Drops and jets of complex fluids - 4: Delayed capillary breakup of falling viscous jets

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Drops and jets of complex fluids

Javadi, A.

Publication date

2013

Link to publication

Citation for published version (APA):

Javadi, A. (2013). Drops and jets of complex fluids.

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4.

Delayed capillary breakup of

falling viscous jets

4.1

Introduction

The breakup of liquid jets due to the action of surface tension is a classic fluid-mechanical instability, first explained theoretically by Plateau (51) and Rayleigh (52). A familiar example of it is a thin stream of water flowing steadily from a faucet, which breaks up into droplets after a distance ≈ 10 cm. If however the water is replaced by a much more viscous fluid like honey, the jet can attain lengths of 10 m or more before breaking. This is paradoxical: theory (53) predicts that the weight of fluid elements in a long viscous jet is balanced by the vertical momentum flux (inertia) over most of the jet’s length, and that the viscous force that resists the stretching of the jet is negligible in comparison. But if this is so, how can the viscosity influence the breakup length? Senchenko and Bohr (54) attempted to answer this question by analyzing the growth of small perturbations of the radius of a viscous jet that is strongly stretched and thinned by gravity. But their conclusion that the growth rate of the perturbations is independent of viscosity only deepens the paradox instead of resolving it.

Here we report the first systematic experimental investigation of the breakup length of falling viscous jets, and propose a new theory that explains how viscosity acts to delay jet breakup. To set the stage, we recall that fluids falling from circular nozzles typically exhibit three distinct regimes as a function of the flow rate (55). At very low flow rates, a ‘periodic dripping’ (PD) regime occurs in which drops of constant mass detach periodically at a downstream distance comparable to the nozzle diameter. As the flow rate increases, a transition to a ‘dripping faucet’ (DF) regime occurs in which the mass of the detaching drops varies quasi-periodically or chaotically. Finally, as the flow rate is increased further, a transition occurs to a ‘jetting’ (J) regime in which a steady jet emerges from the nozzle and breaks up

further downstream. Our focus here is on the length of the intact portion of the jet in this regime.

In the literature, the dependence of the breakup length on the flow rate and fluid properties such as the surface tension has been extensively studied for high-speed jets in quiescent or co-flowing fluids (56; 57). By contrast, viscous jets falling under gravity have been the subject of only a few experimental (58; 59) and theoretical (54; 60; 61; 59) studies, none of which arrived at a prediction for the breakup length as a function of the flow rate and the fluid properties.

But before we get to that, a more detailed summary of the previous works for viscous and inviscid breakup of jets may prove useful for some readers.

4.2

History and backgrounds

The earliest study of jets and breakup was by Leonardo da Vinci (62), who notes correctly that the detachment of a drop falling from a tap is governed by the condition that gravity overcomes the cohesive (surface tension) forces. However, he then incorrectly goes on to assume that the same principle also governs the separation of the drop itself, which occurs once a sufficiently extended fluid neck has formed.

The same flawed argument was elaborated later by Mariotte (63), who claims that gravity is responsible for drop breakup (a jet which is projected upward does not break up); breakup occurs when the fluid thread has become as ‘thin as a hair’. Thus both authors view the breakup of liquids and solids as related phenomena. Mariotte argues more quantitatively that a falling jet acquires a speed correspond-ing to free fall v = √2gz at a distance z from the nozzle (neglecting the initial speed). Then if Q is the flow rate, mass conservation gives

h =  _Q

π√2gz 

(4.1)

for the radius h of the jet, making it increasingly thin as it falls. The ideas of da Vinci and Mariotte suggest that cohesive forces provide a certain tensile strength T of water, which has to be overcome by gravity for the jet to break. Taking the value for the tensile strength of glass T = 108_Nm−2_{as a conservative estimate, the}

thread can no longer support a drop of 1 ml volume when its cross sectional area has become A = 10−5cm2. Taking Q = 1 ml s−1in (4.1), the fluid thread can formally extend to a length of 100 km before it breaks! In 1804-1805 Laplace (64) and Young (2) exhibited the crucial role of the mean curvature, made up of contributions from both the axial and the radial curvature. The subtle point which leads to the fallacy of earlier authors is that surface tension can act in two different ways: while for a hanging drop it indeed acts like an elastic membrane, once a cylindrical shape is reached the radial curvature is driving the breakup! Namely, the system is driven towards a state having a smaller surface area, and thus towards a smaller jet radius, which eventually goes to zero. Thus paradoxically, the greater the cohesion between

4.2. History and backgrounds

Figure 4.1: Sketch by Leonardo da Vinci illustrating the impact of jets (62).

the particles (and thus the surface tension γ), the faster breakup becomes. Namely, if the fluid viscosity is neglected, by dimensional analysis a characteristic timescale of the motion is τ = ρh 3 0 γ 1/2 , (4.2)

where h0 is the initial radius of the fluid cylinder. The time (4.2) estimates the

total time for breakup to occur.

Figure 4.2: A figure from Savart’s original paper (65), showing the breakup of a liquid jet 6 mm in diameter. It clearly shows the succession of main and satellite drops as well as drop oscillations.

Experimentally studying liquid jets, Savart noted that breakup occurs sponta-neously, independent of any external force or the direction in which the jet is pro-jected, and thus must be a feature intrinsic to jet dynamics. To confirm breakup, Savart found that a ‘thin object’ could be made to pass through the jet without getting wet. For more quantitative observation, Savart developed a stroboscopic technique, which allowed him to produce images such as Fig. 4.2. To this end he employed a continuous moving tape painted with alternating black and white stripes, against which the jet was viewed. In particular, Savart noticed the appear-ance of a smaller ‘satellite’ drop between two main ones.

The most unstable mode of breakup is best excited by allowing the vibrations pro-duced by the impact of drops to be fed back to the vessel out of which the jet is

flowing. This produces a definite resonance frequency f of the system. With further investigations of this frequency (65) he could confirm the existence of a characteris-tic wavelength λopt, corresponding to resonance. Indeed, some years later Plateau,

observing the decay of columns of fluid in density matched surroundings (the so-called Plateau tank (66)) found that perturbations are unstable if their wavelength λ is greater than a critical one λcr , whose value lies between 6h0 and 7.2h0 (67).

Plateau also found (68), λcr/h0= 2π ≈ 6.28.

Plateau’s observation is that a jet is unstable to any perturbation which reduces the surface area, thus finally recognizing the crucial role of surface energy (or surface tension) for jet breakup. However, the value of the ‘optimal’ wavelength λopt Plateau deduces from Savart’s measurements is λopt = 8.76h0, significantly

greater than 2π ≈ 6.28. Only Rayleigh (52; 69) realized that to understand this ‘overstretching’, the jet dynamics has to be taken into account: among all unstable wavelengths λ > λcr, the one with the fastest growth rate is selected. For inviscid

jet dynamics, this gives λopt = 9.01h0, in good agreement with Savart’s data.

Rayleigh thus introduced the extremely useful method of linear stability to jet breakup, which will be the topic of the next section. At the same time, high speed photography was used to study complex features of drop breakup .

4.3

Rayleigh Instability

We first turn to the problem of inviscid jet flow with a free surface (Fig. 4.3). We assume axisymmetry, so the boundary is given by a curve in the r − z plane. If the velocity field is irrotational initially, and viscosity can be neglected, it will remain so for the rest of its evolution (70). Thus the velocity potential

v(r, z) = ∇φ, (4.3)

obeys the Laplace equation

∇2φ = 0. (4.4)

If x(z, t) = h(z, t)ˆr + z ˆz is the position of the surface as a function of z, the surface moves according to

Dtx(z, t) = v|∂Ω. (4.5)

The Bernoulli equation for evolution of φ can be written as (71)

∂tφ(z, t) = v2/2 −

γ

ρκ on ∂Ω. (4.6) Here, κ is the mean curvature

κ = 1

h(1 + h02₎1/2 −

h00

(1 + h02₎3/2, (4.7)

4.3. Rayleigh Instability

Figure 4.3: Sketch of a typical flow geometry. Rotational symmetry around the z axis is assumed. The velocity field inside the fluid is v(r, z) = vz(r, z)ˆz + vr(r, z)ˆr .

According to linear instability method, Rayleigh introduced small perturbations, with wavelength λ, to the jet of radius h0, as

h(z, t)/h0= 1 + η(z, t) = 1 + A(t)cos

 2πz λ



. (4.8)

Note that, the amplitude of the perturbations is a function of time. The non-dimensional velocity ˆφ potential can also be written as

φ(r, z, t) = h 2 0 τ ˆ φ(r, z, t). (4.9)

Now, by introducing the non-dimensional parameters

ζ = z h0 , t =˜ t τ, r =˜ r h0 , x = 2πh0 λ , (4.10) and the boundary conditions as

η(ζ, 0) = cos(xζ), ∂˜tη(ζ, 0) = 0, (4.11)

to the equations. (4.4,4.5,4.6), one can solve them to find

η(ζ, ˜t) = cosh(ω˜t)cos(xζ) (4.12) ˆ

φ(˜r, ζ, ˜t) = B(˜t)f (˜r)cos(xζ) = ω xI1(x)

sinh(ω˜t)I0(x˜r)cos(xζ), (4.13)

with non-dimensional growth rate of perturbations ω

ω2=xI1(x) I0(x)

Here In is the modified Bessel function of order n.

ω is real for x < 1, so perturbations grow exponentially, making the jet unstable against arbitrarily small perturbations. For x > 1, on the other hand, the interface performs oscillations, which will eventually be damped by viscosity. The inviscid dispersion relation (4.14) is plotted in Fig. 4.4. The most unstable mode, cor-responding to the largest ω, occurs at xR = 0.697. This is the famous Rayleigh

mode, which has a wavelength λopt= 9.01r0.

Figure 4.4: The dimensionless growth rate of sinusoidal perturbations on a cylinder as a function of the dimensionless wave number. The solid line represents Rayleighs theory for inviscid flow, the squares are data from Donnelly and Glaberson (1966) (72), triangles are from Goedde and Yuen (1970) (73).

4.4

Long wave-length approximation

For the viscous filament, the Navier-Stokes and the continuity equations can be written as

∂tvr+ vr∂rvr+ vz∂zvr= −∂rp/ρ + ν(∂2rvr+ ∂z2vr+ ∂rvr/r − vr/r2), (4.15)

∂tvz+ vr∂rvz+ vz∂zvz= −∂zp/ρ + ν(∂r2vz+ ∂z2vz+ ∂rvz/r), (4.16)

∂rvr+ ∂zvz+ vr/r = 0 (Continuity), (4.17)

where 0 6 r < h(z, t). The balance of normal and tangential forces on the surface gives

n.σ.n = −γκ, Normal forces (4.18) n.σ.t = 0, Tangential forces (4.19) where the stress tensor is σ = −pI + η[(∇v) + (∇v)T_{], n is the outward normal}

and t is the tangential unit vector to the jet’s surface. Finally, the surface has to move with the velocity field at the boundary:

4.4. Long wave-length approximation

Since we are going to look at thin columns of fluid relative to their elongation, we expand in a Taylor series with respect to r. Therefore we have (74)

vz(z, r) = v0(z) + v2(z)r2+ · · · (4.21) vr(r, z) = − 1 2v 0 0r − 1 4v 0 2r 3_{− · · · (4.22)} p(z, r) = p0+ p2r2+ · · · . (4.23)

Introducing (4.21-4.23) to (4.15-4.20) and dismissing the subscript 0 for v0, we

arrive at the one-dimensional equations

∂tv = −vv0+ 3ν  v00+2v 0_h0 h  −γ ρ ∂ ∂zκ, (4.24) ∂th = −vh0− 1 2v 0_{h. (4.25)}

Which are the equations for the ‘long-wavelength’ approximation.

Figure 4.5: Dispersion curve −iω(k) of equation (4.26) for increasing Ohnesorge number Oh−1= 100, 5, 1, 0.2, 0.05. (5)

Now, the linear instability method suggests putting h(z, t) = h0+ ε(t)cos(kz) with

ε(t) = e−iωt, in equations (4.24,4.25). Hence

(−iω)τ = r 1 2(x 2_{− x}4_{) +}9 4Oh 2_x4₋3 2Ohx 2_{, (4.26)}

with x = kh0 and k = 2π/λ. Oh is the Ohnesorge number

Oh = ν r _ρ

γh0

. (4.27)

From (4.26) the growthrate of the most amplified wave-number, xm= √ 1 2+3√2Oh,

is

−iω(xm)τ =

1

2√2 + 6Oh. (4.28) Fig. 4.5 shows the deformation of the dispersion curve as the Ohnesorge number

Figure 4.6: Dimensionless growth rate −iω(k)(ηh0/γ) of sinusoidal perturbations

on a viscous cylinder for Oh = 0.58 as a function of the dimensionless wave number kh0. The solid line represents the theory (73).

is varied, and Fig. 4.6 presents a comparison with experiments.

4.5

Experiments of the viscous falling jet

We used silicone oils with densities ρ = 963-974 kg m−3, surface tension coefficient γ = 0.021 N m−1, and viscosities ν = 50-27800 cS. A thin vertical jet was generated by ejecting the oil downward through a nozzle of diameter 2r0 = 2 − 4 mm at a

constant flow rate with a range Q = 0.0036 − 1.4 ml/s, using either a syringe pump controlled by a stepper motor or an open reservoir with an adjustable valve at the bottom. The reservoir was sufficiently large (14 cm × 14 cm wide and 20 cm deep) that the flow rate was constant to within ±2% during all of the experiments. To eliminate the influence of air drag on longer jets (breakup length lb > 2.5 m), we enclosed the nozzle and the jet in a cylindrical vacuum chamber

with inner diameter 19 cm and length ≤ 7.5 m. The bottom portion (2 m) of the cylinder was transparent to permit observation. A partial vacuum was created inside the cylinder using a Siemens rotary vacuum motor, allowing even the longest jets (lb= 7.5 m) to remain perfectly straight.

We observed three distinct regimes of behavior of the ejected fluid, including the PD regime at very low flow rates and the J regime at high flow rates. At intermediate

4.6. Theoretical investigation

flow rates, however, we did not observe the DF regime, but rather an oscillatory ‘pulsating’ regime. Here the jet had a reasonably steady shape, especially near the nozzle, and broke up at a well-defined distance that greatly exceeded the nozzle diameter. However, small periodic oscillations of the jet’s shape about the mean diameter occurred, corresponding to the absolute instability identified by (59). Buggisch and Sauter (59) found an expression for a line of marginal stability of a long but finite liquid jet with viscosity, inertia, surface tension and gravity. Below this marginal line the falling jet pulsates while above it we are in the stable jetting regime. For a jet with nozzle cross section area a∗₀and exit velocity u∗₀, the marginal line is given by

a0,crit = 0.025u−2.510 , (4.29)

where a0 is a dimensionless cross section area (at the nozzle) defined as

a0=

 ν2_g2_ρ3

π3/2_γ3

2/3

a∗₀, (4.30)

and u0= (νg)−1/3u∗0. Fig. 4.8 shows a0and u0for all our experiments along with

the marginal stability line (solid line).

We measured the breakup length lbfor a total of 87 experiments, including 67 in the

jetting regime and 20 in the pulsating regime. In some experiments, we detected the point of breakup by moving a rapid camera step by step along the jet as it thins. In most cases, however, we first located the breakup point approximately by eye with the help of a stroboscope, and then used a rapid camera at this location to make a more precise measurement. Fig. 4.11 shows lbas a function of flow rate

for three different viscosities. As one expects intuitively, the breakup length is an increasing function of both the flow rate and the viscosity.

4.6

Theoretical investigation

The theory in this section is mostly the works of Jens Eggers and Neil Ribe (75).

4.6.1

Dimensional analysis

The first step towards a more quantitative understanding is a dimensional analysis. The breakup length lb depends on the viscosity ν, the surface tension coefficient

γ, the density ρ, the flow rate Q, the gravitational acceleration g, and the nozzle radius r0. Buckingham’s Π-theorem (76) then implies Πb= fct(Πη, Πγ, Πr) where

Πb= lb  g Q2 1/5 , Πη = η γ(Qg 2₎1/5_, Πγ= γ ρ(Q 4_g3₎−1/5_{, Π} r= r0  g Q2 1/5 (4.31) where η = ρν.

v/v0

(z - z1)

l

Figure 4.7: Left: A jet of silicone oil with viscosity ν = 3500 cS falling at a volumetric rate Q = 0.29 ml s−1 from a nozzle with internal diameter d = 5 mm. Right: Axial velocity v(z) in a falling jet without surface tension (53), where lv =

(ν2/g)1/3 and z = z1 ≡ −(6νv0/g)1/2 is a virtual origin above the nozzle. Short

horizontal lines indicate the boundary between the viscosity-dominated (above) and inertia-dominated (below) parts of the jet.

4.6. Theoretical investigation

Figure 4.8: a0 versus u0 for all our experiments. The solid line is the marginal

Figure 4.9: Some pictures of the vacuum setup with height = 4.3 m. c) The observation part was darkened by a black curtain to allow better observations with stroboscope.

4.6. Theoretical investigation

Syringe Pump

Vacuum

Pump

2 m

5.5

_m

Figure 4.10: Schematic view of the falling viscous jet setup in vacuum. The vacuum motor is placed at the bottom to minimize unwanted upward air flow near the nozzle. The inner diameter is 19 cm. The upper portion in grey is opaque. The bottom portion is a 2 m cylinder of transparent plexi-glass.

Q (cm3s-1)

l

b

(m)

ν = 1000 cS

474 cS

50 cS

Figure 4.11: Breakup length L in the jetting regime as a function of flow rate Q for three different viscosities and the same nozzle diameter: 2r0= 2 mm. For clarity,

4.6. Theoretical investigation

4.6.2

WKB analysis

Because the structure of the jet varies slowly in the axial direction, the growth of perturbations can be treated using a WKB-type approach in which disturbances locally have the form of plane waves (61). The starting point is the equations governing plug flow in a slender vertical jet of viscous fluid (74):

∂tA + (Av)0 = 0 (4.32a)

ρA(∂tv + vv0) = 3η(Av0)0+ ρgA − γAκ0, (4.32b)

κ = 1

r(1 + r02₎1/2 −

r00

(1 + r02₎3/2. (4.32c)

where r(z, t) is the jet’s radius, A = πr2_{, v(z, t) is the axial (vertical) velocity, and κ}

is the mean curvature of the jet’s outer surface. Primes denote differentiation with respect to the distance z beneath the nozzle. Eqns. (4.32a) and (4.32b) express conservation of mass and momentum, respectively. The three terms on the right side of (4.32b) represent the viscous force that resists stretching, the weight of the fluid, and the surface tension force, respectively, all per unit length of the jet. In the absence of perturbations, the steady flow of the jet is governed by (4.32) with ∂t= 0. A general analytical solution of these equations was obtained by (53)

in the limit of no surface tension (γ = 0). The corresponding axial velocity v(z) is shown in Fig. 4.7 for two values of the normalized ejection speed ˆv0= v0/(3gν)1/3.

A clear distinction is evident between the jet’s upper part, where the weight of the fluid is balanced primarily by the viscous force that resists stretching, and its lower part where the weight is balanced by inertia. The boundary between the two is the point where the viscous and inertial terms in (4.32b) are equal, and occurs at a distance B(ν2_/g)1/3 _{from the nozzle, where B = B(ˆv}

0) ≤ 5.0. Because lb ≈

50-500(ν2/g)1/3 in our experiments, breakup always occurs in the inertia-dominated part of the jet. The prefactor B drops to zero for ˆv0 ≥ 1.219, meaning that the

weight is then balanced primarily (> 50 %) by inertia everywhere in the jet. The analytical solution of (53) can also be used to estimate the magnitude of the neglected surface tension term in (4.32b) relative to inertia and the viscous force. For the parameter values of our experiments, it turns out that surface tension is negligible in the inertia-dominated part of the jet, but not in the viscosity-dominated part. However, because the viscosity-viscosity-dominated part of the jet is very short compared to the breakup length, we are justified in using the solution of (53) as our base state.

To model the fluctuating environment surrounding the jet, we introduce small perturbations with different initial wavenumbers k0 at different points along the

jet, i.e., at different times t0 since the fluid element in question exited the nozzle.

Each of these perturbations will grow to O(1) amplitude at some distance zb(k0, t0)

from the nozzle, at which point the jet will break. We posit that the observed breakup length lb is the minimum value of the function zb(k0, t0).

Next we recall that the exponential growth rate σ of a small long-wavelength per-turbations of wavenumber k on the surface of a jet of constant radius r is (74)

σ =  _γ r3_ρ 1/2   ˆ_k2_{(1 − ˆk}2₎ 2 + 9 4Π 2 Ohkˆ 4 !1/2 −3 2ΠOh ˆ k2   (4.33)

where ˆk = kr and ΠOh= νpρ/γr is the Ohnesorge number. We now assume that

(4.33) applies at all points along the jet if r = r(z) and k = k(z) are interpreted as the local values of the jet radius and perturbation wavenumber, respectively. Because the base flow stretches fluid elements at a rate A−1A, r(z) and k(z) are˙ related to their values at the initial position z0 by k(z)r2(z) = k(z0)r2(z0).

To obtain the total growth of the perturbation, the growth rate (4.33) must be integrated along Lagrangian paths, taking into account the variation of r and k. The integrated growth rate is (5)

Z t t0 σ(k0, τ )dτ = π Q Z z z0 r2(ζ)σ (r(ζ), k(ζ)) dζ ≡ s(k0, z0, z), (4.34)

where k0= k(z0) and the time integral has been transformed to a spatial one using

dτ = dζ/v ≡ πr2_{dζ/Q. The total growth of perturbations is the exponential of}

(4.34).

Now suppose that breakup occurs when the quantity s(k0, z0, z) reaches a

crit-ical value scr. Because zb = zb(k0, z0), we have the implicit equation scr =

s(z0, k0, zb(z0, k0)) ≡ ¯s(z0, k0). Differentiating this equation with respect to z0

we obtain ∂s ∂z0 = ∂s ∂z0 + ∂s ∂zb ∂zb ∂z0 = 0. (4.35)

However, the condition of minimal breakup length requires ∂zb/∂z0 = 0, and so

(4.35) implies ∂s/∂z0= 0. This derivative can now be evaluated using the definition

(4.34) for s, noting that the dependence on z0 enters only through the lower limit

of integration. We thereby find that the optimal growth rate at the initial position z0 is σ(r(z0), k0) = 0, whence (4.33) implies that the optimal initial wavenumber

is k0= 1/r(z0). We can therefore write the integrated growth rate as a function of

z0and zb alone, viz.,

s(z0, r−1(z0), zb) ≡ S(z0, zb). (4.36)

Next, differentiate the equation ¯s(z0, k0) = scrwith respect to k0to obtain

∂s ∂k0 = ∂s ∂k0 + ∂s ∂zb ∂zb ∂k0 = 0. (4.37)

However, the condition of minimal breakup length requires ∂zb/∂k0 = 0, whence

(4.37) reduces to ∂s/∂k0= 0. apply the minimal breakup length condition ∂zb/∂k0=

4.6. Theoretical investigation

the most dangerous perturbation therefore reduces to solving the two simultaneous equations

S (z0, zb) = scr,

∂S ∂z0

(z0, zb) = 0 (4.38)

for z0and zb, which we did using standard Matlab routines.

Fig. 4.12 shows the theoretically predicted dimensionless breakup length Πb as a

function of Πη for scr = 8.86 and several combinations of values of Πγ and Πr.

Πb does not depend on Πr when Πr ≥ 1, indicating that the nozzle radius r0

is irrelevant because it is much smaller than the other characteristic lengthscales. However, Πb depends significantly on Πγ when Πη ≤ 3.

Fig. 4.13 compares the theoretically predicted breakup length for Πγ = 1.0, Πr=

1.5 and scr = 8.86 (solid line) with our experimental observations (circles). The

above values of Πγ and Πr are the (logarithmic) mean values of those parameters

for all 87 of our experiments. The value scr= 8.86 was chosen to give the best fit

of the theoretical prediction to the experimental data for log₁₀Πγ ≤ 1.3. The fit

is good for those values of Πγ (68 experiments), but poor for the 19 experiments

with larger values, all but three of which correspond to long jets with lb> 3 m. We

speculate that this is due to the extreme thinness (as low as r = 7 µm) attained by these long jets, which may render them more susceptible to perturbations than the theory predicts.

4.6.3

High-viscosity limit

A simple scaling argument yields an asymptotic expression for the breakup length in the high-viscosity limit ΠOh 1, where the Rayleigh-Plateau growth rate (4.33)

reduces to σ ∼ γ/(ηr). Now because most of the jet is in the inertia-dominated regime (free fall), its radius at a distance H below the nozzle is r ∼ Q2_/gH
1/4

. The growth rate at this distance is therefore σ ∼ (γ/η) gH/Q2
1/4

. Now the time required for a fluid element to fall through the distance H is τ ∼ (H/g)1/2. Breakup occurs at the distance H ≡ lbwhere the Rayleigh growth time σ−1becomes smaller

than the fall time, viz.,

lb= C

 gQ2_η4

γ4

1/3

or Πb= CΠ4/3η (4.39)

where C is a constant. C can be determined from our WKB analysis by ex-panding the integral expression for S in the limit ΠOh  1. This permits (4.38)

to be solved analytically for lb, yielding an expression of the form (4.39) with

C = (9scr)4/3/(2π2/3) ≡ 4.36s 4/3

cr . The dashed line in fig. 4.13 shows the

4

3

log

₁₀

Π

_η

log

10

Π

b

Π

γ

Π

r

1

2

3

4

−2

−1

0

1

2

Figure 4.12: Theoretically predicted breakup length Πb= lb g/Q2
1/5

as a func-tion of Πη = (η/γ)(Qg2)1/5 for scr = 8.86 and different combinations of values of

4.6. Theoretical investigation

2

3

4

−1

0

1

2

log

₁₀

Π

_η

log

10

Π

b

log

₁₀

Π

_γ

Figure 4.13: Experimentally observed dimensionless breakup lengths Πbas a

func-tion of Πη. The solid curve is the theoretical prediction for Πγ = 1.0, Πr = 1.5

and scr= 8.86. Colors indicate the values of Πγ for the different experiments. The

4.7

conclusion

The resolution of the paradox pointed out in the introduction is now clear: viscosity plays completely independent roles in the axial momentum balance of the steady basic state and in the growth of perturbations about that state. The analytical solution of (53) for the basic state shows that viscous forces are negligible in the inertia-dominated part of the jet z  (ν2_/g)1/3_{, which represents > 90% of the}

jet’s length in most of our experiments. However, this does not imply that the effect of viscosity can be neglected in the expression (4.33) for the Rayleigh-Plateau growth rate. That expression has two limits depending on the Ohnesorge number ΠOh = νpρ/γr: a viscosity-independent limit σ ∼ (γ/ρr3)1/2 for ΠOh  1, and

a (less familiar) viscosity-dominated limit σ ∼ γ/ηr for ΠOh  1. To determine

which limit is relevant for our experiments, we used the solution of (53) to calculate ΠOh(r = rb) ≡ ΠbOh for each experiment, where rbis the jet radius at the distance

z = zbwhere the jet breaks up. We thereby find that ΠbOh∈ [0.65, 2160], and that

66 of our 87 experiments have Πb

Oh > 10. It is therefore not surprising that the