Linear Stability Analysis of an Electrified Viscoelastic Liquid Jet

A linear instability analysis method has been used to investigate the breakup of an electrified viscoelastic liquid jet. The liquid is assumed to be a dilute polymer solution modeled by the linear viscoelastic constitutive equation. As for its electric properties, the liquid is assumed to be of perfect electrical conductivity. The axisymmetric and nonaxisymmetric disturbance wave growth rate has been worked out by solving the dispersion equation of an electrified viscoelastic liquid jet, which was obtained by combining the linear instability model of an electrified Newtonian liquid jet with the linear viscoelastic model. The maximum growth rate and corresponding dominant wavenumbers have been observed. The electrical Euler number, non-Newtonian rheological parameters and some flow parameters have been tested for their influence on the instability of the electrified viscoelastic liquid jet. The results show that the disturbance growth rate of electrified viscoelastic liquid jets is higher than that of Newtonian ones for axisymmetric mode disturbance and almost the same for the nonaxisymmetric mode. The growth rate of the axisymmetric mode is greater than that of the nonaxisymmetric mode for large wavenumbers, and the trend is opposite in the small wavenumber range. The ratio of gas to liquid density, electrical Euler number, and elasticity number can accelerate the breakup of the electrified viscoelastic liquid jet for both modes. The increase of the time constant ratio, zero shear viscosity, and jet radius can decrease the growth rate of the axisymmetric mode; however, their effects on the nonaxisymmetric mode are different. As for the effect of surface tension and jet velocity, there is a critical value. The variation trend is opposite when the surface tension or jet velocity is larger or smaller than the critical value.

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Electrohydrodynamic stability: effects of charge relaxation at the interface of a liquid jet

Journal of Fluid Mechanics ◽

10.1017/s0022112071001873 ◽

1971 ◽

Vol 48 (4) ◽

pp. 815-827 ◽

Cited By ~ 56

Author(s):

D. A. Saville

Keyword(s):

Growth Rate ◽

Growth Rates ◽

The Other ◽

Liquid Jet ◽

Relaxation Phenomena ◽

Viscous Effects ◽

Axisymmetric Mode ◽

Charge Relaxation ◽

Small Fields ◽

Interfacial Charge

The interactions between electrical tractions at the interface of a liquid jet and instability phenomena are studied with emphasis on effects due to interfacial charge relaxation. Charge relaxation causes the oscillatory growth of a perturbation. When viscous effects are small, small fields tend to decrease the growth rate of the axisymmetric mode, up to a point, and precipitate instability of the non-axisymmetric modes. Still larger field strengths increase the growth rates of asymmetric as well as axisymmetric modes. Instabilities characterized by highfrequency oscillations appear to persist even though the charge relaxation phenomena may be quite rapid. When, on the other hand, viscous effects predominate the only unstable disturbance form is the axisymmetric one, although the manner of growth may be oscillatory.

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Combined Effect of Viscosity, Surface Tension and Compressibility on Rayleigh-Taylor Bubble Growth Between Two Fluids

Journal of Fluids Engineering ◽

10.1115/1.4027655 ◽

2014 ◽

Vol 136 (9) ◽

Cited By ~ 3

Author(s):

Sourav Roy ◽

L. K. Mandal ◽

Manoranjan Khan ◽

M. R. Gupta

Keyword(s):

Surface Tension ◽

Growth Rate ◽

Bubble Growth ◽

Critical Value ◽

Combined Effect ◽

Bubble Velocity ◽

Wave Number ◽

Damping Factor ◽

Perturbation Wave ◽

Damped Oscillation

The combined effect of viscosity, surface tension, and the compressibility on the nonlinear growth rate of Rayleigh-Taylor (RT) instability has been investigated. For the incompressible case, it is seen that both viscosity and surface tension have a retarding effect on RT bubble growth for the interface perturbation wave number having a value less than three times of a critical value (kc=(ρh-ρl)g/T, T is the surface tension). For the value of wave number greater than three times of the critical value, the RT induced unstable interface is stabilized through damped nonlinear oscillation. In the absence of surface tension and viscosity, the compressibility has both a stabilizing and destabilizing effect on RTI bubble growth. The presence of surface tension and viscosity reduces the growth rate. Above a certain wave number, the perturbed interface exhibits damped oscillation. The damping factor increases with increasing kinematic viscosity of the heavier fluid and the saturation value of the damped oscillation depends on the surface tension of the perturbed fluid interface and interface perturbation wave number. An approximate expression for asymptotic bubble velocity considering only the lighter fluid as a compressible one is presented here. The numerical results describing the dynamics of the bubble are represented in diagrams.

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Capillary Instability of Liquid Jets Issuing From Elliptic Nozzles

ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting: Volume 1, Symposia – Parts A, B, and C ◽

10.1115/fedsm-icnmm2010-30549 ◽

2010 ◽

Cited By ~ 1

Author(s):

Ghobad Amini ◽

Ali Dolatabadi

Keyword(s):

Surface Tension ◽

Growth Rate ◽

Aspect Ratio ◽

Stokes Equations ◽

Practical Importance ◽

Nozzle Geometry ◽

Navier Stokes ◽

Liquid Jets ◽

Liquid Jet ◽

Breakup Mechanism

Breakup of a liquid jet issuing from an orifice is one of the classical problems in fluid dynamics due to its theoretical and practical importance. The main application of the process is in spray and droplet formation, which is of interest in the combustion in liquid-fuelled engines, ink-jet printers, coating systems, medical equipment, and irrigation device. The complexity of the breakup mechanism is due to the large number of parameters involved such as the design of injection nozzle, and thermodynamic states of both liquid and gas. In addition, different combinations of surface tension, inertia, and aerodynamic forces acting on the jet, define main breakup regimes. Effects of nozzle geometry on the behavior of liquid jets have been overlooked in the literature. Elliptic jets have never been investigated theoretically since mostly circular jets or liquid sheets have been analyzed; while experiments have shown that by using elliptical nozzles, entrainment and air mixing of fuel in combustion will be increased. In this article, instability of an elliptic liquid jet under the effect of inertia, viscous, and surface tension forces has been studied using temporal linear analyses. The effects of the gravity and the surrounding gas have been neglected. 1-D Cosserat equation (directed curve) has been used which can be considered as simplified form of Navier-Stokes equations. Results are comparable with classical Rayleigh mode of circular jet when the aspect ratio (ratio of major to minor axis) is one. Growth rate of instability on an elliptic liquid jet under various conditions has been compared with those of a circular jet. Results show that in comparison with a circular jet, the elliptic jet is more unstable and by increasing the aspect ratio the instability grows faster. In addition, similar to the circular case, the effect of viscosity is diminishing the growth rate for the elliptic jet.

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Theoretical Analysis of Surface Waves on Liquid Jets in Crossflows with Deformation

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.681.152 ◽

2013 ◽

Vol 681 ◽

pp. 152-157

Author(s):

Shao Lin Wang ◽

Yong Huang ◽

Fang Wang ◽

Zhi Lin Liu

Keyword(s):

Growth Rate ◽

Theoretical Analysis ◽

Surface Waves ◽

Cross Section ◽

Aerodynamic Force ◽

Linear Instability ◽

Wave Number ◽

Liquid Jets ◽

Liquid Jet ◽

The Cross

Liquid jets in cross air flows are widely used and play an important role in propulsion systems, such as ramjet combustors. Surface waves on the liquid jets in gaseous crossflows have been observed in numerous experiments. Especially for lower gas Webber number, liquid jets breaks up due to the surface waves. However compared with injecting into gas coaxial flow, liquid jet will be deformed in crossflow due to the transverse aerodynamic force. Deformation of jet is investigated by analyzing stress force equilibrium of the cross-section. Though linear instability analysis, dispersion relation and growth rate of surface waves of liquid jet with deformation were derived. According to the present theoretical analysis, the cross-section shape can be deformed to stable ellipse only if the gas velocity was lower than 9m/s for 1mm diameter jet. The maximum growth rate of disturbances takes place at wave number 0.7 approximately, and it will decrease with increasing the jet diameter. The range of instable wave number will expand and the most instable wave number will grow for the deformed jets.

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Pearling instability of a cylindrical vesicle

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.34 ◽

2014 ◽

Vol 743 ◽

pp. 262-279 ◽

Cited By ~ 14

Author(s):

G. Boedec ◽

M. Jaeger ◽

M. Leonetti

Keyword(s):

Surface Tension ◽

Growth Rate ◽

Asymptotic Behaviour ◽

Critical Value ◽

Instability Growth Rate ◽

Instability Growth

AbstractA cylindrical vesicle under tension can undergo a pearling instability, characterized by the growth of a sinusoidal perturbation which evolves towards a collection of quasi-spherical bulbs connected by thin tethers, like pearls on a necklace. This is reminiscent of the well-known Rayleigh–Plateau instability, where surface tension drives the amplification of sinusoidal perturbations of a cylinder of fluid. We calculate the growth rate of perturbations for a cylindrical vesicle under tension, considering the effect of both inner and outer fluids, with different viscosities. We show that this situation differs strongly from the classical Rayleigh–Plateau case in the sense that, first, the tension must be above a critical value for the instability to develop and, second, even in the strong tension limit, the surface preservation constraint imposed by the presence of the membrane leads to a different asymptotic behaviour. The results differ from previous studies on pearling due to the consideration of variations of tension, which are shown to enhance the pearling instability growth rate, and lower the wavenumber of the fastest growing mode.

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Spatial-Temporal Stability of an Electrified Viscoelastic Liquid Jet

Journal of Fluids Engineering ◽

10.1115/1.4024265 ◽

2013 ◽

Vol 135 (9) ◽

Cited By ~ 15

Author(s):

Qing-fei Fu ◽

Li-jun Yang ◽

Pi-min Chen ◽

Yu-xin Liu ◽

Chen Wang

Keyword(s):

Growth Rate ◽

Reynolds Number ◽

Convective Instability ◽

Density Ratio ◽

Temporal Stability ◽

Viscoelastic Liquid ◽

Liquid Jet ◽

Absolute Growth Rate ◽

Temporal Instability ◽

Absolute Growth

This paper presents theoretically the spatial-temporal instability behavior of an electrified viscoelastic liquid jet. Dimensionless parameters have been tested for their influence on the transition of absolute and convective instability for the electrified viscoelastic liquid jet. The results show that larger electrical Euler and Weber numbers can change the flow to convectively unstable. The increase of Reynolds number can decrease the absolute growth rate. Variations of time constant and density ratio rarely change the spatial-temporal instability behavior of the jet. The disturbance wavelength changes very little with these parameters when the flow is absolutely unstable.

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A new upper bound for the largest growth rate of linear Rayleigh–Taylor instability

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02613-y ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Changsheng Dou ◽

Jialiang Wang ◽

Weiwei Wang

Keyword(s):

Surface Tension ◽

Growth Rate ◽

Upper Bound ◽

Viscous Fluids ◽

Instability Condition ◽

Interface Surface ◽

Incompressible Viscous Fluids ◽

Rayleigh Taylor Instability ◽

Surface Tensor ◽

Rt Instability

AbstractWe investigate the effect of (interface) surface tensor on the linear Rayleigh–Taylor (RT) instability in stratified incompressible viscous fluids. The existence of linear RT instability solutions with largest growth rate Λ is proved under the instability condition (i.e., the surface tension coefficient ϑ is less than a threshold $\vartheta _{\mathrm{c}}$ ϑ c ) by the modified variational method of PDEs. Moreover, we find a new upper bound for Λ. In particular, we directly observe from the upper bound that Λ decreasingly converges to zero as ϑ goes from zero to the threshold $\vartheta _{\mathrm{c}}$ ϑ c .

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Stability of a Viscous Liquid Jet in a Coaxial Twisting Compressible Airflow

Processes ◽

10.3390/pr9060918 ◽

2021 ◽

Vol 9 (6) ◽

pp. 918

Author(s):

Li-Mei Guo ◽

Ming Lü ◽

Zhi Ning

Keyword(s):

Viscous Liquid ◽

Axial Velocity ◽

Liquid Velocity ◽

Velocity Ratio ◽

Dominant Mode ◽

Liquid Jet ◽

Axisymmetric Mode ◽

Jet Instability ◽

Jet Stability ◽

Jet Breakup

Based on the linear stability analysis, a mathematical model for the stability of a viscous liquid jet in a coaxial twisting compressible airflow has been developed. It takes into account the twist and compressibility of the surrounding airflow, the viscosity of the liquid jet, and the cavitation bubbles within the liquid jet. Then, the effects of aerodynamics caused by the gas–liquid velocity difference on the jet stability are analyzed. The results show that under the airflow ejecting effect, the jet instability decreases first and then increases with the increase of the airflow axial velocity. When the gas–liquid velocity ratio A = 1, the jet is the most stable. When the gas–liquid velocity ratio A > 2, this is meaningful for the jet breakup compared with A = 0 (no air axial velocity). When the surrounding airflow swirls, the airflow rotation strength E will change the jet dominant mode. E has a stabilizing effect on the liquid jet under the axisymmetric mode, while E is conducive to jet instability under the asymmetry mode. The maximum disturbance growth rate of the liquid jet also decreases first and then increases with the increase of E. The liquid jet is the most stable when E = 0.65, and the jet starts to become more easier to breakup when E = 0.8425 compared with E = 0 (no swirling air). When the surrounding airflow twists (air moves in both axial and circumferential directions), given the axial velocity to change the circumferential velocity of the surrounding airflow, it is not conducive to the jet breakup, regardless of the axisymmetric disturbance or asymmetry disturbance.

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