Theoretical analysis of linear stability of electrified jets flowing at high velocity inside a coaxial electrode

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Abstract

In this article we develop a temporal linear stability analysis of a circular electrified jet flowing inside a cylindrical coaxial electrode. The problem of a high-velocity jet going in a gaseous atmosphere is examined and we analyze the results in order to bring out the influence of the electrification, the surface tension, the velocity and other parameters on the stability of the jet. From this theory we finally state the changes in the breakup phenomena that are expected to be observed when electrifying these jets.

Introduction

Usually, liquid jets appear in industrial processes where it is desired to increase the surface/volume ratio of the liquid (combustion, chemical reactors, etc.) or to cover a target with a liquid in droplets form (painting, pesticide applications, ink jets printers, etc). The electrification of the jets has enabled a higher control of these processes as the electric forces may change the stability of the jet or the trajectory of the charged droplets created by the jet disintegration.

In nonelectrified jets, we can observe different regimes when increasing the jet velocity (Rayleigh regime, wind regimes and atomization) [1]. Though these regimes are quite different from each other, from all of them we can have a rough idea of the breakup process of the liquid jet into droplets by means of a linear stability analysis 2, 3, 4, 5, 6, 7, 8.

The stability analysis is concerned with the development in space and time of infinitesimal perturbations in a given basic flow. As a function of the kind of perturbation imposed to the flow, two different problems, the spatial and the temporal ones, have been usually considered.

The spatial problem considers that the perturbation is applied in a certain region and it follows a given function in time. A special case of this problem is the signaling problem where a perturbation is periodically forced at a specific location. The development in Fourier series of this kind of perturbation gives components with real values of frequencies ω and their variation in space is obtained with the complex wave numbers k(ω)=kr(ω)+iki(ω) deduced from the dispersion equation D(k,ω)=0. The temporal problem considers that the perturbation is applied to the flow in a certain region of the space at time equal to zero. The development in Fourier series of this kind of perturbation gives components with real values of the wave number k and their variation in time is obtained with the complex frequencies ω(k)=ωr(k)+iωi(k), roots of the dispersion equation. In general, both analyses give the same information only when the conditions specified in Gaster’s theorem are verified [9]. Otherwise, in most cases we cannot extrapolate results from one analysis to the other.

The spatial analysis has usually been used to describe the response or “receptivity of the flow” to different excitation frequencies. The downstream development of the perturbation is considered as a set of spatially growing waves of various frequencies. With this analysis, we can specify which are the amplifying waves, that is to say those whose amplitude increase from the source of perturbation, and which are the evanescent ones, those whose amplitude decrease. Among others, Melcher [10], Crowley 11, 12, 13 and Atten 14, 15, have undertaken this analysis in electrified jets with the excitation frequency imposed by electric forces through coaxial electrodes.

A large part of previous research has also been devoted to the temporal theory, and one of the first work has been carried out by Rayleigh [16]. Other researchers starting from different hypotheses have studied the effect on the stability of the jet of a radial electric field 17, 18, 19, 20, 21, 22, 23 of an axial electric field 24, 25 or both 26, 27.

The temporal analysis appears to be quite important as it enables to predict if the flow is linearly stable (temporal growth rate positive) and to establish the interval of frequencies of the waves susceptible of being amplified in a signaling problem. Also, if there is no preferential excitation of the jet, the linear temporal stability appears to be a good tool to give the first approach of the breakup phenomena of an electrified liquid jet. For instance, the wavelength of the most unstable wave (the one with the highest growth rate) can be associated with the mean droplet diameter produced by the disintegration of the jet.

To our knowledge, most of the previous research has dealt with jets issuing from a nozzle at the Rayleigh regime (low velocity). In this situation a hypothesis usually accepted is that the effect of the surrounding atmosphere on the stability of the jet can be neglected. At high velocities this hypothesis is very strong, as the breakup phenomenon differs from the Rayleigh regime mainly because of the interaction of the liquid jet with the surrounding gas.

The stability analysis of high-velocity jets shows that the flow is unstable for waves with short wavelengths compared to jet radius and for waves with a large growth rate compared to the ones of Rayleigh regimes. Hence, from a mathematical point of view some simplifications for long waves can no longer be valid for these regimes and most of the salient features of capillary instabilities (Rayleigh regime) are not preserved. Also, as the perturbation has a large growth rate the assumption of a liquid jet with the surface at constant potential must be analyzed carefully because of the rapid movement of the interface.

The effect of finite electrical conductivity on the instability of electrified jets has been studied by different authors 27, 28, 29. These works shed light on the problems of low-velocity jets but it does not seem appropriate to extend their results to high-velocity jets.

In this work, we do not undertake this study and we limit ourselves to two simple ideal situations, the surface of the jet being an equipotential and the surface of the jet moving so rapidly that charge position on the jet surface is governed only by fluid motion.

As a summary, in this article we propose to undertake a linear temporal analysis of an electrified liquid jet flowing at high velocity inside a cylindrical coaxial outer electrode. We look for predictions in the two extreme electrical situations above cited and from these theoretical results we propose to obtain some insight on the breakup process of the jet.

Section snippets

Problem description and assumptions

Let us consider a liquid jet flowing vertically downwards out from an injector and into a gas at room pressure. Depending on the velocity of the liquid, one obtains different kinds of jets. The one of interest here corresponds to the second wind regime or to the regime of atomization, that is to say for very high velocity (about 100 m/s). For these regimes the aspect of the jet looks like a pulverization shaped as a cone composed of sparse droplets in most of its volume, except in the region of

The results

In the next section, we determine the value of the corresponding ω for a given real wave number k and a given mode number n. If ωr is positive, the jet is unstable and the greater ωr is, the more unstable the jet is. On the other hand, if ωr is negative or null then the jet is stable. As regards ωi, we will just mention that ∂ωi∂k is linked to the propagation velocity of the perturbation. In order to give an idea of the variation with ak we show some results in Section 3.1.

To better understand

Conclusion

We solved the dispersion equation for different values of the parameters to determine their influence on the stability of an electrified jet flowing at high velocity in a gaseous atmosphere.

The first important thing that we showed is that the electrification acts on the stability of a jet in a different way depending on the electrical state of the surface of the jet. The response is not the same whether the surface is considered equipotential or nonequipotential.

For the equipotential situation,

Acknowledgements

This paper relates a part of G. Artana’s Doctoral Thesis which Dr. H. Romat and Prof. G. Touchard directed together at the University of Poitiers with the help of Dr. P. Atten to whom our thanks are due.

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    1

    Now at: Conicet-Dep. de Mecanica Aplicada, Facultad de Ingenieria, Universidad de Buenos Aires, Paseo Colon 850 (1063), Buenos Aires, Argentine.

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