Viscoelastic effects on the jetting–dripping transition in co-flowing capillary jets

J. Fluid Mech. (2008), vol. 610, pp. 249–260. c 2008 Cambridge University Press doi:10.1017/S0022112008002681 Printed in the United Kingdom 249 Viscoelastic effects on the jetting–dripping transition in co-flowing capillary jets J. M. MONTANERO 1AND A. M. G A ˜ N´ AN-CALVO 2 1Departamento de Ingenier´ ıa Mec´ anica, Energ´ etica y de los Materiales, Universidad de Extremadura, E-06071 Badajoz, Spain 2Departamento de Ingenier´ ıa Aeroespacial y Mec´ anica de Fluidos, Universidad de Sevilla, E-41092 Sevilla, Spain (Received 31 March 2008 and in revised form 2 June 2008) Linear hydrodynamics stability analysis is used to determine the influence of elasticity on the jetting–dripping transition and on the temporal stability of non-axisymmetric modes in co-flowing capillary jets. The critical Weber number for which axisymmetric perturbations undergo a transition from convective to absolute instability is calculated from the spatio-temporal analysis of the dispersion relation for Oldroyd-B liquids, as a function of the density and viscosity ratios, and the Reynolds and Deborah numbers. Elasticity increases the critical Weber number for all cases analysed and, consequently, fosters the transition from jetting to dripping. The temporal analysis of the dispersion relation for the m= 1 lateral mode shows that elasticity does not affect its stability. 1. Introduction The controllable production of small flowing particles has been extensively investigated because of its relevance to an immense field of practical and industrial applications, ranging from chemical engineering to bio-industry. Different mechanisms to produce dispersions of drops have been considered (Basaran 2002), including the use of surfactants (Eggleton, Tsai & Stebe 2001), electrical forces (de la Mora 2007), extensional flows (Taylor 1934; Cohen & Nagel 2002), thermal gradients, and, more recently, concentrated photon irradiation (Schroll et al. 2007). Appropriate extensional flows can be obtained by a co-flowing outer fluid, which reduces the volume of the issuing drops and results in near monodispersity. Among the extensional flows analysed, flow focusing (Ga˜ n´ an-Calvo 1998) is attractive because of its purely hydrodynamic nature and continuous high-rate production. In the original flow focusing device, a moderately high Reynolds-number liquid jet focused by a gas breaks up into droplets about two orders of magnitude smaller than the source liquid stream. Several alternatives to the original design have been considered over the last few years. In particular, a double flow focusing arrangement has recently been proposed to produce submicrometre liquid jets yielding nanoparticles (Ga˜ n´ an- Calvo et al. 2007). A coherent understanding of the micro- and nano-scale interfacial flows appearing in all these configurations is essential for a rational design and manufacturing strategy. When a liquid is injected from a nozzle into a fluid bath or stream, several breakup modes are observed. For low injection velocities, droplets are formed periodically at the nozzle and no jet is observed (dripping). As the flow rate increases, dripping gives https://doi.org/10.1017/S0022112008002681 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 09 Nov 2020 at 16:01:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. 250 J. M. Montanero and A. M. Ga˜ n´ an-Calvo way to jetting. While dripping produces drops at the nozzle, jetting features a long jet that may extend several orifice diameters downstream and then breaks into droplets through Rayleigh instability. Steady micro-jetting gives rise to a good productivity and well-controlled small drop size. On the other hand, dripping gives rise to much larger droplets under comparable Reynolds and Weber number conditions. Dripping usually yields a highly monodisperse spray, but it may also exhibit bidisperse or polydisperse distributions (Coullet & Mahadevan 2005). Therefore, it is of interest and importance to understand the mechanisms of instability underlying the jetting–dripping (J–D) transition, which would allow one to map that transition and determine the jetting conditions yielding the smallest possible jet diameter. Although the mechanism of jet breakup is highly nonlinear, a linear hydrodynamic stability analysis may, however, be a helpful tool to predict breakup and instability transition occurrence. Among the different oscillation modes with different azimuthal wavenumbers mcontained in natural small disturbances of capillary jets, axisymmetric (m= 0) modes are generally responsible for instability transitions and breakage processes associated with surface tension (Leib & Goldstein 1986; Lin 2003). If a jet issues from a stationary source, its behaviour is essentially determined by whether the perturbations are convected downstream for all wavenumbers (convective instability), or some of them can travel upstream while growing (absolute instability). The relationship between these two scenarios (convective and absolute instability) and the jetting and dripping regimes has been already reasonably established (Huerre & Monkewitz 1990; Lin 2003; Guillot et al. 2007). The calculation of the parameter values for which a basic flow undergoes a convective/absolute (C/A) instability transition can be made analytically tractable at the expense of geometrical simplifications. If the model includes all significant effects (viscosity, inertia, interfacial tension forces, etc.), and the basic flow around which infinitesimal perturbations are considered is realistic, then this approach provides reliable predictions for the J–D transition. Consider a manifold capillary comprising a liquid stream that envelopes a much thinner inner capillary jet of a second liquid immiscible with the first. Assume that the basic flow in the manifold capillary is characterized by constant velocity and pressure, and can be considered as infinite in terms of the inner jet radius. These conditions are approximately satisfied in several co-flowing configurations (Suryo & Basaran 2006; Guillot et al. 2007). An example is the flow beyond the orifice of a double flow focusing arrangement (Ga˜ n´ an-Calvo et al. 2007), where the manifold capillary is in turn focused by a third current of a gas forced through a small round orifice. An analysis of the C/A instability transition for axisymmetric (m= 0) perturbations in the limit of small Reynolds number showed that those perturbations are convected downstream provided that the jet velocity exceeds a certain threshold (Ga˜ n´ an-Calvo et al. 2007). Interestingly, this threshold depends on the viscosities and interfacial tension but is independent of the jet radius (we call this ‘unconditional jetting’). This theoretical prediction was confirmed by experiments using a double flow focusing device (Ga˜ n´ an-Calvo et al. 2007). It is well-known that non-Newtonian liquids can exhibit responses that differ drastically from those of Newtonian liquids in most free-surface flows. Even for small flow rates, the small dimensions of the capillary jets and the very large frequencies arising, for instance, in flow focusing devices cause large strain rates and extremely fast events capable of distorting the microstructure of the liquids (Eggers & Villermaux 2008), and thus inducing viscoelastic effects. For this reason, increasing attention is being paid to non-Newtonian fluids in microfluidics. Indeed, the study of rheological https://doi.org/10.1017/S0022112008002681 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 09 Nov 2020 at 16:01:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. Viscoelastic effects on the jetting–dripping transition 251 effects is currently a vigorous area of microfluidic research (Stone, Stroock & Adjari 2004; Eaggers & Villermaux 2008). The numerous rheological effects can probably be best elucidated by dissecting, isolating and analysing them separately. In the present paper, we mainly focus our attention on the influence of elasticity on the J–D transition appearing in co-flowing jets of Oldroyd-B liquids. In the Oldroyd-B model (Joseph 1990), the elastic effects increase with the Deborah number, which is defined as the ratio of the stress relaxation time to the characteristic time. The characteristic time takes very small values for micrometre and submicrometre jets produced by flow focusing or microfluidic devices, and thus one obtains large values of the Deborah number even for small stress relaxation times. Therefore, one may expect a significant influence of elasticity on the J–D transition in flow focusing and microdevices operating with viscoelastic liquids. Temporal linearized stability analysis for axisymmetric disturbances shows that an Oldroyd-B jet exhibits a faster growth rate than a Newtonian one with the same Ohnesorge number, indicating that non-Newtonian jets are more unstable than their Newtonian counterparts (Brenn, Liu & Durst 2000; Funada & Joseph 2003). The most unstable wavelength on a viscoelastic jet is shorter than on a comparable Newtonian one, which implies that smaller drops would form for viscoelastic liquids. This theoretical result has been confirmed by simulations (Zhou, Yue & Feng 2006). The effect of a small elasticity on non-axisymmetric modes evolving in a single column (Bauer 1986) and in a jet moving in an inviscid gaseous environment (Liu & Liu 2006) has also been examined. The results indicated that elasticity does not modify the stability of these latter modes though, and only slightly affects their frequencies and damping factors. The J–D transition is controlled by the ratio of the time scale required for pinch-off to a convective time scale. For large flow velocities, the convective time scale is smaller than that required for pinch-off, transport dominates, and a jet forms. The fact that viscoelastic properties accelerate the pinch-off in the linear regime (Brenn et al. 2000; Funada & Joseph 2003) suggests that those properties make jetting take place at jet velocities larger than in the Newtonian case (which might be the case in (Anna & Mayer 2006). A spatiotemporal analysis is necessary to confirm that expectation, and to determine the critical values of the Weber number for which the J–D transition occurs. In the present paper, we describe a linear spatiotemporal stability analysis for axisymmetric (m= 0) perturbations in a co-flowing system of two viscoelastic liquids moving with the same velocity to determine the parametric values for which a C/A (J– D) instability transition takes place. Our study extends the previous one by Funada & Joseph (2003) in two directions: a spatiotemporal analysis is carried out, and elastic effects on both the inner jet and the co-flowing liquid are considered. In addition, we evaluate the influence of elasticity on the damping factor characterizing the evolution of the lateral (m= 1) mode. 2. Linear stability Consider a cylindrical jet of radius Rmoving in the axial direction into another liquid co-flowing with the same velocity V(see e.g. Ga˜ n´ an-Calvo et al. 2007). The flow configuration considered is sketched in figure 1. The perturbed shape of the jet, velocity, and pressure may be expressed as r=f(θ,z,t), v(r,t)≡[u(r,t),v(r,t),w(r,t)], and p(r,t), respectively. In what follows, we shall make all the variables dimensionless using the inner jet radius R, velocity V, time R/V , and momentum ρ1V2as the https://doi.org/10.1017/S0022112008002681 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 09 Nov 2020 at 16:01:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. 252 J. M. Montanero and A. M. Ga˜ n´ an-Calvo r V R z θ Figure 1. Sketch of the flow configuration considered. The circled region is where the analysis presented here applies. characteristic length, velocity, time, and pressure, respectively, where ρ1is the inner liquid density. Using a Lagrangian frame of reference moving with both fluids, we propose the following dependence for the hydrodynamic fields and the interface position: vj(r,t)=ε{Uj(r),V j(r),W j(r)}ei(mθ+kz−ωt)+c.c.,(2.1a) pj(r,t)−δj1 We =εPj(r)ei(mθ+kz−ωt)+c.c.,(2.1b) f(θ,z,t)−1=εFei(mθ+kz−ωt)+c.c.,(2.1c) where j= 1 and 2 stand for the inner jet and co-flowing liquid, respectively, mis the azimuthal wavenumber, k=kr+ikithe axial wavenumber, ω=ωr+iωithe frequency, We =ρ1V2R/σ the Weber number, σthe interfacial tension, and δij the Kronecker delta. The rheological behaviour of a viscoelastic material can be described by the Oldroyd-B model (Joseph 1990): (1+τG)σik =2μ0(1 + λG)Dik,(2.2) where G[A]≡dA/dt−wA +Aw is the upper convected derivative operator, d/dtis the material derivative, wis the antisymmetric part of D,σik is the (non-equilibrium) stress tensor, μ0the Newtonian viscosity, and Dik the strain rate tensor. The model also involves the two rheological parameters τand λ. For small perturbations around our motionless (in the Lagrangian frame of reference) basic flow, the upper convected derivative operator Gcan be approximated by the partial derivative ∂/∂t, and thus the Oldroyd-B model yields σik +τ∂σik ∂t =2μ0Dik +λ∂Dik ∂t ,(2.3) where τand λare the (dimensionless) stress and strain relaxation times. These parameters are also known as the Deborah number and the (dimensionless) retardation time, respectively. For λ= 0, (2.3) reduces to the Maxwell model, widely used to describe a viscoelastic material in the linear response regime. If one consistently assumes a temporal dependence of the form e−iωt for σik and Dik to the order ε,then https://doi.org/10.1017/S0022112008002681 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 09 Nov 2020 at 16:01:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. Viscoelastic effects on the jetting–dripping transition 253 the constitutive relation (2.3) to that order is σik =2μ0 1−iλω 1−iτωDik.(2.4) The parametric values for which the system becomes temporally unstable are obtained from the dispersion relation by imposing the condition of zero growth rate (ωi=0), and thus the critical motion corresponds to an harmonic oscillation of frequency ωr. If the oscillation period is large in terms of τand λ(|ωr|τ,|ωr|λ1), the liquid can be regarded as Newtonian. On the other hand, if the oscillation period is of the same order as or even larger than the stress relaxation time (|ωr|τ,|ωr|λ∼1or |ωr|τ,|ωr|λ>1), the elastic effects are significant and may affect the J–D transition. It must be noted that large values of |ωr|lead to large values of the strain rate Dik and, consequently, may yield a failure of the linear model (2.4). If one introduces (2.1) and (2.4) into the equations of conservation of mass and momentum, and neglects terms in ε2, one obtains U j+Uj/r +imVj/r +ikWj=0,(2.5a) −ρδj2iωUj+P j=μδj2 Re 1−iλjω 1−iτjω[U j+U j/r−(m2+1)Uj/r2−k2Uj−2imVj/r2],(2.5b) −ρδj2iωVj+imPj/r =μδj2 Re 1−iλjω 1−iτjω[V j+V j/r −(m2+1)Vj/r2−k2Vj+2imUj/r2], (2.5c) −ρδj2iωWj+ikPj=μδj2 Re 1−iλjω 1−iτjω(W j+W j/r −m2Wj/r2−k2Wj),(2.5d) where ρ≡ρ2/ρ1and μ≡μ2/μ1are the ratios between the density ρ2and Newtonian viscosity μ2of the co-flowing fluid and the corresponding values ρ1and μ1of the inner jet, and Re =ρ1VR/μ 1is the Reynolds number. The general solution to (2.5), satisfying the regularity conditions U1(0) = V1(0) = W 1(0) = 0 for m=0,U 1(0) + iV1(0) = W1(0) = 0 for m=1, (2.6a) U1(0) = V1(0) = W1(0) = 0 for m>2,U 2=V2=W2=P2= bounded at r→∞, (2.6b) can be written as Uj(r)=icj1Θ m(kr)+icj2Θ m(kjr)+imc j3 Θm(kjr) kjr,(2.7a) Vj(r)=−mc j1 Θm(kr) kr −mc j2 Θm(kjr) kjr−cj3Θ m(kjr),(2.7b) Wj(r)=−cj1Θm(kr)−cj2kj Θm(kjr) k,(2.7c) Pj(r)=−ρδj2ωc j1 kΘm(kr).(2.7d) Here, Θmis the modified Bessel function of the first kind Imif j= 1, and of the second kind Kmif j=2,k1=±[k2−iωRe (1 −iτ1ω)/(1 −iλ1ω)]1/2,k2=[k2−iωReρ/μ (1 −iτ2ω)/(1 −iλ2ω)]1/2,and{cj1,c j2,c j3}are six arbitrary constants. Note that (2.7) https://doi.org/10.1017/S0022112008002681 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 09 Nov 2020 at 16:01:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. 254 J. M. Montanero and A. M. Ga˜ n´ an-Calvo is symmetric with respect to k1, while the real part of kmust be positive to satisfy (2.6b). The non-slip condition {U1=U2,V 1=V2,W 1=W2}at the interface r= 1 yields a linear system of equations that allows one to obtain {c21,c 22,c 23}in terms of {c11,c 12,c 13}. The condition of dynamical equilibrium at the interface r=1 for the normal and two tangential components leads to P1−P2+i(1 −m2−k2) We ωU1=2 Re 1−iλ1ω 1−iτ1ωU 1−μ1−iλ2ω 1−iτ2ωU 2,(2.8a) 1−iλ1ω 1−iτ1ω(W 1+ikU1)=μ1−iλ2ω 1−iτ2ω(W 2+ikU2),(2.8b) 1−iλ1ω 1−iτ1ω(imU1+V 1−V1)=μ1−iλ2ω 1−iτ2ω(imU2+V 2−V2),(2.8c) where use has been made of the kinematic compatibility condition F=iUj(1)/ω to eliminate Ffrom (2.8a). The set of equations (2.8) constitutes a linear system of equations for {c11,c 12,c 13}. The solvability condition Det(Δij ) = 0, with Δij being the 3×3 matrix associated with that system of equations, yields the dispersion relation Sm(k,ω,ρ,μ,Re,We ,τ 1,τ 2,λ1,λ2)=0.(2.9) A practical explicit expression of (2.9) cannot be provided here. A Mathematica notebook containing the dispersion relation can be obtained upon request from the authors. The most important conclusion drawn from the temporal analysis presented in Montanero & Ga˜ n´ an-Calvo (2008) for Newtonian liquids is that non-axisymmetric (m= 0) perturbations possess negative growth factors over the whole parameter space. Therefore, a transition from convective (jetting) to absolute instability for m= 0 is not possible for the basic flow analysed. A natural question is whether elasticity could modify the above result. To answer this question, we here perform a temporal analysis (kreal) of (2.9) for the lateral mode m= 1. For this purpose, we re-write (2.9) in the form Qm(k,  ω, ρ, μ, Oh,τ 1,τ 2,λ1,λ2)=0,(2.10) where  ω= ωr+i  ωi≡We1/2ωand Oh ≡We1/2/Re is the Ohnesorge number. This is equivalent to choosing the capillary velocity Vc≡(σ/ρ1R)1/2instead of Vas the characteristic velocity, an appropriate choice when the flow is described in our Lagrangian frame of reference. It must be noted that this simplification is exclusive to the basic flow considered here, where both the inner jet and the co-flowing liquid move with the same velocity V. Solutions to (2.10) in the complex plane ( ωr, ωi)werefoundnumericallyfor0<k<1 and fixed values of the rest of the parameters. Because we were mainly interested in finding possible transitions from stable ( ωi<0) to unstable ( ωi>0) perturbations, (2.10) was carefully explored close to the  ωi= 0 axis, while less attention was paid to other regions of the ( ωr, ωi)-plane. The roots of (2.10) were found by means of the Newton–Raphson method, using as initial guess the value obtained from a linear extrapolation of the two previous solutions. We considered as solutions those satisfying the condition |Q1|<10−10. In this paper, we mainly focus attention on the J–D transition in the co-flowing jets, explained in terms of the C/A instability transition for the axisymmetric mode https://doi.org/10.1017/S0022112008002681 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 09 Nov 2020 at 16:01:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. Viscoelastic effects on the jetting–dripping transition 255 (m= 0). In this case, an explicit expression for the dispersion relation (2.9) can be provided: S0≡iωN(k,k1,k 2,μ ∗) M(k,k1,k 2,μ ∗)+2(1−μ∗)−Re∗ We (1 −k2)=0,(2.11) N≡2kμ∗k1k2[K0(k2)I1(k1)k1+I0(k1)K1(k2)k2]+kk2(μ∗−1) −k2 1+μ∗k2 22 ×I0(k)I1(k1)K0(k)K1(k2)+4k3k1k2(μ∗−1)2I0(k1)I1(k)K0(k2)K1(k) −k2I1(k1)K0(k2)k4+k2 1k2 2+k2k2 1−k2 2μ∗I1(k)K0(k) +k4 1+k4(1 −2μ∗)2−2k2k2 1(μ∗−1)I0(k)K1(k) −k1I0(k1)K1(k2)k4(μ∗−2)2+2k2k2 2μ∗(μ∗−1) + μ∗2k4 2I1(k)K0(k) +k2k2−k2 1+k2 2k2+k2 1μ∗I0(k)K1(k),(2.12) M≡k[k2K0(k2)K1(k)−kK0(k)K1(k2)]k2 1−k2I1(k)I1(k1) +μ∗[k1I0(k1)I1(k)−kI0(k)I1(k1)]k2 2−k2K1(k)K1(k2).(2.13) In the above expressions, k1=±√k2−iωRe∗,k2=k2−iωRe∗ρ/μ∗,Re∗≡ Re(1 −iτ1ω)/(1 −iλ1ω), and μ∗=μ(1 −iτ1ω)/(1 −iτ2ω)(1 −iλ2ω)/(1 −iλ1ω). To determine the parameter values for which the basic flow undergoes a C/A instability transition, one should explore the response of the system to perturbations characterized by a complex axial wavenumber k(spatio-temporal analysis), observed by a fixed observer anchored at the nozzle (Huerre & Monkewitz 1990). To change the frame of reference from a travelling observer to a fixed one, we just need to replace the wave frequency ωby ω−kin the dispersion relation (2.11) (Leib & Goldstein 1986). A localized initial distortion spreads in the fluid domain bounded by two fronts moving with velocities v∗ +and v∗ −. If the distortion is the superposition of perturbations proportional to ei(kz−ωt)=e ki(vt−z)ei(krz−ωrt),thenv∗ ±are calculated as the extremal values of the envelope velocity v=ωi/kiof those perturbations (van Saarloos 1987, 2003; Guillot et al. 2007). The C/A instability transition occurs for parameter points for which the velocity v∗ −of the back front vanishes. As demonstrated in the Appendix, this criterion is equivalent to the so-called saddle-point rule, which establishes that the C/A instability transition occurs when there is at least one solution of the dispersion relation satisfying dω/dk= 0 (zero group velocity) with ω i= 0 (Briggs 1964; Huerre & Monkewitz 1990). Once the parameters {ρ,μ,Re,τ 1,τ 2,λ1,λ2}are set, one has to find the value of the Weber number for which v∗ −= 0 or, equivalently, dω/dk=ω i=0.The latter condition can be formulated in terms of four coupled transcendental equations Fi(kr,k i,ω  r,We)=0 (i=1,2,3, and 4) for the set of variables {kr,k i,ω  r,We}.Inthis work, the roots of that system of equations were found numerically by the secant method within the variable space volume delimited by the previous solution and that obtained from a linear extrapolation of the two previous ones. We considered as solutions those satisfying the condition |Fi|<10−10 (i=1,2,3, and 4). 3. Results and conclusions Viscoelastic liquids in co-flowing devices usually have similar densities (ρ≃1) and retardation times at least one order of magnitude less than the relaxation times (λjτj) (see e.g. Brenn et al. 2000 or Funada & Joseph 2003 and references therein). https://doi.org/10.1017/S0022112008002681 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 09 Nov 2020 at 16:01:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. 256 J. M. Montanero and A. M. Ga˜ n´ an-Calvo 0,10,100 103103 100 0,10 0,10,100,1030,10,100,103 10–2 10–3 10–4 10–5 10–2 (b)(a) 10–3 10–4 10–5 0 0.2 0.4 0.6 k 0.8 1.0 0 0.2 0.4 0.6 k 0.8 1.0 –ω ~i Figure 2. Growth factor  ωias a function of kfor μ=1 and for Oh = 10 (solid lines) and Oh = 100 (dashed lines): (a)τ2= 0 and different values of τ1,(b)τ1= 0 and different values of τ2,as indicated by the labels. The black curves correspond to overdamped motions, while the grey lines labelled 103correspond to oscillatory motions. Because of the high dimension of the parameter space, we explored it partially by considering the cases ρ=1andλj= 0 (Maxwell model). The temporal analysis of (2.10) for the lateral mode m= 1 is presented in §3.1, while in §3.2 we show the spatiotemporal analysis of (2.11) for the axisymmetric mode m=0. 3.1. Temporal analysis of the lateral mode For non-surrounded jets (ρ=μ= 0), it is well-known that axisymmetric (m=0) perturbations are stable for k>1 and unstable for 0 <k<1. For this reason, in the present study we calculate the growth factors characterizing the dominant lateral motions of co-flowing jets within the interval 0 <k<1. The values of the Ohnesorge number considered correspond to high-Newtonian-viscosity liquids, as are usually employed in experiments. The most important conclusion drawn from the results shown in figure 2 (and others not presented here) is that lateral perturbations possess negative growth factors within the region of parameter space explored, and hence elasticity does not modify their stability. For Oh = 100, the influence of elasticity is negligible even for large values of the Deborah number. The damping factor slightly decreases as τjincreases for Oh = 10. A transition from overdamped to oscillatory motion takes place between τj= 100 and 103for that value of the Ohnesorge number. 3.2. Spatio-temporal analysis of the axisymmetric mode The main aim of the present work was to determine the influence of elasticity on the J–D transition in a capillary flow consisting of two co-flowing liquid jets moving with the same velocity. For this purpose, the critical Weber number for which the C/A instability transition occurs is obtained as a function of the other parameters. When working with millimetric jets, the Deborah number usually takes values less than unity. Figure 3 shows the ratio between the critical Weber number We for 0 6τj61 and its value We0for the Newtonian case (τ1=τ2= 0). Elasticity slightly increases the critical Weber number. Similar results are obtained when the viscoelastic material is the inner (solid lines) or outer (dashed lines) jet. The effect of elasticity increases as the Reynolds number decreases. For micrometre and submicrometre jets, the Deborah number may take large values even for small stress relaxation times. Figure 4 shows the J–D transition curves for τj=0,10,102,103,and10 4, and different values of the Newtonian viscosity ratio μ. The main result is that elasticity increases the critical Weber number significantly https://doi.org/10.1017/S0022112008002681 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 09 Nov 2020 at 16:01:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. Viscoelastic effects on the jetting–dripping transition 257 0 0.2 0.4 0.6 0.8 1.0 1.00 1.05 1.10 1.15 1.20 100 10 1 τ We We0 10–6 Figure 3. Ratio between the critical Weber number We and its value We0for the Newtonian case (τ1=τ2= 0) both calculated for μ= 1. The solid (dashed) lines indicate the dependence on τ1(τ2)forτ2=0 (τ1= 0). The labels are the values of the Reynolds number. 10–6 10–4 10–2 100102 10–6 10–4 10–2 100102 10–6 10–4 10–2 100102 10–6 10–4 10–2 100102 10–7 10–5 10–3 10–1 101 10–7 10–5 10–3 10–1 101 μ = 0.1 μ = 1 μ = 10 μ = 10 μ = 1 104 Ca 10–7 10–6 10–4 10–2 100102 10–6 10–4 10–2 100102 10–5 10–3 10–1 101 Ca 10–7 10–5 10–3 10–1 101 Ca 10–7 10–5 10–3 10–1 101 Ca 103 0 100 10 (a)(b) 10–7 10–5 10–3 10–1 101 Ca 10–7 10–5 10–3 10–1 101 Ca We 10–7 10–5 10–3 10–1 101 We 10–7 10–5 10–3 10–1 101 We 10–7 10–5 10–3 10–1 101 We 10–7 10–5 10–3 10–1 101 We 10–7 10–5 10–3 10–1 101 We jetting dripping μ = 0.1 104 103 0 100 10 jetting dripping 104 103 0 100 10 jetting dripping 10 10 0 100 10 jetting dripping 104 103 0 100 10 Re jetting dripping 104 103 0 100 10 Re jetting dripping Figure 4. Critical value of the Weber (solid lines) and capillary (dashed lines) numbers as a function of the Reynolds number for μ=0.1, 1, and 10: (a)forτ2= 0 and different values of τ1,(b)τ1= 0 and different values of τ2, as indicated by the labels. https://doi.org/10.1017/S0022112008002681 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 09 Nov 2020 at 16:01:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms.