U ηλ 4a U a+ + ηλ +a U a U 5 Uξ +U 4 +3U Uξ. Again, aming λ<< one can neglect ome term which can in fact be kept, bt that make the analyi more complicated and obtain the eqation d U dξ = U 5 β U 3 + U δu a U 5 Uξ +U 4 +3U Uξ, with β = λ 3 c. In the particlar cae a = we can write β U 3 δu a+ =

26 Vol Break-p and no break-p in a family of model Figre 3. Repreentation of one tationary oltion exhibiting bead-on-tring trctre. β δ U 3 o that we can generate the ame oltion a the one repreented in figre 3 jt by chooing β δ =.395, that i, c = λ 3 δ.395λ 3 = η.395λ 3 η. The propagation velocity in thi cae wold be c ± η. The exitence of travelling wave moving in both direction i, in fact, an experimental fact that wa alo oberved in [] where the bead of the beadon-tring trctre move along the jet and ometime coalece with other bead. Here we have conidered only the propagation of a bmp or a periodic array of bmp. The coalecence of bmp i left for ftre work. It i worth noticing that thee tationary and travelling wave oltion are alo preent in the model for µ and µ. Thi pport the validity of thee model to decribe the evoltion of jet. 7. Conclion In thi paper we have conidered the evoltion of jet of polymeric oltion. We have dedced and tdied two model baed on the amption that the olvent in

27 M. A. Fontelo ZAMP the polymer oltion i very high µ or that it i very low µ. In both cae we have amed D. The firt limit wa dedced in a impler verion which neglect term of the crvatre by Renardy, while the econd ha never been conidered before. For the firt model we howed that break-p cannot happen at finite time for ome range of the contittive parameter a and in ome other range we have fond elf-imilar breaking-p oltion. In the econd model, we jt tdy elf-imilar oltion and find a imilar conclion: break-p or not break-p depending on the vale of a. The fact that thi parameter a i critical for the exitence of break-p and the dependence of the break-p profile on it cold be intereting in order to determine it vale for a particlar flid experimentally. The determination of contittive contant from experiment i a claical problem in rheology. We have alo worked ot the tationary oltion exhibiting bead-on-tring trctre by integrating a dynamical ytem of dimenion and fond travelling wave oltion modelling drop migration by identical techniqe. An intereting problem now i to decribe the coalecence of two of thee migrating drop. According to the experiment in [] thi phenomenon originate a very pecliar twin droplet trctre and the qetion i wether one-dimenional model are able to explain it or not. Acknowledgement The athor i gratefl for the pport from the Virginia Polytechnic Intitte and State Univerity where part of thi work wa done and ha been partially pported by the Spanih Minitry of Edcation throgh it potdoctoral program. Reference [] G. I. Barenblatt, Scaling, elf-imilarity and intermediate aymptotic. Cambridge Text in Applied Mathematic 4. Cambridge Univerity Pre, 996. [] R. B. Bird, R. C. Armtrong, O. Haager. Dynamic of polymeric liqid, Vol.. Wiley, New York, 977. [3] H. C. Chang, E. A. Demekhin and E. Kalaidin, Iterated tretching of vicoelatic jet, Phy. Flid , [4] C. M. Dafermo, Hyperbolic Conervation Law in Continm Phyic. Grndlehren der mathematichen Wienchaften Vol. 35, Springer-Verlag,. [5] J. Egger and T. F. Dpont, Drop formation in a one-dimenional approximation of the Navier-Stoke eqation, J. Flid Mech., 6 994, 5. [6] J. Egger, Nonlinear dynamic and breakp of free rface flow, Rev. Modern Phy., , [7] M. A. Fontelo and J. J. L. Velázqez, On ome breakp and inglarity formation mechanim for invicid liqid jet, SIAM J. Appl. Math., , [8] M. A. Fontelo and J. J. L. Velázqez, Fractal-like inglaritie for an invicid onedimenional model of flid jet, Ero Jnl. of Applied Mathematic,, 9-6.

28 Vol Break-p and no break-p in a family of model [9] F. J. García and A. Catellano, One-dimenional model for lender axiymmetric vico liqid jet, Phy. Flid, 6 994, 676. [] M. Goldin, J. Yerhalmi, R. Pfeffer and R. Shinnar, Breakp of a laminar capillary jet of a vicoelatic flid, J. Flid Mech., , [] S. L. Goren and M. Gottlieb, Srface-tenion-driven breakp of vicoelatic liqid thread, J. Flid Mech., 98, [] T.A. Kowalewki, On the eparation of droplet from a liqid jet, Flid Dyn. Re., 7 996,. [3] R. G. Laron, Intabilitie in vicoelatic flow, Rheologica Acta, 3 99, [4] M. Renardy, Some comment on the rface-tenion driven break-p or lack of it of vicoelatic jet, J. Non-Newtonian Flid Mech., 5 994, 97. [5] M. Renardy, A nmerical tdy of the aymptotic evoltion and breakp of Newtonian and vicoelatic jet, J. Non-Newtonian Flid Mech., , [6] L. Ting and J. B. Keller, Slender jet and thin heet with rface tenion, SIAM J. Appl. Math., 5 99, pp [7] D. T. Papageorgio, On the breakp of vico liqid thread, Phy. Flid, 7 995, [8] A. L. Yarin, Free liqid jet and film: hydrodynamic and rheology. Interaction of Mechanic and Mathematic Serie. Longman Scientific & Technical, Harlow,. Departamento de Matemática Aplicada Univeridad Rey Jan Carlo C/ Tlipán S/N 8933-Mótole, Madrid Spain Received: Febrary, To acce thi jornal online: