Investigation of the Structure of Waves Generated by a δ-perturbation of the Surface of a Capillary Jet Текст научной статьи по специальности «Физика»

Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 3, pp. 367-378. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220303

NONLINEAR PHYSICS AND MECHANICS

MSC 2010: 76E30

Investigation of the Structure of Waves Generated by a 6-perturbation of the Surface of a Capillary Jet

A. A. Safronov

The wave capillary flow of the surface of an inviscid capillary jet, initiated by a single ¿-perturbation of its surface, is studied. It is shown that the wave pattern has a complex structure. The perturbation generates both fast traveling damped waves and a structure of nonpropa-gating exponentially growing waves. The structure of self-similar traveling waves is investigated. It is shown that there are three independent families of such self-similar solutions. The characteristics of the structure of nonpropagating exponentially growing waves are calculated. The characteristic time of formation of such a structure is determined.

Keywords: instability, capillary flow, nonviscous jet

1. Introduction

The liquid jet is unstable with respect to long-wave perturbations of its surface. This phenomenon is used to generate droplet flows in many technical applications, including chemical technology [1, 2], heat removal systems [3-6], inkjet printers [7-9], engine nozzles [10-13], etc. For theoretical modeling of the process of capillary disintegration of jets into drops, approximate quasi-one-dimensional equations are used, obtained by asymptotic expansion of the system of Navier-Stokes equations. The development of long-wavelength (wavelength exceeds the jet radius) axisymmetric perturbations in a free inviscid jet can be described by two dimensionless equations relating the jet radius R and the velocity of matter in it u [7] (the derivation of the equations from the Navier-Stokes system is given in the Supplementary Information):

dtu + u dxu = —j dxR + dxxxR, dtR + u dxR = — ^R dxu,

Received December 20, 2021 Accepted May 13, 2022

Andrey A. Safronov a.a.safr@yandex.ru

Keldysh Research Center

ul. Onezhskaya 8, Moscow, 125438, Russia

where x is the axial coordinate and t is the time. In this case, the radius of the unperturbed jet

/ pri\ 1/2

r0 is used as the length scale, i0 = ( ^ ) use<^ as scale> where p and a are the

density and surface tension of the liquid, and the velocity is the ratio v0 = j1: v0 = •

Below, we assume that the perturbations of the jet surface are small, and it suffices to use the linear approximation to describe them. Assuming that R = 1 + h, we get

dt u — dxh + dxxxh,

i (i.i) dth = ~2dxu-

The dispersion relation of this system for waves h, u ~ exp(ikx — iwt) has the form

1

w

(k) = —=kVk2 - 1. (1.2)

It follows from the dispersion relation that waves with k > 1 are traveling and do not change their amplitude during propagation. On the contrary, waves with k < 1 are nonpropagating and grow exponentially with time (see Supplementary Information).

In early theoretical studies, the solution of the problem of capillary disintegration of a jet was carried out using the absolute instability model [8]. In this case, the growth of periodic perturbations with k < 1 was studied, and it was assumed that the perturbation amplitude depends only on time, but not on the coordinate. The results of such an analysis correctly describe the regularities of capillary decay in the case when the jet outflow velocity V from the capillary nozzle significantly exceeds the characteristic capillary velocity v0:

M = vJ^> 10.

a

For a theoretical description of the regularities of the decay of a jet flowing out at a lower velocity, it is necessary to use the theory of convective instability [3]. In this case, it is assumed that perturbations of the jet surface in one way or another (with the help of acoustic or electromagnetic fields, nozzle vibration, periodic change in the fluid outflow rate, etc.) are generated in a small vicinity of the capillary nozzle. The theory of convective instability makes it possible to simulate the capillary breakup of a jet with M ^ 5.

Experiments on the capillary breakup of slow jets with M < 5 have shown that the regularities of their breakup can be described using models of neither absolute nor convective jet instability [10-15]. This is explained by the fact that, at small M, traveling capillary waves with a dimensionless wave number k > 1 begin to play a key role in the process of capillary decay. Such waves are generated as a result of detachment of drop nuclei from the jet [10-13]. The resulting waves move against the flow to the point where the fluid exits the capillary nozzle. When reflected due to the Doppler effect, the wave number decreases. Reflected waves with k < 1 grow according to the laws described by the theory of convective jet instability. The development of perturbations leads to the disintegration of the jet into drops, which results in the formation of new traveling capillary waves.

A theoretical description of this mechanism of the capillary disintegration of a jet requires a model for the generation of waves that arise when the jet breaks. The wave structure arising on the jet surface is calculated using the Green's function method. However, in the works [10-13] devoted to the study of capillary decay of slow jets, the Green's function was calculated for

a simplified system of equations: the term dhx, which describes the influence of the jet radius on the capillary pressure, is excluded from the first equation of system (1.1). In this case, the dispersion relation (1.2) is simplified and takes the form w ~ k2. The Green's function calculated in this approximation does not describe the perturbations growing with time observed in the experiment [14]. At the same time, in [15], the Green's function was calculated for the complete system of equations describing the behavior of a capillary jet as applied to the problem of capillary meniscus compression. However, the calculation was carried out only for the near field of the generated waves and does not describe solutions propagating and growing with time. At the same time, growing capillary waves generated by ¿-perturbation of the jet surface are in some cases the key process that determines the patterns of drop formation [14].

In this paper, we study the wave structure that appears after a ¿-perturbation of the surface of an inviscid capillary jet.

2. Formulation of the problem

It is considered that the dynamics of capillary waves is described by the dispersion relation (1.2) obtained from Eq. (1.1).

An infinite homogeneous jet is considered. The initial condition is

h(x, t = 0) = 5(x), u(x, t = 0) = 0. (2.1)

The problem of developing the initial perturbation was solved in the Fourier representation. Since the initial perturbation of the surface has the form of a ¿-momentum, the formal solution of the problem can be represented as

<x

h(x't]=S e"dk■

0

The dispersion relation (1.2) describes traveling waves of constant amplitude for k > 1 and nonpropagating growing waves for k < 1. Taking this into account, we divide the last relation into two parts:

1 <x

h(x, t) = j cos(kx)ef(k)t dk + j eikx-iu(k)t dk, (2.2)

01

where

1

f(k) = - k2. (2.3)

The first term in relation (2.2) describes the growing nonpropagating waves, the second describes the traveling ones.

3. Traveling waves with k > 1

We will calculate the contribution due to traveling waves with k > 1 using the stationary phase method [16]. In this case, the Fourier integral takes the form

<x

h(x, t) = J e-ix(k)t dk, (3.1)

1

where x(k) = oj(k) — kj = oj(k) — kc.

For large values of t, the main contribution to the integral (3.1) comes from the neighborhood of the stationary points of the relation

x' (k*) = u' (k*) - c = 0.

The expression for k* has the form

k*l:2 = -^2 + C2±v/^34).

The value of k* is valid when c ^ 2.

It is easy to see that, when c » 2, the quantity k* tends to the asymptotic values: k\ ^72, k2* ~ 1. The first term of the asymptotic expansion of the integral (3.1) with respect to the velocity c » 2 has the form

h(x, t) — h1(x, t) + h2(x, t),

ma v/tt I cx cH \/2 7r \ i /a \/tt I y/2 x2 7t \ h^x, t) = 2 ' cos —=---= + i—t - - = 2 ' cos ^—t +

r---7= + ~r^ ~ T = 2 ~r cos I ~rL 1--P7" ~ * I' to

s/2 2s/2 4 4 / v/i \ 4 2\/2i 4 J (3.2)

7T / 7T t \ ,- t ( 7T t2

h9(x, t) = \ —r- cos + —--= s/TT-j- cos [ x + — —

2V ' ; V c t v 4 2cJ x3/2 V 4

Consider the case when c œ 2. Let us introduce a parameter £ such that

c = 2 + V-6

Then the first term of the asymptotic expansion of the integral (3.1) obtained by the stationary phase method has the form

h(x, t) = y^cos - £) j = 7^(6t(x - 2t))~1^ cos - 0 j . (3.3)

It can be seen that the perturbation amplitude rapidly increases when approaching the point x = 2t.

Relation (3.3) obtained by the stationary phase method does not describe the solution of

1 /2

the problem for c = 2, which corresponds to the wavenumber kA = (f) . This is because at this point the second derivative of the phase function x''(k^) = u''(kA) = 0. To calculate the solution, it is necessary to take into account the value of the third derivative of the phase function x(k). In this case, a wave structure of the Airy wave type [17] is formed, the shape of which is described by the relation

=^Ai s - 0i2/3) v I (* -1) ' (3-4)

where Ai(z) = ^ f cos (zp + |p2) dp is the Airy function.

— <x

At t ^ <x>, the Airy wave is the dominant wave with k > 1. It follows from relations (3.2) and (3.4) that the ratio of the amplitude of the Airy wave with c — 2 to the amplitude of

the waves with c » 2 increases with time in proportion to t1/6. However, as follows from the results obtained below, the solution of problem (1.1), (2.1) is of greatest practical interest at t ~ 15-20. At such times, the amplitude of the Airy wave (3.4) turns out to be comparable with the amplitude of the waves described by relation (3.2).

As noted above, in [10-13], the calculation of the shape of the jet surface was carried out for a simplified system of equations, for which the dispersion relation has the form w ~ k2. In this case, to calculate the shape of the jet surface, the following relation was obtained [10-13]:

h(x, t) ~ cos

Experiments show that this approximation can be used for some practical applications [10-13]. At the same time, one of the components of the solution obtained in this work, relation (3.2) for h1, is consistent with the results obtained in [10-13] and refines them.

0.3 0.2 0.1 0

-0.1 -0.2 -0.3

r hl

3C

40

Fig. 1. Dependence h1(x) for t = 15

Figure 1 shows the graph of the function h1(x) for t = 15. At x > 50, where the assumptions in which relation (3.2) is obtained are applicable, there is a modulation of the function h1 (x).

4. Growing waves with k < 1

The calculation of the contribution to the solution, determined by growing waves with k < 1, was carried out for the case when t is large enough to assume that the main contribution to the integral

1

h(x, t) = j cos(kx)ef (k)t dk (4.1)

0

comes from a small neighborhood of the point k*, defined by the relation

f ' (k*)=0.

This equation has one solution: k* =2 1/2. The Taylor series expansion of the function f in the vicinity of the point k* gives

f(k) = -^=- V2(k- k*)2 -2(k- *•)» - (k - *•)< + O ((fc - F)5). (4.2)

In the first approximation, the integral (4.1) can be estimated using only information about the second derivative f:

h(x, t) & / cos(kx)e

dk.

(4.3)

Therefore,

h(x, t)

21/4

7T -L=t 1 / x e2v'2 —cos

(4.4)

To calculate the higher terms of the expansion, we represent the integral (4.1) in the following form, using the notation ( = k — k*:

h{x, t) œ e2V2

3-V2C 2i

dk.

(4.5)

We expand the expression under the integral in square brackets into a Taylor series in (. The zero order of expansion corresponds to expression (4.4). The first and third expansion orders give zero contribution to the integral. The solution up to the fourth order of expansion has the form

1 x2

h4 (x, t)

21/4

7T -L=t 1

e2v'2 —:

1

t

cos

(4.6)

This relation allows us to estimate the minimum value of t at which estimates (4.4) and (4.6) make sense:

1 2 t > —=x .

The value of the "propagation velocity" of a perturbation in the form (4.6) can be estimated as follows:

x = 2- 21/4 • yft. (4.7)

Based on the last relations, assuming that, x = |r, we can estimate the formation time of an exponentially growing solution t as follows:

2n2 ~ 14.

(4.8)

5. Discussion

The wave pattern arising in a capillary jet after ¿-perturbation of its surface has a complex structure. Far from the point of disturbance propagate waves traveling at a speed c> 2, described by relation (3.2), whose amplitude A decreases with time according to the law At ~ t-1/2. They

are followed by an Airy wave with a wave number kA = (f)1^2 with a speed c = 2, whose _RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2022, 18(3), 367 378_

amplitude decreases with time according to the law Aa ~ t-1/3. For smaller values of x, the wave pattern is composed of waves with a spatial frequency k* — 2-1/2, the amplitude of which increases with time according to the law (4.5). In this case, the formation of an exponentially growing structure in the solution of the system of equations (1.1) takes a dimensionless time of the order of t ~ 14 starting from the moment when the S-impact was exerted on the jet surface.

The results presented above do not allow us to describe the wave structure in the interval between the Airy wave with speed c — 2 and perturbations with wave number k* — 2-1/2, which propagate by diffusion according to the law (4.7). In order to study the structure of waves in this interval, a numerical solution of the problem was carried out. To increase the stability of the numerical solution, viscosity was added to the model equations. The following system was solved (see the derivation in the Supplementary Information):

dt u — dxh + dxxxh + 3Oh dxxu, dth = ~^dxu,

where Oh is the Ohnesorge number, an analogue of the Reynolds number for capillary waves, which in the numerical calculation was Oh — 0.05 (a characteristic value for a jet flowing from the capillary hole of the print head of an inkjet printer). The solution was carried out by the finite difference method using an explicit calculation scheme of the first order of accuracy. The delta function in the initial condition was set as a n-shaped pulse at the origin.

Figure 2 presents the numerical solution of this system of equations with the initial condition (2.1) at various times (the calculation was carried out in the region of space [—100^; 100^]; at the boundaries of the interval, the amplitude of perturbations was considered equal to zero). Due to the presence of viscosity, the form of the far field of the wave structure changes. However, the patterns of development of growing perturbations with a wavenumber close to 1 are preserved.

30 40

i= 14.9

Fig. 2. Dependence h(x) at different times

From Fig. 2 one can see that at t < 14 the wave pattern changes rapidly. In this case, the amplitude of the waves turns out to be much less than unity. And at t > 14, perturbations arise

in the solution that grow exponentially with time. This result agrees well with estimate (4.8) obtained analytically.

An analysis of the numerical calculation results shows that near the origin, the wave number of growing perturbations is k* = 2-1/2. And in the interval between the Airy wave with the wave number kA = (f)1^2, whose velocity in the numerical experiment with a good approximation was equal to two, and perturbations with k* = 2-1/2, a transitional wave structure is formed. The local wave number, determined from the distance between neighboring extrema, smoothly decreases from kA to k*. And the amplitude of the perturbation grows with a decrease in the wave number. We can propose the following relation, which locally describes the structure of the transition layer:

h(x) = const • cos(kx)e'

at—bx

(5.1)

By substitution, one can make sure that relation (5.1) is the exact solution of the system of equations (1.1) in the case when

1

b =

1 + 3k 1

Vl + 3 k

VkV 1 + k (4k2 - 1), Vk3 + 3fc2 -k- 1.

(5.2)

0.7

0.8

(b)

0.9

Fig. 3. Dependences a(k), b(k), cp(k) and c (k)

Relations (5.1)—(5.2), which describe the structure of growing waves, have a physical meaning for nonpropagating waves with k < 1. Figure 3 shows the dependences a(k), b(k), as well as the phase cp(k) = | and group cg(k) = || perturbation velocities (5.1). It can be seen that, there is some value of the wavenumber kgp for which cp(kgp) = cg(kgp). If k > kgp, then for waves of the form (5.1) cg > cp. For k < kgp — cg < cp. That is, in the wave pattern of long-wave growing perturbations, shorter-wave perturbations with a wave number close to unity "break away" from longer-wave ones with a wave number close to k* = 2-1/2 and move in the form of separate wave crests k in the direction of Airy wave, "fading" as it propagates. This result agrees with numerical calculations.

a

From the presented results, it follows that growing perturbations propagate in a capillary jet at a speed not less than cgr = 1.5 (due to various nonlinear effects, which were not taken into account in this work, the propagation velocity of such perturbations can increase significantly). At the same time, the dimensionless velocity of the terminal drop formed at the edge of the jet after its break is u < 1. Thus, the appearance of a propagating growing structure of waves in a capillary jet can in some cases be the determining mechanism of its breakup into drops.

6. Supplementary Information

An axisymmetric jet of a viscous liquid is considered (Fig. 4). It is assumed that the fluid flow in the jet is not bending (the jet axis is straight). It is believed that the jet is in a vacuum, and body forces do not act on it.

Fig. 4. Axisymmetric jet

Axisymmetric perturbations in the jet are considered. The equations of the Navier-Stokes system in cylindrical coordinates (the x-axis coincides with the jet axis) are written as follows:

dtvr + vr drvr + vT d„vr = — + (drVr + dlvr + ^r ^

ytLr ' 1 r Ur 1 r ' X X 1 r ' 1 Ur 1 r ' UX 1 r ' 2 In

r \ ' ' J p

dtvx + vr drvx + vx dxvx = — + (d2vx + dxvx + ^r%x ' ^

r J p

v

drvr + dxvx + — = 0,

where t is the time, p is the pressure, ¡i and p are the dynamic viscosity and density of the fluid, and vr and vx are the radial and axial components of the velocity vector.

The dynamic boundary condition at the interface R(x, t) for the normal and tangential components of the stress tensor [19] is

1 + (dx R)

2 dxRdrvr + (drvx + dxvr) 1 — (dxR)2 — 2 dxvx dxR^j

= 0,

r=R

where a is the surface tension of the liquid, and tt— =-" „ and 77- =--0 1/0 are

h ' rc1 (l+dxR2) /2 Rc2 h(l+dxR2)1/2

the principal values of jet surface curvature. The kinetic boundary condition is

dtR + vx dx R = vr\r=R .

Of practical interest are perturbations of the jet surface with a wavelength significantly exceeding its radius (the jet is unstable when the perturbation wavelength is greater than 2nr0).

Using this, we expand the system of Navier-Stokes equations into a power series with respect to the small parameter r. Taking into account the axial symmetry of the problem and the continuity equation, the values vr, vx and p take the form

vx(x, r) = v0 + v2r2 + v4r4 + o(r4),

1 1 3 3

vr(x, t) = --rdxv0 - -r dxv2 + o{r ),

p(x, r) = p0 + p2r2 + o(r2),

where o is the designation of o small. Because of the axial symmetry, the axial fluid velocity depends on the radius only in the second order of expansion.

To solve the problem, the first order of expansion of unknown functions in a power series along the radius is used:

vx(x, r) = V0,

vr(x, t) = -~rdxv0, p(x, r) = Po.

In this approximation, the system of Navier-Stokes equations with boundary conditions is reduced to two equations:

%v0 + VndrVn — — dT ( -—-q~777--ttt I H ~—~—„o

4 0 0 x 0 P x\(1 + dxR2f/2 h(1 + dxR2)1/2l P R2

dtR + v0dxR = ~^Rdxv 0. To pass to dimensionless variables, we choose the radius of the undisturbed jet as the unit

of length and time r0, and t0 = \J dimensionless velocity u = In the new variables, the system takes the form [7]

dxxR 1 V, dx R2 dxu)

dtu + ucLu = cL ----+ 3 Qh-

(1 + dx r2)3/2 h(1 + dx R2)1/2J R2

i

dtR + udxR = --R dxu,

where Oh = is the Ohnesorge number, which describes the effect of viscous forces on

v 0p

capillary flow.

If the perturbation amplitude is sufficiently small (less than half the radius of the unperturbed jet r0), the following approximation of the model equations gives the accuracy sufficient for practical applications:

1 dx (R2 dxu)

xx

dtu + u dTu = —r dxR + dxxxR + 3Oh R2

dtR + udrR = — \R dru.

JtUl ' UiUxu, uxlv 1 ^xxxJ-v 1 j-^2

1

T

Assuming that R = 1 + h,

dtu = dx h + dxxxh + 3Ohdxxu, dth = ~dxu.

Assuming that the perturbation of the jet surface from the state h = 0, u = 0 is small and proportional to exp(ikx — iwt), we obtain the following dispersion relation:

u(k) = -loh • k2i ± -Oh • k2J-^\ - 1. V ' 2 2 V 9 Oh2k2

One can see from the dispersion relation that, for k > 1, the waves are traveling and damping. And for k < 1, the waves are nonpropagating and growing at a speed

The growth of perturbations occurs due to the fact that the area and, after it, the surface energy of the perturbed jet is smaller than the area of the jet in the unperturbed state — in the case when the perturbation period exceeds the value A* = 2nr0. As a result of the growth of perturbations, the jet breaks up into drops.

In some cases important for applications, it is possible to neglect the viscosity of the liquid. So, for a water jet with a diameter of 1 mm, Oh ~ 10_3, and for a jet with a diameter of 10 ¡m, Oh ~ 10"2. It follows from the dispersion relation that in these cases, at times of the order of several tens of t0, the attenuation of traveling capillary waves can be neglected. Then, to describe small capillary waves, it is possible to use the approximate model system of equations

dt u — dxh + dxxxh,

dth = ~dxu

and the dispersion relation uj(k) = -^kyjk2 — 1.

Conflict of interest

The author declares that he has no conflicts of interest.

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