Investigation of the Linear Stability Problem of Electrified Jets, Inviscid Analysis

2012 ◽  
Vol 134 (9) ◽  
Author(s):  
Serkan Özgen ◽  
Oguz Uzol
Keyword(s):  
Surface Tension ◽  
Dispersion Relation ◽  
Linear Stability ◽  
Equations Of Motion ◽  
Density Ratio ◽  
Linear Stability Theory ◽  
Wave Number ◽  
Liquid Jet ◽  
Liquid Density ◽  
Electrified Jets

The instability characteristics of a liquid jet discharging from a nozzle into a stagnant gas are investigated using the linear stability theory. Starting with the equations of motion for incompressible, inviscid, axisymmetric flows in cylindrical coordinates, a dispersion relation is obtained, where the amplification factor of the disturbance is related to its wave number. The parameters of the problem are the laminar velocity profile shape parameter, surface tension, fluid densities, and electrical charge of the liquid jet. The dispersion relation is numerically solved as a function of the wave number. The growth of instabilities occurs in two modes, the Rayleigh and atomization modes. For rWe<1 (where We represents the Weber number and r represents the gas-to-liquid density ratio) corresponds to a Rayleigh or long wave instability, where atomization does not occur. On the contrary, for rWe>>1 the waves at the liquid-gas interface are shorter and when they reach a threshold amplitude the jet breaks down or atomizes. The surface tension stabilizes the flow in the atomization regime, while the density stratification and electric charges destabilize it. Additionally, a fully developed flow is more stable compared to an underdeveloped one. For the Rayleigh regime, both the surface tension and electric charges destabilize the flow.

Linear stability analysis of an electrified incompressible liquid sheet streaming into a compressible ambient gas

2016 ◽  
Vol 231 (10) ◽  
pp. 1849-1858 ◽  
Author(s):  
Yuxin Liu ◽  
Chaojie Mo ◽  
Lujia Liu ◽  
Qingfei Fu ◽  
Lijun Yang
Keyword(s):  
Stability Analysis ◽  
Electric Field ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Density Ratio ◽  
Sheet Thickness ◽  
Liquid Sheet ◽  
Wave Number ◽  
Liquid Density ◽  
Ambient Gas

This article presents the linear stability analysis of an electrified liquid sheet injected into a compressible ambient gas in the presence of a transverse electric field. The disturbance wave growth rates of sinuous and varicose modes were determined by solving the dispersion relation of the electrified liquid sheet. It was determined that by increasing the Mach number of the ambient gas from subsonic to transonic, the maximum growth rate and the dominant wave number of the disturbances were increased, and the increase was greater in the presence of the electric field. The electrified liquid sheet was more unstable than the non-electrified sheet. The increase of both the gas-to-liquid density ratio and the electrical Euler number accelerated the breakup of the liquid sheet for both modes; while the ratio of distance between the horizontal electrode and the liquid-sheet-to-sheet thickness had the opposite effect. High Reynolds and Weber numbers accelerated the breakup of the electrified liquid sheet.

Convective and absolute instability of falling viscoelastic liquid jets surrounded by a gas

2020 ◽  
Author(s):  
A Alhushaybari ◽  
J Uddin
Keyword(s):  
Dispersion Relation ◽  
Density Ratio ◽  
Viscoelastic Liquid ◽  
Liquid Jets ◽  
Mapping Technique ◽  
Liquid Jet ◽  
Absolute Instability ◽  
Liquid Density ◽  
Complex Frequency ◽  
Instability Analysis

Abstract We examine the convective and absolute instability of a 2D axisymmetric viscoelastic liquid jet falling vertically in a medium of an inviscid gas under the influence of gravity. We use the upper-convected Maxwell model to describe the viscoelastic liquid jet and together with an asymptotic approach, based on the slenderness of the jet, we obtain steady-state solutions. By considering travelling wave modes, and using linear instability analysis, the dispersion relation, relating the frequency to wavenumber of disturbances, is derived. We solve this dispersion relation numerically using the Newton–Raphson method and explore regions of instability in parameter space. In particular, we investigate the influence of gravity, the effect of changing the gas-to-liquid density ratio, the Weber number and the Deborah number on convective and absolute instability. In this paper, we utilize a mapping technique developed by Afzaal (2014, Breakup and instability analysis of compound liquid jets. Doctoral Dissertation, University of Birmingham) to find the cusp point in the complex frequency plane and its corresponding first-order saddle point (the pinch point) in the complex wavenumber plane for absolute instability. The convective/absolute instability boundary is identified for various parameter regimes along the axial length of the jet.

scholarly journals Shear instability of an axisymmetric air–water coaxial jet

2018 ◽  
Vol 843 ◽  
pp. 575-600 ◽  
Author(s):  
Jean-Philippe Matas ◽  
Antoine Delon ◽  
Alain Cartellier
Keyword(s):  
Surface Tension ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Convective Instability ◽  
Scaling Laws ◽  
Shear Instability ◽  
Liquid Jet ◽  
Absolute Instability ◽  
Instability Waves

We study the destabilization of a round liquid jet by a fast annular gas stream. We measure the frequency of the shear instability waves for several geometries and air/water velocities. We then carry out a linear stability analysis, and show that there are three competing mechanisms for the destabilization: a convective instability, an absolute instability driven by surface tension and an absolute instability driven by confinement. We compare the predictions of this analysis with experimental results, and propose scaling laws for wave frequency in each regime. We finally introduce criteria to predict the boundaries between these three regimes.

scholarly journals On the deflection of a liquid jet by an air-cushioning layer

2018 ◽  
Vol 846 ◽  
pp. 711-751 ◽  
Author(s):  
M. R. Moore ◽  
J. P. Whiteley ◽  
J. M. Oliver
Keyword(s):  
Surface Tension ◽  
Equations Of Motion ◽  
Constant Thickness ◽  
Liquid Jet ◽  
Pressure Jump ◽  
Length Scales ◽  
Small Surface ◽  
Numerical Studies ◽  
Impact Problems ◽  
Systematic Derivation

A hierarchy of models is formulated for the deflection of a thin two-dimensional liquid jet as it passes over a thin air-cushioning layer above a rigid flat impermeable substrate. We perform a systematic derivation of the leading-order equations of motion for the jet in the distinguished limit in which the air pressure jump, surface tension and gravity affect the displacement of the centreline of the jet, but not its thickness or velocity. We identify thereby the axial length scales for centreline deflection in regimes in which the air layer is dominated by viscous or inertial effects. The derived length scales and reduced equations aim to expand the suite of tools available for future analyses of the evolution of lamellae and ejecta in impact problems. Assuming that the jet is sufficiently long that tip and entry effects can be neglected, we demonstrate that the centreline of a constant-thickness jet moving with constant axial speed is destabilised by the air layer for sufficiently small surface tension. Expressions for the fastest-growing modes are obtained in both the viscous-dominated air and inertia-dominated air regimes. For a finite-length jet emanating from a nozzle, we show that, in one particular asymptotic limit, the evolution of the jet centreline is akin to the flapping of an unfurling flag above a thin air layer. We discuss the distinguished limit in which tip retraction can be neglected and perform numerical investigations into the resulting model. We show that the cushioning layer causes the jet centreline to bend, leading to rupture of the air layer. We discuss how our toolbox of models can be adapted and utilised in the context of recent experimental and numerical studies of splash dynamics.

scholarly journals Effect of Pressure Gradient on the Development of Görtler Vortices

2019 ◽  
Author(s):  
Leandro Marochio Fernandes ◽  
Marcio Teixeira de Mendonça
Keyword(s):  
Pressure Gradient ◽  
Linear Stability ◽  
Boundary Layers ◽  
Adverse Pressure Gradient ◽  
Base Flow ◽  
Linear Stability Theory ◽  
Wave Number ◽  
Local Mode ◽  
Effect Of Pressure ◽  
Centrifugal Instability

Boundary layers over concave surfaces may become unstable due to centrifugal instability that manifests itself as stationary streamwise counter rotating vortices. The centrifugal instability mechanism in boundary layers has been extensively studied and there is a large number of publications addressing different aspects of this problem. The results on the effect of pressure gradient show that favorable pressure gradients are stabilizing and adverse pressure gradient enhances the instability. The objective of the present investigation is to complement those works, looking particularly at the effect of pressure gradient on the stability diagram and on the determination of the spanwise wave number corresponding to the fastest growth. This study is based on the classic linear stability theory, where the parallel boundary layer approximation is assumed. Therefore, results are valid for Görtler numbers above 7, the lower limit where local mode linear stability analysis was identified in the literature as valid. For the base flow given by the Falkner-Skan solution, the linear stability equations are solved by a shooting method where the eigenvalues are the Görtler number, the spanwise wavenumber and the growth rate. The results show stabilization due to favorable pressure gradient as the constant amplification rate curves are displaced to higher Görtler numbers, with the opposite effect for adverse pressure gradient. Results previously unavailable in the literature identifying the fastest growing mode spanwise wavelength for a range of Falkner-Skan acceleration parameters are presented.

The formation of drops through viscous instability

1995 ◽  
Vol 289 ◽  
pp. 351-378 ◽  
Author(s):  
Silvana S. S. Cardoso ◽  
Andrew W. Woods
Keyword(s):  
Surface Tension ◽  
Linear Stability ◽  
Intermediate Layer ◽  
Nonlinear Effects ◽  
Linear Stability Theory ◽  
Relevant Parameter ◽  
One Dimensional ◽  
Linear Mode ◽  
Outer Interface ◽  
The Stability

The stability of an immiscible layer of fluid bounded by two other fluids of different viscosities and migrating through a porous medium is analysed, both theoretically and experimentally. Linear stability analyses for both one-dimensional and radial flows are presented, with particular emphasis upon the behaviour when one of the interfaces is highly stable and the other is unstable. For one-dimensional motion, it is found that owing to the unstable interface, the intermediate layer of fluid eventually breaks up into drops.However, in the case of radial flow, both surface tension and the continuous thinning of the intermediate layer as it moves outward may stabilize the system. We investigate both of these stabilization mechanisms and quantify their effects in the relevant parameter space. When the outer interface is strongly unstable, there is a window of instability for an intermediate range of radial positions of the annulus. In this region, as the basic state evolves to larger radii, the linear stability theory predicts a cascade to higher wavenumbers. If the growth of the instability is sufficient that nonlinear effects become important, the annulus will break up into a number of drops corresponding to the dominant linear mode at the time of rupture.In the laboratory, a Hele-Shaw cell was used to study these processes. New experiments show a cascade to higher-order modes and confirm quantitatively the prediction of drop formation. We also show experimentally that the radially spreading system is stabilized by surface tension at small radii and by the continual thinning of the annulus at large radii.

Linear stability analysis of a Newtonian ferrofluid cylinder surrounded by a Newtonian fluid

2021 ◽  
Vol 927 ◽  
Author(s):  
Romain Canu ◽  
Marie-Charlotte Renoult
Keyword(s):  
Magnetic Field ◽  
Stability Analysis ◽  
Dispersion Relation ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Density Ratio ◽  
Bond Number ◽  
Wire Radius ◽  
Azimuthal Magnetic Field ◽  
Relative Magnetic Permeability

We performed a linear stability analysis of a Newtonian ferrofluid cylinder surrounded by a Newtonian non-magnetic fluid in an azimuthal magnetic field. A wire is used at the centre of the ferrofluid cylinder to create this magnetic field. Isothermal conditions are considered and gravity is ignored. An axisymmetric perturbation is imposed at the interface between the two fluids and a dispersion relation is obtained allowing us to predict whether the flow is stable or unstable with respect to this perturbation. This relation is dependent on the Ohnesorge number of the ferrofluid, the dynamic viscosity ratio, the density ratio, the magnetic Bond number, the relative magnetic permeability and the dimensionless wire radius. Solutions to this dispersion relation are compared with experimental data from Arkhipenko et al. (Fluid Dyn., vol. 15, issue 4, 1981, pp. 477–481) and, more recently, Bourdin et al. (Phys. Rev. Lett., vol. 104, issue 9, 2010, 094502). A better agreement than the inviscid theory and the theory that only takes into account the viscosity of the ferrofluid is shown with the data of Arkhipenko et al. (Fluid Dyn., vol. 15, issue 4, 1981, pp. 477–481) and those of Bourdin et al. (Phys. Rev. Lett., vol. 104, issue 9, 2010, 094502) for small wavenumbers.

Nonlinear Propagation of Wave-Packets on Fluid Interfaces

1976 ◽  
Vol 43 (4) ◽  
pp. 584-588 ◽  
Author(s):  
A. H. Nayfeh
Keyword(s):  
Surface Tension ◽  
Differential Equations ◽  
Multiple Scales ◽  
Wave Packets ◽  
Density Ratio ◽  
Capillary Waves ◽  
Nonlinear Propagation ◽  
Wave Number ◽  
Wide Range ◽  
Cutoff Wave Number

The method of multiple scales is used to derive two partial differential equations which describe the evolution of two-dimensional wave-packets on the interface of two semi-infinite, incompressible, inviscid fluids of arbitrary densities, taking into account the effect of the surface tension. These differential equations can be combined to yield two alternate nonlinear Schro¨dinger equations; one of them contains only first derivatives in time while the second contains first and second derivatives in time. The first equation is used to show that the stability of uniform wavetrains depends on the wave length, the surface tension, and the density ratio. The results show that gravity waves are unstable for all density ratios except unity, while capillary waves are stable unless the density ratio is below approximately 0.1716. Moreover, the presence of surface tension results in the stabilization of some waves which are otherwise unstable. Although the first equation is valid for a wide range of wave numbers, it is invalid near the cutoff wave number separating stable from unstable motions. It is shown that the second Schro¨dinger equation is valid near the cutoff wave number and thus it can be used to determine the dependence of the cutoff wave number on the amplitude, thereby avoiding the usual process of determining a new expansion that is only valid near the cutoff conditions.

Azimuthal shear instability of a liquid jet injected into a gaseous cross-flow

2015 ◽  
Vol 767 ◽  
pp. 146-172 ◽  
Author(s):  
M. Behzad ◽  
N. Ashgriz ◽  
A. Mashayek
Keyword(s):  
Boundary Layer ◽  
Stagnation Point ◽  
Linear Stability ◽  
Density Ratio ◽  
Cross Flow ◽  
Shear Instability ◽  
Liquid Jet ◽  
Flow Density ◽  
Azimuthal Shear ◽  
Continuous Formulation

AbstractWe investigate azimuthal instabilities which exist on the periphery of a non-turbulent liquid jet injected transversely into a gaseous cross-flow. We predict that the temporal growth of such instabilities may lead to the formation of interface corrugations, which are eventually sheared off of the jet surface (known as the jet ‘surface breakup’). In this study we employ temporal linear stability analyses to understand the nature of these instabilities. The analysis is based on a continuous formulation of momentum equations in which the jet and cross-flow are considered to be slightly miscible at the vicinity of the interface. We identify the shear instability as the primary destabilization mechanism in the flow. This inherently inviscid mechanism opposes the previously suggested mechanism of surface breakup (known as ‘boundary-layer stripping’), which is based on a viscous interpretation. The results show that the wavelengths of instabilities increase by moving away from the jet windward stagnation point toward the leeward point. We also investigate the influence of the jet-to-cross-flow density ratio on the flow stability and find that a higher ratio leads to formation of instabilities with higher wavenumbers on the jet surface. The results show that the density may have a non-monotonic stabilizing/destabilizing effect on the flow.

scholarly journals Numerical Study of Density Ratio Influence on Global Wave Shapes Before Impact

2018 ◽  
Author(s):  
Stéphane Etienne ◽  
Yves-Marie Scolan ◽  
Laurent Brosset
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Gas Flow ◽  
Numerical Study ◽  
Mechanical Energy ◽  
Density Ratio ◽  
Free Surface Flows ◽  
Liquid Density ◽  
The Impact ◽  
Wave Shapes

The influence of the gas-to-liquid density ratio (DR) on the global wave shape before impact is studied through numerical simulations of the propagation of two different waves in a rectangular wave canal. Two different codes are used: the first one, named FSID, is a highly non-linear 2D bi-fluid potential code initially developed in the frame of SLOSHEL JIP (Kaminski et al. (2011)) to simulate incompressible inviscid free-surface flows without surface tension thanks to a desingularized technique and series of conformal mappings; the second one, named CADYF, is a bi-fluid high-fidelity front-tracking software developed by Ecole Polytechnique Montreal to simulate separated two-phase incompressible viscous flows with surface tension. The first studied wave leads to a flip-through impact while the second one leads to a large gas-pocket impact. Each condition is studied with water and three different gases with increasing densities corresponding to DR = 0.001, 0.003 and 0.005. The global wave shapes are compared a few tenths of second before the impact, before free surface instabilities triggered by the shearing gas flow have developed and also before any gas compressibility matters. Both codes give precisely the same global wave shapes. Whatever the condition studied, it is shown that DR has an influence on these global wave shapes. The trends observed from the simulations are the same as those described in Karimi et al. (2016) obtained from sloshing model tests with Single Impact Waves (SIW) in a 2D tank with a low filling level. A small part of the mechanical energy of the liquid is progressively given to the gas. The larger the DR, the larger this transfer of energy from the liquid to the gas. This explains an increasing delay of the wave front for increasing DRs.

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