Elsevier

European Journal of Mechanics - B/Fluids

Volume 89, September–October 2021, Pages 191-202
European Journal of Mechanics - B/Fluids

Temporal instability of surfactant-laden compound jets with surface viscoelasticity

https://doi.org/10.1016/j.euromechflu.2021.05.009Get rights and content

Highlights

  • Surface viscoelastic effects are considered in a compound jet flow.

  • The increase of surface viscoelastic parameters will make the system more stable.

  • The increase of surface shear viscosity weakens the influence of surface elasticity.

  • Long-wave assumption is adopted to acquire simplified model.

Abstract

When compound jets are covered with surfactant on both interfaces, the property of those interfaces will be viscoelastic, unlike the Newtonian fluid interface. In this paper, the effects of surface viscoelasticity are principally studied through linear stability analysis and calculated using the Chebyshev spectral collocation method. Both surface elasticity and viscosity have properties to make the jet more stable, but the physical effects behind the two influencing mechanisms are different. When the inner and outer interfaces are separately covered with surfactant, the variations of surface viscoelastic parameters affect the growth rate and dominant wavenumber differently. In addition, the sensitivity of the maximum growth rate to the surface viscoelastic parameters depends on which interface the surfactant covers. A significant increase in the surface shear viscosity reduces the influence of surface elasticity on the maximum growth rate. The long-wave assumption was adopted to derive a one-dimensional dispersion equation in an algebraic form. The comparison between the simplified expression and the numerical results validated the applicability of the one-dimensional model in the vicinity of small wavenumber.

Introduction

Rayleigh [1] was the first to theoretically investigate liquid columnar jets. This fluid system can be applied in various fields: ink-jet printing [2], [3], biotechnology (targeted drug delivery [4], particle sorting [5]), industrial fabrication (fiber spinning [6]) and liquid rocket engines (fuel atomization and spray [7]). Surface tension is responsible for the onset of instability in liquid jets. Although Rayleigh’s derivation did not include the effect of viscosity, and it neglected the presence of the surrounding gas, his work has played an essential role in the exploration of jet instability. Before Rayleigh’s theory, Tomotika [8] considered the effect of a viscous immiscible phase when a liquid thread was examined during linear stability analysis. Weber [9] extended Rayleigh’s theory to an arbitrarily viscous situation. Instability occurring on the surface of liquid jets was accepted as surface tension at a relatively low speed, in which the dominant mode was symmetric. To study instability growth rates under axisymmetric disturbances, García [10] used the derivations in a set of simplified models for slender viscous liquid jets. His formulaic conclusion agreed with Weber’s theoretical predictions. These pioneering works significantly enriched the mechanism of jet instability.

The incipient evolution of a perturbed jet can be regarded as a linear regime, in which the stage linear perturbation theory can be used to study the evolution of the surface wave. Compound jets can be departed into two fluid regions and interfaces: fluid 1 denotes the core region of the jet; fluid 2 wraps the core in an annular region. The inner interface separates fluid 1 and fluid 2, while the outer interface is the free surface of fluid 2. The Stretching mode becomes predominant for a wide range of physical parameters; the squeezing mode is another instability mode that maintains relatively small growth rate. The instability of compound jets was first studied by Hertz and Hermanrud [11], in their search for an effective method to solve the colored ink-jet printing problem. Sanz and Meseguer [12] used a one-dimensional model to capture the dynamic characteristic of compound capillary jets without viscosity. Chauhan et al. [13] explored the capillary instability of inviscid compound jets in time and space. The jet became absolutely unstable when its base velocity was below a certain value, while convective instability occurred at a relatively large velocity. Subsequently, the maximum growth rate and the dominant wave number as well as the maximum amplitude ratio of the annular to the core were investigated for the relevance of system parameters: the outer-to inner-ratios of density, viscosity and surface tension (Chauhan et al. [14]). Among these parameters, the density ratio was found to be the most sensible factor to change amplitude ratio and growth rate. Alsharif et al. [15] examined the instability of viscoelastic compound jets using an asymptotic approach, as well as by estimating the breakup length. Ye et al. [16] performed an analysis of the energy budget of a viscoelastic compound under axisymmetric destabilization condition. They discussed two parameters, stress relaxation time and time constant ratio, affect growth rate of the jet in inverse ways.

Because the surface tension of fluids can be altered by surfactant, it has been widely applied in diverse fields, such as fiber spinning, stabilizing emulsions, moderating transport conditions, etc. The presence of surfactant breaks the uniformity of the surface tension, leading to surface tension gradient as a result of reverse flow, known as Marangoni effect (Scriven [17]). The work of Boussinesq [18] defined two useful parameters named surface shear viscosity and surface dilatational viscosity to describe the characteristics of this behavior in interfacial flow. Timmermans et al. [19] performed a linear stability analysis and a one-dimensional nonlinear model to capture the dynamics of a surfactant-laden liquid in the absence of ambient gas. The dimensionless number β, defined as a parameter to measure the strength of surfactant, played a significant role in distinguishing two regimes: when β <1, the instability regime of the jet is in extension mode which is inherently controlled by inertia; when β >1, the shear regime became dominant, and the entire flow was supported by viscous shear stress resulting from the surface tension gradient. Hameed et al. [20] theoretically and experimentally investigated the surfactant-coated inviscid liquid jets surrounded by viscous fluids; they showed that surfactant hindered the breakup process. Stone et al. [21] reported that surfactants increased the deformation of drops during the breakup process, which was confirmed analytically and numerically at low capillary numbers. Craster et al. [22] focused on the breakup and satellite formation process of viscid liquid threads in the presence of an insoluble surfactant. They adopted a nonlinear analysis approach, and their calculations showed that the disturbance growth rate decreased as the surfactant concentration increased. In a sequent study, they extended their work to a compound thread (Craster et al. [23]). A wide range of system parameters was discussed, and due to the additional interface, more such cases should be considered. They concluded that the dominant stretching mode became unstable when the ratio of surface tension increased and the ratio of radii, viscosity, and surfactant concentration decreased. However, they did not incorporate the effect of surface viscosity. Dravid et al. [24] used a two-dimensional finite element method to investigate the breakup and satellite droplet formation process of a viscous filament covered with surfactant. Ye et al. [25] studied the linear instability of a surfactant-laden compound jet, neglecting the effect of the surrounding phase. Their approach differed from Craster et al. [23] in their consideration of the radial effect of the general variables. Ye et al. [25] used long-wave approximation methods to examine the disturbed axial velocity field in both stretching and squeezing modes. Their final conclusions provided an analytical explanation for the uniformity of leading-order axial velocity under the stretching mode, manifested in the applicability of the asymptotic model. Ponce-Torres et al. [26] concentrated on the breakup of a pendant drop covered with surfactant, to experimentally and theoretically explore the additional term of surface viscosity affecting surfactant concentration of satellite drops.

To maintain the stable shape of the jet during the fabrication of compound fibers, beaded morphologies should be avoided (McKee et al. [27], Shenoy et al. [28]). To achieve this, Ye et al. [25] proposed using surfactants to lower the liquid–liquid interfacial tension. This work explores viscoelasticity property on a compound jet surface to determine the dependence of physical parameters on the disturbance growth rate of the system, as well as the dominant wave number, which is an extended work based on the analysis of Martínez-Calvo et al. [29]. For the sake of simplicity, this work also formulated a one-dimensional model to recognize the parameter relevance. The remainder of this paper is organized as follows: Section 2 describes the physical model with governing equations and boundary conditions. Section 3 deals with the linear equation derived in Section 2, utilizing Chebyshev spectral collocation method to determine the relevance of the surface parameter on growth rate. Conclusions are presented in Section 4.

Section snippets

Mathematical formulation

This work considers an axisymmetric compound jet covered with insoluble surfactants in a vacuum environment; both liquids are considered to be incompressible Newtonian fluids with constant density and viscosity. The physical model of a compound jet appears in Fig. 1, which uses a cylindrical coordinate system (r, θ, z) to describe the problem. Here, the effects caused by gravity are neglected. In the following context, ρi stands for the density in bulk, μi for viscosity in bulk, σi is surface

Linear analysis

In this work, the Chebyshev spectral collocation method was employed to acquire growth rate as the eigenvalue of the system. The spectral collocation method has been accepted by more researchers [36], [37], [38], [39] because it offers greater accuracy than the finite difference method and it is more efficient for calculation. A detailed discussion of the Chebyshev spectral collocation method was offered by Hussaini et al. [40]. Acceptably accurate results can be obtained by taking 20

Conclusion

This work considered temporal instabilities in a compound jet under surfactant-coated conditions and the insoluble surfactant behaviors on viscoelasticity of both surfaces. The Chebyshev spectral collocation method was applied to solve the hydrodynamic problem numerically and to investigate the effect of surface viscoelastic parameters on the co-flowing system. Variations in the dispersion curve when the surface viscoelastic parameters changed were discussed under three conditions respectively:

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.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11922201), the National Science and Technology Major Project, China (2017-III-0004-0028).

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