Electrohydrodynamic instability of confined viscoelastic liquid jets

https://doi.org/10.1016/j.jnnfm.2020.104453Get rights and content

Highlights

  • Linear stability analysis for a viscoelastic jet under radial electric field.

  • Stability of m=0 and 1 modes are studied under creeping-flow limit.

  • For m=0, critical electrode to jet radius ratio is not affected by elasticity.

  • The growth rate diverges above a critical Deborah number for Maxwell fluid.

  • For m=1, the growth rates are unbounded even in the Newtonian limit.

Abstract

Linear stability analysis is performed to study the electrohydrodynamic instability of viscoelastic jets subjected to axisymmetric (m=0) and first non-axisymmetric (m=1) perturbations in the creeping-flow limit. The viscoelastic liquid jet is under the influence of a radially applied electric field induced by a concentrically placed electrode located at a finite gap width from the jet. The leaky dielectric model is used to account for the finite conductivity of the fluid. The gap between the liquid jet and the electrode is assumed to be occupied by a hydrodynamically passive gas. The influence of the applied electric potential, electrode width, electrical properties, fluid elasticity, and solvent viscosity on the stability of the jet is analyzed comprehensively. For m=0 mode, the electric field has a dual effect on the stability of a Newtonian jet above a critical electrode width to jet radius ratio (Rcr). Here, the dual effect means that the electric field has a stabilizing effect for low wavenumbers (k) and a destabilizing effect for higher k. However, for R<Rcr, the electric field has a uniformly destabilizing effect, and the dual nature disappears. In the limit of perfectly conducting Newtonian jet, Rcr = 2.7. The maximum growth rate increases with an increase in Deborah number (De), which is a dimensionless relaxation time of the fluid, but the range of unstable wavenumbers remains unaffected due to an increase in De. For m=0 mode, there exists a critical Deborah number (Decr), for given values of other physical parameters, above which the growth rate diverges. This behavior is similar to the previous results for the planar geometry and is caused due to the neglect of inertia. As the conductivity of the liquid increases, the Decr decreases and reaches an asymptotic value in the limit of a perfect conductor. The inclusion of elasticity has no effect on the Rcr value, thus suggesting that the dual nature of the electric field is independent of fluid elasticity. The singularity in the growth rate is shown to be mitigated by introducing solvent viscous stresses into the model. Thus, the present study shows that finite electrode distance and elasticity have important consequences on the stability of the liquid jet for the axisymmetric mode. For m=1 mode, the growth rate is singular at low wavenumbers which is due to the neglect of inertia of the system. For the Newtonian case, the range of wavenumbers where the growth rates are singular increases with either increasing the electric potential or decreasing the electrode width. Inclusion of elasticity increases the range of wavenumbers where the growth rates are singular, whereas the range of unstable wavenumbers remains unchanged. Therefore, the neglect of inertia gives rise to non-physical growth rates even in the Newtonian flow limit for m=1 mode, compared to m=0 mode where the singularity of growth rate was observed for the Maxwell fluid above Decr.

Introduction

Modulation of free-flowing liquid jets using the influence of an applied electric field is important to a number of applications. Electro-spinning is used for the production of micro- and nano-scale fibers, while electro-spraying and ink-jet printing exploit the phenomenon of disintegration of jets under the influence of electric field [1], [2], [3]. Much research has been carried out to understand the instability characteristics of Newtonian jets under the influence of an applied electric field (for review, see [4], [5], [6], [7]). With the advances being made in polymer synthesis, manufacturing of polymeric fibers and droplets has gained popularity and use over the last few decades [8], [9], [10]. However, the role of electrical forces on the instability of free jet for polymeric materials has not been explored extensively in the existing literature. In this section, a literature review of the previous works carried out for neutral (in the absence of electric fields) and electrified jets (under the influence of axial and radial electric fields) are provided, and finally, the objective of the present study will be discussed.

The study of capillary instability arguably began in the 19th century with the work of Rayleigh [11], who analyzed the stability of an inviscid liquid jet flowing through a vacuum. Rayleigh’s work was later extended by considering the outside atmosphere to be an inviscid fluid [12]. Tomotika [13] studied the capillary instability of Newtonian viscous jet including the effect of another surrounding liquid. Later work done by Yuen [14] explored the nonlinear capillary instability behavior of Newtonian jet to study the effect of finite amplitude disturbances. An extensive review of the dynamic and breakup of neutral Newtonian liquid jets can be found in Eggers and Villermaux [15]. For viscoelastic jets, there have been many studies on the instability characteristics in the absence of electric field. Middleman [16] carried out a linear stability analysis of viscoelastic jet ejected into an inviscid fluid. This work was analogous to the investigation carried out by Weber [12] for Newtonian jet by replacing the simple stress–strain rate relation for Newtonian fluid with a linear viscoelastic model [17]. Kroesser and Middleman [18] extended the work of Middleman [16] by studying the breakup of a viscoelastic jet at low horizontal speed. They presented theoretical predictions on the breakup lengths for different molecular weights and concentrations of polymers. Goldin et al. [19] found that the elastic effect of the polymeric material enhances the instability for axisymmetric perturbations. Goren and Gottlieb [20], using linear stability analysis, found that the elasticity does not make the system less stable for every condition, and if the axial tension is not zero there is a possibility that the elasticity will have a stabilizing role. Brenn et al. [21] included the effect of outside inviscid gas moving with constant axial velocity. It was shown that increasing gas to liquid density ratio and the relative velocity of jet with respect to air promotes interfacial instability for axisymmetric perturbations. Liao et al. [22] carried out linear stability analysis for the viscoelastic jets in the presence of a swirling air stream and found that air swirl suppresses the instability for both axisymmetric and non-axisymmetric modes.

The investigation of the stability of charged liquid jets started with the early work of Basset [23], where the effect of the radial electric field on conducting inviscid jets was studied. Taylor [24], after correcting a mistake in the work by Basset [23], observed good agreement between the experimental results and calculations conducted by Basset [23]. Schneider et al. [25] studied the instability and breakup of charged Newtonian jets and the mechanism of droplet formation. Theoretical calculations were performed to analyze the role of various properties of the droplets in terms of jet parameters and the applied electric field. Huebner and Chu [26] investigated the stability of a charged jet for axisymmetric and non-axisymmetric perturbations by considering a finite electrode distance in their formulation. Later, Setiawan and Heister [27] extended this analysis, but only for axisymmetric perturbations, by studying the nonlinear evolution of the perturbations. Artana et al. [28] performed linear stability analysis of a circular electrified inviscid jet flowing inside a cylindrical coaxial electrode. The problem of a high-velocity jet flowing in a gaseous atmosphere was examined and the influence of electrification, surface tension, velocity, and other parameters on the stability of the jet was studied. López-Herrera et al. [29] studied temporal stability for axisymmetric perturbations in imperfectly conducting liquid jets. The outside air was considered to be inviscid with constant axial velocity. The effect of conductivity, permittivity, and electrode width on the stability were discussed for axisymmetric perturbations. They observed a so-called ‘dual effect’ of the electric field, where the electric field is stabilizing for the low wavenumbers (k) and destabilizing for high k. Li et al. [30] conducted stability analysis of an electrified coaxial jet under the influence of the radial electric field from a coaxial electrode. They studied the characteristics of axisymmetric and helical modes for varying parameters which include electrostatic force, diameter ratios, density ratio, etc. They did not observe the dual nature of the electric field and found that reducing the confinement always increases the growth rate. Collins et al. [31] studied the stability and breakup of electrified perfectly conducting jets under the influence of the radial electric field. They developed two algorithms to investigate the dynamics of jet breakup starting from a quiescent jet to a pinch-off state. These algorithms were then also compared with linear stability results for early times. Li et al. [32] carried out linear stability analysis of a viscous coaxial jet in the presence of a radial electric field. Both axisymmetric and non-axisymmetric perturbations were analyzed using the Chebyshev spectral collocation method. The outside liquid was taken to be leaky dielectric and the inner to be perfect dielectric. All the non-axisymmetric modes were found to be stable and para-sinuous mode was most unstable in the absence of electric field. The helical mode was found to be most unstable at sufficiently high electric fields. Wang et al. [33] conducted long-wave nonlinear stability analysis for axisymmetric perturbations of viscous conducting liquid jets in the presence of radial electric field originating from the concentrically placed electrode of a given radius. Wang [34] studied the instability behavior of viscous jet, surrounded by another viscous jet for axisymmetric perturbations in the presence of a finite concentric electrode. Non-linear evolution of the instability is also studied using direct numerical simulation based on the boundary integral method. Effect of radial time-periodic ac electric field on the stability of Newtonian jet has been studied by González et al. [35], [36].

Compared to Newtonian fluids, the theoretical/numerical work on the role of applied electric fields on viscoelastic jets is rather limited. This is generally attributed to the challenge in dealing with the nonlinear constitutive equations for viscoelastic materials. Reneker et al. [37] and Yarin et al. [38] studied the bending instability of a electrified polymeric jet which is the key instability in electrospinning process. They provided an explanation for this (bending) instability using a mathematical model to simulate the nonlinear behavior of the polymeric jet. Zhang et al. [39] used a one-dimensional Giesekus constitutive equation to study the axisymmetric instability of a straight electrically driven jet. In this model, the viscosity ratio of polymer to solvent is considered as an important parameter. It was found that with an increase in viscosity ratio the instability region becomes wider for finite electric field, once the instability occurs. Li et al. [40] showed a comparison between axisymmetric and non-axisymmetric instability of an electrically charged viscoelastic liquid jet. The Oldroyd-B constitutive equation was used to model the polymeric material and the Taylor–Melcher leaky dielectric model was used with regard to the electrical properties of the material. It was found that elasticity destabilizes the axisymmetric mode more than the non-axisymmetric modes. With an increase in the electric field, the non-axisymmetric mode becomes unstable and dominates over the axisymmetric mode. Ruo et al. [41] employed the Chebyshev collocation method to study three-dimensional instability behavior of electrified non-Newtonian jet. The leaky-dielectric model was used to study the charge transport process and the Oldroyd-B model was used to describe the viscoelastic fluid. Aerodynamic interaction, arising from the surrounding gas was also considered and was found to have a destabilizing role. Fu et al. [42] performed a spatio-temporal analysis for the case of electrified viscoelastic jet for axisymmetric perturbations. It was shown that the flow undergoes a transition from convective to absolute instability for a certain range of non-dimensional flow parameters.

Early works on the study of the effect of the axial electric field on the stability of charged liquid jets were by Taylor [24] and Saville [43]. Saville [44] carried out theoretical work describing the instability behavior of electrically charged viscous cylinders under the axial electric field. The effect of the electric field and the viscosity of the fluid on the instability behavior were investigated. It was found that viscosity of the liquid suppresses the instability of axisymmetric mode more than the sinuous mode and that the electric field can greatly make the nonaxisymmetric mode unstable such that the growth rate of this mode is comparable to that of the axisymmetric mode. Mestel studied the stability of charged liquid jets under the influence of the axial electric field for slightly viscous jets [45] and highly viscous jets [46]. Finite values of the electrical relaxation time were considered and the studies were made for both axisymmetric as well as asymmetric modes. Hohman et al. [47] studied the effect of the axial electric field on liquid jets to understand the electrospinning process. A slender fluid model was employed for the jet and an asymptotic analysis was carried out to study the axisymmetric and sinuous modes. Feng [48] studied the stretching of a charged polymeric jet of Geisekus fluid using a slender-body theory. The role of strain-hardening as a rheological property was found to be most significant to determine the level of stretching. Deshawar and Chokshi [49] conducted linear stability analysis of electrified liquid jet for axisymmetric perturbations by imposing a thinning jet profile to the jet in its base state. The thinning jet was found to have a lower unstable growth rate than the uniform radius jet. Li et al. [50] investigated the spatiotemporal instability of charged viscous jet under the influence of both axial and radial electric fields. They studied the convective-absolute transition for the axisymmetric and first non-axisymmetric modes.

Deshawar and Chokshi [51] carried out linear stability analysis of a polymeric electrified jet for axisymmetric perturbations with application to electrospinning process. They analyzed the stability for conditions corresponding to two different types of polymeric fluids, PIB-based Boger fluid, and PEO solution where the former has lower electrical conductivity than the latter. Later, Deshawar and Chokshi [52] also studied the stability of an electrified polymeric thinning jet under the non-isothermal condition for more realistic electrospinning condition.

In this study, the stability of axisymmetric (m=0) and first non-axisymmetric (m=1) modes of a viscoelastic jet under the influence of radial electric field are investigated. The m=0 and m=1 modes are closely related to electrospraying and electrospinning processes, respectively [32]. For axisymmetric mode, there have been various studies exploring the effect of confinement on the stability of Newtonian liquid jets under applied radial electric field [26], [28], [29], [30], [34], [36], but arguably no study has been conducted for the viscoelastic fluids. González et al. [36], López et al. [29] and Wang [34], for axisymmetric perturbations in Newtonian charged jet, showed that the role of the confinement is especially important because of the variation in the dual nature of the applied electric field with changing confinement widths. González et al. [36] provides a physical explanation for this dual effect of the electric field, while exploring the stability of a perfect conducting jet. Using asymptotic analysis, Wang [34] found the critical value of the electrode radius/jet radius where the dual nature of the electric field disappears for a poorly conducting jet. Although there have been previous efforts to understand the effect of confinement, there is a lack of a systematic study exploring the combined effects of electrical properties and confinement ratio, even for the Newtonian jets.

Therefore, the first objective of this paper is to carry out a systematic investigation of the effect of electrode widths on the stability of jet for varying values of electrical properties of the fluid (from poorly conducting case to perfectly conducting case). Next, the effect of elasticity on the confined jet, under the influence of the radial electric field, will be studied. To the best of our knowledge, this is the first work to study this phenomenon. A fundamental understanding of the stability of viscoelastic jets is of relevance to processes such as electrospraying, ink-jet printing, and electrospinning. Although most of the practical processes are highly nonlinear, the current linear study provides some insight into the initial stage of instability near the upstream, and will aid nonlinear simulations in choosing appropriate parametric regimes where the jet is unstable.

For m=1 mode, a few studies have been carried out to investigate the stability of a viscoelastic jet under the influence of the radial electric field [40], [41], but none of them either explored the confinement effects or the system where inertia is negligible. Saville [53] explored the stability of charged Newtonian viscous cylinders under an applied radial electric field. It was observed that for the case when the viscous effect dominates, the m=1 is the most unstable mode. It was also argued that the neglect of inertia in the formulation is non-physical for the m=1 mode. In this paper, the study by Saville [53] is extended by considering the effects of electrode width and, electrical properties and elasticity of the liquid.

The viscoelasticity of the liquid is modeled using the Jeffreys constitutive relation which can be simplified both to the pure polymer case and the Newtonian fluid case [54], [55]. The effect of the electric field on the jet is modeled using the Taylor–Melcher leaky-dielectric model [56]. This model is used for imperfect conductors which can asymptotically reach the case of a perfect conductor in the case of very high conductivity. In this study, the creeping-flow limit is employed which is a valid assumption for the jets of a very small radius. For example, for a jet made of 4000 ppm PIB (Mol. wt. = 106), Carroll and Joo [57] obtained the typical Reynolds number in the order of 10−2−10−3. The effects of applied electric potential, electrode width, permittivity, conductivity, fluid elasticity and solvent viscosity are discussed for the case of axisymmetric (m = 0) and first non-axisymmetric (m = 1) modes.

The rest of the paper is organized as follows: the problem formulation is discussed in Section 2. Section 3 discusses the linear stability analysis carried out in the present work. Section 4 shows the results and discussions obtained and finally, the conclusions of the present work are discussed in Section 5.

Section snippets

Geometry

We consider an infinitely long cylindrical liquid jet of radius a surrounded by a concentric electrode of radius b as shown in Fig. 1. The viscoelastic liquid is assumed to be incompressible. The gap between the jet and the electrode is occupied by a gas which is considered here to be hydrodynamically passive. The effects of gravity and temperature are neglected. The jet and the surrounding are denoted by subscript “I“ and subscript “II”, respectively. Cylindrical coordinates (r, θ, z) are

Linear stability analysis

In linear stability analysis (LSA), the governing equations and the boundary conditions are linearized about the base state. A small perturbation is imposed on each dynamical variable f as: f=f̄+f,where f̄ is base-state variable (steady state variable) and f is the perturbation variable. The analysis is conducted to first order in perturbation quantities. The perturbation is expanded in a normal mode form as: f=fˆ(r)exp{i(kz+mθ)+st},where k and m are the non-dimensional axial and azimuthal

Scope of the present study

The analytical dispersion relation is obtained after solving the linearized governing equations and boundary conditions directly using Mathematica. In the following discussion, the effects of various non-dimensional physical parameters i.e. Eu, R, σ, ϵr, De and δ on the instability is investigated for m = 0 and 1 modes. The values for the physical properties of the fluid is taken by considering the fluid to be made of 4000 ppm PIB (Mol. wt. = 106), as studied by Carroll and Joo [57]. The

Conclusion

Axisymmetric (m = 0) and first non-axisymmetric (m = 1) modes of instability of a viscoelastic free jet under the effect of radial electric field for finite electrode width has been studied using linear stability analysis in the creeping-flow limit. Effects of different physical parameters namely electric potential, electrode distance, electrical properties, fluid elasticity and solvent viscosity have been extensively studied using various dimensionless parameters. The axisymmetric mode is

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    Present address: School of Engineering, University of Liverpool, Liverpool L69 3GH, UK.

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