Viscoelastic Potential Flow Analysis of Capillary Instability

Transcription

1 VE-CapInstab/viscoelastic-Rev2-NNFM.tex Viscoelastic Potential Flow Analysis of Capillary Instability T. Funada a and D.D. oseph b a Department of Digital Engineering, Numazu College of Technology, Ooka 3600, Numazu, Shizuoka, apan b Department of Aerospace Engineering and Mechanics, University of Minnesota 0 Union St. SE, Minneapolis, Minnesota USA Revised November 2002 Draft printed anuary 2, 2003 Abstract Analysis of the linear theory of capillary instability of threads of Maxwell fluids of diameter D is carried out for the unapproximated normal mode solution and for a solution based on viscoelastic potential flow. The analysis here extends the analysis of viscous potential flow (Funada and oseph 2002) to viscoelastic fluids of Maxwell type. The analysis is framed in dimensionless variables with a velocity scale based on the natural collapse velocity V = fl=μ (surface tension/liquid viscosity). The collapse is controlled by two dimensionless parameters, a Reynolds number = V Dρ=μ = ρfld=μ 2 = (Oh) 2 where Oh is the Ohnesorge number, and a Deborah number Λ = V=D where is the relaxation time. The density ratio ρ a =ρ and μ a =μ are nearly zero and do not have a signicant effect on growth rates. The dispersion relation for viscoelastic potential flow is cubic in the growth rate ff and it can be solved explicitly and computed without restrictions on the Deborah number. On the other hand, the iterative procedure used to solve the dispersion relation for fully viscoelastic flow fails to converge at very high Deborah number. The growth rates in both theories increase with Deborah number at each xed Reynolds number, and all theories collapse to inviscid potential flow for any xed Deborah number as the Reynolds number tends to innity. Keywords: Instability, capillary, viscoelastic, viscous, inviscid, Oldroyd. Corresponding author. Tel , fax address: joseph@aem.umn.edu.

2 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 2 Introduction Capillary instability of a liquid cylinder of mean radius R leading to capillary collapse can be described as a neckdown due to surface tension fl in which fluid is ejected from the throat of the neck, leading to a smaller neck and greater neckdown capillary force as seen in the diagram in gure. The dynamical theory of instability of a long cylindrical column of liquid of radius R under the action of capillary force was given by Rayleigh (879) following earlier work by Plateau (873) who showed that a long cylinder of liquid is unstable to disturbances with wavelengths greater than 2ßR. Rayleigh showed that the effect of inertia is such that the wavelength corresponding to the mode of maximum instability is = 4:5 2R; exceeding very considerably the circumference of the cylinder. The idea that the wave length associated with fastest growing growth rate would become dominant and be observed in practice was rst put forward by Rayleigh (879). The analysis of Rayleigh is based on potential flow of an inviscid liquid neglecting the effect of the outside fluid. (Looking forward, we here note that it is possible and useful to do an analysis of this problem based on the potential flow of a viscoelastic fluid.) r = R+η R u Capillary Force γ/r Figure : Capillary instability. The force fl=r forces fluid from the throat, decreasing r leading to collapse. An attempt to account for viscous effects was made by Rayleigh (892) again neglecting the effect of the surrounding fluid. One of the effects considered is meant to account for the forward motion of an inviscid fluid with a resistance proportional to velocity. The effect of viscosity is treated in the special case in which the viscosity is so great that inertia may be neglected. He shows that the wavelength for maximum growth is very large, strictly innite. He says,... long threads do not tend to divide themselves into drops at mutual distances comparable to with the diameter of the cylinder, but rather to give way by attenuation at few and distant places.

3 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 3 Weber (93) extended Rayleigh s theory by considering an effect of viscosity and that of surrounding air on the stability of a columnar jet. He showed that viscosity does not alter the value of the cut-off wavenumber predicted by the inviscid theory and that the influence of the ambient air is not signicant if the forward speed of the jet is small. Indeed the effects of the ambient fluid, which can be liquid or gas, might be signicant in various circumstances. The problem, yet to be considered for liquid jets, is the superposition of Kelvin- Helmholtz and capillary instability. Tomotika (935) considered the stability to axisymmetric disturbances of a long cylindrical column of viscous liquid in another viscous fluid under the supposition that the fluids are not driven to move relative to one another. He derived the dispersion relation for the fully viscous case (his (33)); he solved it only under the assumption that the time derivative in the equation of motion can be neglected but the time derivative in the kinematic condition is taken into account (his (34)). These approximations lead herein to the asymptotic solution in the limit of! 0. The effect of viscosity on the stability of a liquid cylinder when the surrounding fluid is neglected and on a hollow (dynamically passive) cylinder in a viscous liquid was treated briefly by Chandrasekhar (96). The parameter flrρ=μ 2 which can be identied as a Reynolds number based on a velocity fl=μ and the viscosity of the liquid μ, appears in the dispersion relation derived there.. Eggers (997) has given a comprehensive review of nonlinear dynamics and breakup of free surface flows of Newtonian fluids. All of the references just mentioned are discussed in the Eggers review. Tomotika s problem was studied by Lee and Flumerfelt (98) without making the approximations used by Tomotika, focusing on the elucidation of various limiting cases dened in terms of three dimensionless parameter, a density ratio, a viscosity ratio and the Ohnesorge number Oh p = ρfld=μ = =2. It is perhaps necessary to call attention to the fact it is neither necessary or desirable to put the viscosities to zero when considering potential flows. The Navier-Stokes equations, and the equations for many models of a viscoelastic fluid, are satised by potential flow; the viscous term is identically zero when the vorticity is zero but the viscous stresses are not zero (oseph and Liao 994). It is not possible to satisfy the no-slip condition at a solid boundary or the continuity of the tangential component of velocity and shear stress at a fluid-fluid boundary when the velocity is given by a potential. The viscous stresses enter into the viscous potential flow analysis of free surface problems through the normal stress balance at the interface. Viscous potential flow analysis gives good approximations to fully viscous flows in cases where the shears from the gas flow are negligible; the Rayleigh-Plesset bubble is a potential flow which satises the Navier-Stokes equations and all

4 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 4 the interface conditions. oseph, Belanger and Beavers (999) constructed a viscous potential flow analysis of the Rayleigh-Taylor instability which can scarcely be distinguished from the exact fully viscous analysis. Similar agreements were demonstrated for viscoelastic fluids by oseph, Beavers & Funada (2002). In a recent paper, Funada and oseph (200) analyzed Kelvin-Helmholtz instability of a plane gas-liquid layer using viscous potential flow. Funada and oseph (2002) did a viscous potential flow analysis of capillary instability. Results of linearized analysis based on potential flow of a viscous and inviscid fluid are compared with the unapproximated normal mode analysis of the linearized Navier-Stokes equations. The growth rates for the inviscid fluid are largest, the growth rates of the fully viscous problem are smallest and those of viscous potential flow are between. They found that the results from all three theories converge when is large with reasonable agreement between viscous potential and fully viscous flow with > O(0). The convergence results apply to two liquids as well as to liquid and gas. In this paper we shall extend the results of Funada and oseph (2002) to the case of viscoelastic liquid laments of Maxwell type. oseph, Beavers and Funada (2002) have analyzed Rayleigh-Taylor instability of an Oldroyd fluid using viscoelastic potential flow. They show that the most unstable wave is a sensitive function of the retardation time 2 which ts experiments when 2 = = O(0 3 ). The growth rates for the most unstable wave are much larger than for the comparable viscous drop, which agrees with the surprising fact that the breakup times for viscoelastic drops are shorter. They also do an analysis of Rayleigh-Taylor instability based on viscoelastic potential flow which gives rise to nearly the same dispersion relation as the unapproximated analysis. The linear stability analysis of the capillary instability of a viscoelastic fluid has been done by Middleman (965) and Goldin, Yerushalmi, Pfeffer and Shinnar (969). They showed that the growth rates are larger for the viscoelastic fluid. The analysis done here of viscoelastic potential flow has not been done before. Chang, Demekhin and Kalaidin (999) did a long wave study of the stretching dynamics of bead-string laments for FENE and Oldroyd-B fluids which extends the results of Bouseld, Keunings, Marrucci and Denn (986). They also do a long wave study of linear stability of the cylinder studied here in the small limit. Strictly speaking, the long wave equations perturb solutions with innitely long wavelengths. The wavelength for the maximum growth rates computed here range from 20ßD to a little less than 2ßD. Bouseld, Keunings, Marrucci and Denn (986) did a nonlinear analysis of capillary instability which captures observed features of the breakup in experiments cited by them better than the linear theory.

5 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 5 A referee of this paper noted that the observed increase of stability of jets with increasing elasticity, which contradicts the results of the linear stability analysis given here and elsewhere, may possibly be explained by linear stability analysis of a stressed lament at rest (see Entov 978 for a simplied analysis). One difculty is that a stressed lament at rest is not a permanent solution. The analysis which is developed in the sections to follow is framed in a general way suitable for an Oldroyd-B lament in another viscous fluid which is designated with a subscript a, for air. The linear theory perturbs a uniform cylinder with capillary forces. It is well known that all small perturbations of motionless states are governed by the constitutive equation of linear viscoelastic fluids (oseph 990, h i + = μ rv + (rv) T h i where is the viscous stress tensor, μ 0 is the viscosity, and rv + (rv) T is the rate of strain tensor. 2 Here we have assumed an Oldroyd model so that the parameters of the general linear viscoelastic fluid are expressed in terms of the model parameters. 2 Linear Stability for Small Disturbances of Fully Viscoelastic Flow In an undisturbed rest state, the long column of liquid (viscoelastic liquid of density ρ and viscosity μ) with mean radius R is put in 0» r < R and the fluid a (air or gas of density ρ a and viscosity μ a )isin R < r <, using cylindrical coordinates (r; ; z). The normal stress balance is given by μp μp a = fl R ; (2) where μp is the liquid pressure, μp a is the gas pressure, and fl denotes the interfacial tension. The stability problem of this state to small axisymmetric disturbances is solved herein. In terms of the diameter of column D, typical time T, typical velocity U = D=T and typical pressure p 0 = fl=d = ρu 2 for which U = p fl=(ρd), we have the normalization: r = D~r; z = D~z; t = T ~t; p = p 0 ~p; ψ = UD 2 ~ ψ; u = U ~u; w = U ~w; = D ~ ; 9 >= >; (3) where ψ is the stream function for axisymmetric flow: u ; w : (4)

6 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 6 The parameters are given by Λ ` = ρ a ρ ; m = μ a μ 0 ; = V Dρ μ 0 = ρfld ; μ 2 0 = V D ; ^ = Λ p ; Λ 2 = 2V D ; ^ 2 = Λ 2 p ; 9 >= >; (5) where V = fl=μ 0 denotes the capillary collapse velocity, and Λ is the Deborah number. It is noted that T may also be dened as T = D=V which is suitable for small, butt = D=U = (D=V )p is suitable for large. Hereinafter, dimensionless quantities are used without a tilde. For small disturbances, the equations for the liquid are given + u r = 0; p ^μ r 2 u u ; r = + where the constitutive equation is expressed symbolically as ^μ p r 2 w; (7) and the equations for fluid a are given by ^μ = + + ^ @u + u a r = 0; (9) `@u + p m r 2 u a u a ; `@w a r + m p r 2 w a : (0) with the Laplacian dened as r : () The kinematic condition at the interface r = R + ß R (R = ) is given for each fluid = u a; (2) and the normal stress balance at r ß R is given by p p a + 2 p a = + : (3) 2 R 2

7 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 7 The velocity components are continuous across the interface, for which u = u a ; w = w a : (4) The tangential stress balance at the interface is given by = @z a : The stability problem for the disturbances is given by the equations (6)-(0) and the boundary conditions (2) (5). The equation of motion gives the pressure p and the equation of vorticity is expressed in terms of = 2 w; D Dψ = 0; with the operator D dened as Dψ 2 2 r The solutions for small disturbances may take the 2 : (7) 2 ψ = [A ri (kr) + A 2 ri (k v r)] exp(fft + kz) + c:c:; (8) ψ a = [B rk (kr) + B 2 rk (k a r)] exp(fft + kz) + c:c:; (9) = H exp(fft + kz) + c:c:; (20) where the modied Bessel functions of the rst order are denoted by I for the rst kind and K for the second kind; ff is the complex growth rate and k is the wavenumber. Substitution of (8)-(20) into (2) (5) leads to the solvability condition, which is given as the dispersion relation of ff: I (kr) I (k v R) K (kr) K (k a R) ki 0 (kr) k v I 0 (k v R) kk 0 (kr) k a K 0 (k a R) 2^μk 2 I (kr) ^μ k 2 + kv 2 I (k v R) 2mk 2 K (kr) m k 2 + ka 2 K (k a R) F F 2 F 3 F 4 = 0; (2) where F -F 4 are dened as F = ffi 0 (kr) + 2 ^μk2 p I 0 (kr) R 2 k2 k ff I (kr); (22)

8 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 8 F 2 k2 = 2 ^μkk v p I 0 (k vr) R k 2 ff I (k v R); (23) F 3 = ff`k 0 (kr) + 2 mk2 p K 0 (kr); (24) F 4 = 2 mkk a p K 0 (k ar); (25) with k v and k a given by k v = s k 2 + p ^μ ff; k a = s k 2 + `p m ff; (26) and the prime denotes the derivative: I 0 (kr) = di (kr). We note again that ^μ = + ^ d(kr) 2 ff = + ^ ff. It is noted that (2) with ^μ = (for ^ = ^ 2 or even for steady case ff = 0) corresponds to (normalized) equation (33) in the paper of Tomotika (935), while (2) reduces to (normalized) (34) if (2) is manipulated by assuming both p =^μ and `p=m are small. Comparing with a Newtonian fluid of same viscosity (^μ = ), we nd readily that ^μ = + ^ 2 ff = + ^ ff < if 0» ^ 2 < ^ and ff > 0, but^μ > if 0» ^ 2 < ^ and ff < 0 within + ^ ff > 0. It may be regarded that the viscoelastic fluid of ^μ < has viscosity less than the Newtonian. Even for the Oldroyd model, instability arises in 0 < kr < ; the critical wavenumber is given by k c = R = 2. Then we have ff = 0 at the critical and at k = 0. Therefore the fluid in consideration behaves as a Newtonian fluid of same viscosity near the stability boundaries, while as a fluid of less viscosity inside those, which we concern here of the onset of instability. For the single column given by ` = 0 and m = 0, (2) reduces to 2^μk 2 I (kr) ^μ k 2 + kv 2 I (k v R) = 0; (27) F F 2 which involves, Λ and Λ 2. This may approximate (2) well for viscoelastic fluids with small values of ` and m as shown in table. Equation (27), as well as (2), is to be solved numerically by an implicit iteration method. For the viscoelastic potential flow, (2) reduces to I (kr) K (kr) F F 3 = 0; (28)

9 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 9 which is arranged as with k2 ff (ff + `ff a ) + 2p k2 k (^μ + m a ) R 2 ff ff = I 0(kR) I ; ff a = K 0(kR) (kr) K ; (kr) Arranging then (29) using ^μ = + ^ 2 ff = = 0; (29) = I 0 (kr) I ; (kr) + ^ ff, we have the cubic equation of ff (the dispersion a = K0 (kr) K (kr) : (30) relation of the viscoelastic potential flow): Aff 2 + ^ ff + 2p k2 (B + B ff) ff + C + ^ ff = 0; (3) with k2 A = (ff + `ff a ) ; B = + m a ; B = ^ 2 + m^ a ; C = R k; (32) 2 it is easy to have explicit expressions of the solutions of (3). When ^μ = for which ^ viscous potential flow) to give the solutions ff: ( ff = k2 + m a ) p ± (ff + `ffa ) = ^ 2, (3) reduces to the quadratic equation (the dispersion relation of the s» 2 k2 ( + m a ) p + (ff + `ffa ) k2 R 2 k (ff + `ff a ) : (33) When the viscosity is dominant, ff for instability (with the upper sign in (33)) may be expressed as k2 p ff = 2 R 2 k ( + m ; (34) a ) whereas when the viscosity is negligible, ff may be expressed as s k2 ff = R k 2 (ff + `ff : (35) a ) The viscous potential flow analysis includes the inviscid potential flow (IPF) (Funada & oseph 2002). 3 Asymptotic Forms in Large, Inviscid Potential Flow Taking the limit! in the system of equations (6) (5), we have the equations for inviscid fluids, which allows, of course, solutions for inviscid potential flow: s k2 ff = ± R k 2 (ff + `ff : (36) a )

10 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 0 For the single column given by ` = 0, (36) has the following asymptotic form: s k2 r k ff = ± R 2 ff ß ± k 2 = ±k: (37) 2R Another limiting case needs to be added as inviscid potential flow. If ^ is huge but ^ 2 is kept xed, ^μ goes to zero. In addition, if m= p is sufciently small, (29) reduces to give the solutions (36). 4 Asymptotic Forms in Small Assuming ff p is sufciently small, manipulating the columns in (2) as was made by Tomotika (935), we have the dispersion relation for small p ff : I (kr) kri 0 (kr) K (kr) krk 0 (kr) I 0 (kr) I 0 (kr) + kri (kr) K 0 (kr) K 0 (kr) + krk (kr) ^μi (kr) ^μkri 0 (kr) mk (kr) mkrk 0 (kr) G G 2 G 3 G 4 = 0; (38) where G -G 4 are dened as G = ^μi 0 (kr) p 2 I (kr) (kr) 2ffR kr ; (39) p G 2 = ^μ I 0 (kr) + kri00 (kr) Λ I 0(kR) 2 (kr) 2ffR I 0 (kr); (40) G 3 = mk 0 (kr); (4) G 4 = m K 0 (kr) + krk00 (kr) + K 0(kR)Λ : (42) In this limit! 0, the inertia terms can be neglected, by which ` is not included in (38)-(42). We may recall that when ^μ = (viscous fluid), (38) reduces to Tomotika s (normalized) (34) which takes the solution of the form ff = (a function of k and m) p. Now that ^μ = + ^ 2 ff = + ^ ff, (38) is the quadratic equation of ff, whose expression is long, thus is omitted here. For the single column given by m = 0, (38) reduces to ^μi (kr) ^μkri 0 (kr) = 0: (43) G G 2

11 VE-CapInstab/viscoelastic-Rev2-NNFM.tex which is arranged as the quadratic equation of ff: Solving this, we have p ff ^μ = 2R ff = f ^ (kr) 2 f (k): (44) (kr) 2 + (kr) 2 2 ff 2^ 2 ± v u ψ t f ^ 2^ 2! 2 + f^ 2 ; (45) when ^ 2 6= 0. The solution with the upper sign becomes ff = f (k) if ^ 2 = ^ (viscous fluid). For k small, f (k) ß p =3 (kr) 2, which corresponds to Rayleigh s case (892). When ^ 2 = 0 in (44), we have ff = f (k)= ^ f (k), which is larger than f (k) if ^ f (k) > 0; this growth rate has the larger value than that of the inviscid potential flow. For the viscoelastic potential flow, (38) reduces to I (kr) K (kr) = 0; (46) G G 3 which is arranged as the quadratic equation of ff: 2p k2 (B + B ff) ff + C + ^ ff = 0: (47) For the single column given by m = 0 and ^ 2 = 0, (47) reduces to linear equation to give the solution ff:» 2k2 ff = p C + ^ : (48) When ^ p = 0, wehaveff = at k = 0 (the maximum growth rate in the limit! 0 for the viscous potential flow; Funada & oseph 2002). 5 Results The data on viscoelastic fluids is provided for the diameter of the column D = cm in table. The peak values and the associated wavenumber are listed in table 2, from the growth rate curves in gure 2 for PAA and gure 3 for PO. The growth rate and the wavenumber are shown in gures 4-7 for various values of Λ. The comparison of VPF and FVF are shown in gures 8-9. The growth rates in both theories increase with Deborah number at each xed Reynolds number. The growth rate and the wavenumber are shown in gures 0- for changing p, and all theories collapse to inviscid potential flow for any xed Deborah number as the Reynolds number tends to innity.

12 VE-CapInstab/viscoelastic-Rev2-NNFM.tex σ VPF VPF (Newtonian) FVF FVF (Newtonian) k Figure 2: The growth rate ff versus k in viscoelastic potential flow (VPF) and fully viscoelastic flow (FVF). PAA as Newtonian and viscoelastic.

13 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 3 2% PAA 2% PO ρ (g cm 3 ) μ 0 (P) ρ a (g cm 3 ) : : μ a (P) :8 0 4 :8 0 4 fl (dyn cm ) (sec) (sec) V (cm sec ) U (cm sec ) (T = D=U) 4: : p ` : : m : : Λ Λ Table : Viscoelastic fluid properties and the parameters. (After oseph, Belanger & Beavers 999)

14 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 4 PAA k m ff m () VPF E E-02 (2) VPF (Newtonian) E E-02 (3) FVF E E-02 (4) FVF (Newtonian) E E-02 (5) IPF E E-0 PO k m ff m () VPF E E-02 (2) VPF (Newtonian) E E-02 (3) FVF E E-03 (4) FVF (Newtonian) E E-03 (5) IPF E E-0 Table 2: Peak values. VPF: viscoelastic potential flow, FVF: fully viscoelastic flow, and IPF: inviscid potential flow. Newtonian means the case for which Λ = 0.

15 VE-CapInstab/viscoelastic-Rev2-NNFM.tex VPF VPF (Newtonian) 0.05 σ FVF FVF (Newtonian) k Figure 3: The growth rate ff versus k in viscoelasic potential flow (VPF) and fully viscoelasic flow (FVF). PO as Newtonian and viscoelastic.

16 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 6 VPF VPF-03 σ VPF k Figure 4: PAA: VPF Λ = 0 6 ο 0 4. Λ is given as Λ = 0 (n 3)=5 for n = ; 2; ; 5, and n is used to label the curves.

17 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 7 VPF VP-03 σ 0.0 VPF k Figure 5: PO: VPF Λ = 0 6 ο 0 4. Λ is given as Λ = 0 (n 3)=5 for n = ; 2; ; 5, and n is used to label the curves.

18 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 8 FVF-05 FV σ FVF FVF k Figure 6: PAA: FVF Λ = 0 6 ο 0 4 Λ is given as Λ = 0 (n 3)=5 for n = ; 2; ; 5, and n is used to label the curves. (curves of No.3-37 are not shown.)

19 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 9 FVF-05 FVF σ 0.0 FVF-030 FVF k Figure 7: PO: FVF Λ = 0 6 ο 0 4. Λ is given as Λ = 0 (n 3)=5 for n = ; 2; ; 5, and n is used to label the curves. (curves of No.3-37 are not shown.)

20 VE-CapInstab/viscoelastic-Rev2-NNFM.tex VPF Λ = σ VPF Λ = FVF Λ = 5 0. FVF Λ = k Figure 8: PAA: Λ =, VPF Λ = 5 σ FVF Λ = VPF Λ = FVF Λ = k 0 Figure 9: PO: Λ =, 5.

21 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 2 0 IPF 6 FVF Λ = 0 σ FVF Λ = 0 0. FVF Λ = IPF FVF Λ = 0 6 k FVF Λ = FVF Λ = Figure 0: The ff m and k m versus p for FVF for Λ =.0, 0, 0 6. The maximum growth rate of IPF is given by ff m = at k m =.3957, which is drawn for reference in the gure.

22 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 22 0 IPF FVF Λ = 0 6 σ FVF Λ = FVF Λ = IPF 6 VPF Λ 0 k VPF Λ = 0 VPF Λ = Figure : The ff m and k m versus p for VPF for Λ =.0, 0, 0 6. The maximum growth rate of IPF is given by ff m = at k m =.3957, which is drawn for reference in the gure.

23 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 23 6 Conclusions and Discussion The following conclusions arise from our study of the linearized theory of capillary instability of a viscoelastic lament of Maxwell type in air. For many viscoelastic fluids the nal stage of collapse is controlled by nonlinear effects not considered here. ffl Capillary collapse is controlled by two parameters, a Reynolds number = fldρ=μ 2 and a Deborah number Λ = fl =μd. The density ratio ` = ρ a =ρ and viscosity ratio m = μ a =μ are small and have only a small effect. ffl The dispersion relation for viscoelastic potential flow can be solved for ff and computed for any value of Λ. ffl The dispersion relation for fully viscoelastic flow (where the shear stress and tangential velocity conditions are enforced) must be solved by iteration which breaks down at high Λ. ffl Following earlier studies, we found that a disturbance will grow more rapidly on a viscoelastic lament than on a Newtonian liquid; the growth rates increase with Λ ataxed. ffl The growth rates for viscoelastic potential flow are several times larger than for fully viscoelastic flow for each xed when is not large. The dispersion relation ff vs. k and the peak value k m and ff m = ff(k m ) are not greatly different. ffl As in our study (Funada and oseph 2002) of capillary instability of viscous threads, all theories collapse to the inviscid theory in the limit of large even when Λ is huge. Inertia beats viscoelasticity when!. The ratio Λ = = μ=ρd 2 = E is the elasticity number. Figures 0 and show that inertia beats elasticity even when E is rather large. ffl The effect of increasing the retardation time Λ 2 (0» Λ 2» Λ ) is to drive growth to Newtonian values. Nonlinear studies of capillary instability are frequently framed in terms of extensional flow. Such studies are particularly popular in the rheology community. Purely extensional flow can be obtained from a potential; it is a potential flow. Potential flows must be irrotational, but strong effects of viscosity and viscoelasticity can be represented. Evaluation of the utility of potential flow solutions requires comparison with exact

24 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 24 results. Here and in the other studies of stability reviewed in the introduction we nd very accurate results from viscous and viscoelastic potential flow when the appropriate Reynolds number is not too large. Even at low, say those for gures 2 and 3 the discrepancy in the peak values is not greater than about two and the wave numbers for maximum growth are not greatly different. In general, the peak values are much closer even for Reynolds numbers in which inviscid potential flow is way off the mark. In the present case, and in many other cases, the potential flow solution can be done analytically, here even in cases in which the numerical simulations giving the exact results fail. The main advantage of the viscous and viscoelastic potential studies is the determination of the conditions under which the flows are very nearly potential flows. This means that vorticity, not viscosity is negligible. For example, we could think of ways to connect the vorticity boundary layers on a flat plate to the potential flow of say glycerin in the free stream. We simplify the analysis using potential flow. The observed dynamics of collapsing Newtonian and viscoelastic laments is determined by nonlinear effects not considered in our linear analysis. Focusing now on the dynamics of thread rupture, in the Newtonian case, we consider the possibility that the main events prior to rupture are governed by viscous potential flow. The nal event is universally, described as a pinch-off and the fundamental physics governing the rupture of the thread is not considered. A pinch-off is a squeezing flow; the radius of the jet at the pinch point collapses, squeezing fluid out as the lament collapses. Here one nds a stagnation point; stagnation point flow is a potential flow and the effects of viscosity in such a flow may be huge, even more so in the viscoelastic case where extensional flow = stagnation point flow is so important. Certainly, potential flow of an inviscid fluid is not the right tool here. We can get results in which viscosity acts strongly using viscous potential flow. The question is not whether viscosity is important, which it is, but whether vorticity is important. Chen, Notz and Basaran (2002) have studied pinch-off and scaling during drop formation using highaccuracy computation and ultra-fast high-resolution imaging. They discuss dynamic transition from potential flow with a 2/3 scaling due to Keller and Miksis (983) to an inertial-viscous regime described by Egger s (993) universal solution. They nd overturn before breakup in experiments in water ( cp) well before the dynamic transition from the potential flow to the inertial viscous regime. On the other hand, an 85 cp glycerol-water solution is said to exhibit this transition. The potential flow solutions discussed by Chen et al (2002) are for inviscid solutions. Of course, water and glycerol are not inviscid. The scaling of Keller and Miksis (983) which gives rise to the 2/3 power collapse law does not work for viscous potential flow.

25 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 25 The spoiler is their equation (3.3) expressing the normal stress balance. To this equation we must add the viscous component n The term (rf) 2 in (3.3) scales like f 2 =L 2 whereas the viscous component scales like f=l 2, so that the similarity transformation does not factor through. Analogies have been put forward between capillary pinch-off of a viscous fluid thread and Van der Waals driven ruptures of a free thin viscous sheet (Vaynblat, Lister and Witelski 200). The observation that a lament under capillary collapse ruptures in a pinch-off does not come to grips with the physics which leads to a loss of the continuum. One idea is that thread breaks under the action of disjoining pressures. Unfortunately a mathematical theory for disjoining pressures for thin threads is not available. Another idea is that the high extensional stresses which are generated in the pinch region at rupture are the cause of the rupture, the rupture may be framed as a stress induced cavitation following ideas introduced by oseph (995, 998). Cavitation will occur in pure extension when the extensional stress is large enough, at high rates of extension. The idea that threads will break under tension at high rates of extension is particularly interesting for viscoelastic fluids which can support large extensional stresses. Small bubbles of vapor and gas could be expected to appear in the solvent of polymeric liquids or polymer melts when the thread enters into tension. The breaking of polymer strands on a spinline observed by Wagner, Schulze and Göttfert (996) was framed in terms of this maximum tension leading to cavitation by oseph (998). Wagner et alclaim that the breaking stress is a pure material constant which in their LDPE and HDPE samples is about 0 6 PA, an order of magnitude larger than atmospheric; the thread is in tension when it breaks. Lundgren and oseph (997) analyzed the breakup of a capillary lament assuming extensional flow at the pinch point (z = r = 0);u z = a(μ; fl; t)z; u r = a(μ; fl; t)r where fl is surface tension and μ is 2 viscosity. This is a viscous potential flow with a potential f = 2 az2 4 ar2 (49) They found that the principal balance is between surface tension producing the neckdown and the viscosity which resists collapse; that the neck is of parabolic shape and its radius collapses to zero in a nite time. During the collapse the tensile stress due to viscosity increases in value until at a certain nite radius, which is about micron for water in air, the stress in the throat passes into tension, presumably inducing cavitation there. The Reynolds number Re = R c flρ=ν 2 based on the velocity =μ of capillary collapse where R c = micron is about 55. The solution of Lundgren and oseph is flawed, it satises normal stress balance only at z = 0 but not elsewhere. However, it should be possible to compute an exact numerical

26 VE-CapInstab/viscoelastic-Rev2-NNFM.tex 26 solution of capillary collapse based on the equations of viscous potential flow in much the same way that capillary breakup was studied by numerical simulation of the equations for potential flow of an inviscid fluid by Mansour and Lundgren (990). The recent literature on capillary collapse is presently dominated by the discovery of self similar, nite time singularity formation. These solutions are discussed in the recent papers Chang et al (999), McKinley and Tripathi (2000), and in the paper of Chen et al (2002). This literature does not treat the physics of rupture or breakup by cavitation and does not compute stresses. All of the above mentioned authors nd that capillary radius decreases to zero linearly in time, but the rate of collapse differs from author to author. McKinley and Tripathi (2000) write the formula R (t) mid = R 2X fl 6 μ t (50) for the neck radius of the collapsing capillary in the stage of nal decay as t increases to tλ when R (tλ) mid = 0. They give the X obtained by different authors in their table, but without the value X = 2 obtained by Lundgren and oseph (997) for viscous potential flow, who give the fastest decay. Eggers (993, 997) obtained X = 0:592 and Papageoriou (995) obtained X = 0:727. The solutions of the two authors last named have vorticity; Papageriou s solution has no inertia. McKinley and Tripathi (2000) note that very close to breakup the solution of Papageriou crosses over to Egger s similarity solution. Maybe the nal breakup is a non-similarity solution which crosses over from Egger s solution to rupture. The solution of Eggers (993) gives rise to a universal scaling law which has been observed for viscous liquids but not in water (see Chen et al2002). The long wave approximation used to derive universal scalings may prevent it from resolving the dynamics of rupture. The similarity solution of Eggers does not lead to a cavitation threshold; in his solution the tension due to extension increases but not fast enough to overcome the compression due to the capillary pressure of the thinning lament. The criterion for the termination of the continuum is probably not a nite time singularity; the thread radius does not go to zero. It comes apart before then. The physics of rupture is up for discussion. Acknowledgement This work was supported by the NSF/CTS , the Engineering Research Program of the Ofce of Basic Energy Sciences at the DOE and by an ARO grant DA/DAAH04.

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