Stability of thin, radially moving liquid sheets

1978 ◽  
Vol 87 (2) ◽  
pp. 289-298 ◽  
Author(s):  
Daniel Weihs
Keyword(s):  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Classical Type ◽  
Fluid Viscosity ◽  
Single Wave ◽  
Nozzle Orifice ◽  
Liquid Sheets ◽  
Special Cases ◽  
The Stability ◽  
The Waves

An analysis of the stability of thin viscous liquid sheets, such as those emitted from industrial spraying nozzles, is presented. These sheets are in the form of a circular sector whose thickness reduces as the distance from the nozzle increases.The Kelvin-Helmholtz type of instability usually observed causes the breakup and atomization of the sheet, as required in most industrial spraying processes. Waviness, like that of a flapping flag, is produced and increasing amplitudes finally cause breakup.An analytical solution in the form of hypergeometric functions for the shape of the sheet and the waves is obtained. This solution includes, as special cases, analyses existing in the literature, in addition to establishing the possibility of a new type of instability dependent on the distance from the nozzle. Also, the classical type of instability, in which the waves increase with time, is examined and relations for unstable waves as a function of parameters such as the fluid viscosity, surface tension and sheet velocity are obtained. It is shown that there is no single wave that has a maximum growth rate, but that the wavenumber for maximum instability increases with the distance from the nozzle orifice.

Non-normal stability analysis of a shear current under surface gravity waves

2008 ◽  
Vol 609 ◽  
pp. 49-58
Author(s):  
D. AMBROSI ◽  
M. ONORATO
Keyword(s):  
Growth Rate ◽  
Modal Analysis ◽  
Gravity Waves ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Surface Gravity ◽  
Horizontal Shear ◽  
Surface Gravity Waves ◽  
Shear Current ◽  
The Stability

The stability of a horizontal shear current under surface gravity waves is investigated on the basis of the Rayleigh equation. As the differential operator is non-normal, a standard modal analysis is not effective in capturing the transient growth of a perturbation. The representation of the stream function by a suitable basis of bi-orthogonal eigenfunctions allows one to determine the maximum growth rate of a perturbation. It turns out that, in the considered range of parameters, such a growth rate can be two orders of magnitude larger than the maximum eigenvalue obtained by standard modal analysis.

Ageostrophic instability of ocean currents

1982 ◽  
Vol 117 ◽  
pp. 343-377 ◽  
Author(s):  
R. W. Griffiths ◽  
Peter D. Killworth ◽  
Melvin E. Stern
Keyword(s):  
Potential Vorticity ◽  
Lower Layer ◽  
Rotation Period ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Inertial Effects ◽  
Mean Flow ◽  
Special Cases ◽  
Free Streamlines ◽  
A Current

We investigate the stability of gravity currents, in a rotating system, that are infinitely long and uniform in the direction of flow and for which the current depth vanishes on both sides of the flow. Thus, owing to the role of the Earth's rotation in restraining horizontal motions, the currents are bounded on both sides by free streamlines, or sharp density fronts. A model is used in which only one layer of fluid is dynamically important, with a second layer being infinitely deep and passive. The analysis includes the influence of vanishing layer depth and large inertial effects near the edges of the current, and shows that such currents are always unstable to linearized perturbations (except possibly in very special cases), even when there is no extremum (or gradient) in the potential vorticity profile. Hence the established Rayleigh condition for instability in quasi-geostrophic models, where inertial effects are assumed to be vanishingly small relative to Coriolis effects, does not apply. The instability does not depend upon the vorticity profile but instead relies upon a coupling of the two free streamlines. The waves permit the release of both kinetic and potential energy from the mean flow. They can have rapid growth rates, the e-folding time for waves on a current with zero potential vorticity, for example, being close to one-half of a rotation period. Though they are not discussed here, there are other unstable solutions to this same model when the potential vorticity varies monotonically across the stream, verifying that flows involving a sharp density front are much more likely to be unstable than flows with a small ratio of inertial to Coriolis forces.Experiments with a current of buoyant fluid at the free surface of a lower layer are described, and the observations are compared with the computed mode of maximum growth rate for a flow with a uniform potential vorticity. The current is observed to be always unstable, but, contrary to the predicted behaviour of the one-layer coupled mode, the dominant length scale of growing disturbances is independent of current width. On the other hand, the structure of the observed disturbances does vary: when the current is sufficiently narrow compared with the Rossby deformation radius (and the lower layer is deep) disturbances have the structure predicted by our one-layer model. The flow then breaks up into a chain of anticyclonic eddies. When the current is wide, unstable waves appear to grow independently on each edge of the current and, at large amplitude, form both anticyclonic and cyclonic eddies in the two-layer fluid. This behaviour is attributed to another unstable mode.

Instabilities of a Baroclinic, Double Diffusive Frontal Zone

2008 ◽  
Vol 38 (4) ◽  
pp. 840-861 ◽  
Author(s):  
W. D. Smyth
Keyword(s):  
Growth Rate ◽  
Frontal Zone ◽  
Baroclinic Instability ◽  
Flux Ratio ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Variable Diffusivity ◽  
The Stability ◽  
Parameter Values ◽  
Double Diffusive

Abstract The linear theory of double diffusive interleaving is extended to take account of baroclinic effects. This study goes beyond previous studies by including the possibility of modes with nonzero tilt in the alongfront direction, which allows for advection by the baroclinic frontal flow. This requires that the stability equations be solved numerically. The main example is based on observations of interleaving on the lower flank of Meddy Sharon, but a range of parameter values is covered, leading to conclusions that are relevant in a variety of oceanic regimes. The frontal zone is treated as infinitely wide with uniform gradients of temperature, salinity, and alongfront velocity. The stationary, vertically symmetric interleaving mode is shown to have maximum growth rate when its alongfront wavenumber is zero, providing validation for previous studies in which this property was assumed. Besides this, there exist two additional modes of instability: the ageostrophic Eady mode of baroclinic instability and a mode not previously identified. The new mode is oblique (i.e., it tilts in the alongfront direction), vertically asymmetric, and propagating. It is strongly dependent on boundary conditions, and its relevance in the ocean interior is uncertain as a result. Effects of variable diffusivity and buoyancy flux ratio are also considered.

Electrohydrodynamic stability of a slightly viscous jet

1994 ◽  
Vol 274 ◽  
pp. 93-113 ◽  
Author(s):  
A. J. Mestel
Keyword(s):  
Surface Charge ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Small Fields ◽  
Weakly Conducting ◽  
Tangential Field ◽  
Tangential Surface ◽  
The Stability ◽  
Electric Effects ◽  
Phase Differences

Many electro-spraying devices raise to a high electric potential a pendant drop of weakly conducting fluid, which may adopt a conical shape from whose apex a thin, charged jet is emitted. Such a jet eventually breaks up into fine droplets, but often displays surprising longevity. This paper examines the stability of an incompressible cylindrical jet carrying surface charge in a tangential electric field, allowing for the finite rate of charge relaxation. The viscosity is assumed to be small so that the shear resulting from the tangential surface stress can be large, even for relatively small fields. This shear can suppress surface tension instabilities, but if too large, it excites electrical ones. For imperfect conductors, surface charge is redistributed by the rapid fluid reaction to variations in tangential stress as well as by conduction. Phase differences between the effects due to the tangential field and the surface charge lead to charge ‘over-relaxation’ instabilities, but the maximum growth rate can still be lower than in the absence of electric effects.

Stability of relativistic transverse cold plasma waves. Part 2. Linearly polarized waves

1979 ◽  
Vol 22 (2) ◽  
pp. 201-222 ◽  
Author(s):  
F. J. Romeiras
Keyword(s):  
Nonlinear Wave ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Transverse Modes ◽  
Electron Positron ◽  
Polarized Waves ◽  
Linearly Polarized ◽  
Linear Differential ◽  
Unmagnetized Plasma ◽  
The Stability

This is part 2 of a paper concerned with the stability against small perturbations of a certain class of nonlinear wave solutions of the equations that describe a cold unmagnetized plasma. It refers to transverse linearly polarized waves in an electron-positron plasma. A numerical method, based on Floquet's theory of linear differential equations with periodic coefficients, is used to solve the perturbation equations and obtain the instability growth rates. All the three possible types of perturbations are discussed for a typical value of the (large) amplitude of the nonlinear wave: electrically longitudinal slightly unstable modes (with maximum growth rate γ approximately equal to 0·07ω0, where ω0is the frequency of the nonlinear wave); purely transverse moderately unstable modes (with γ ≃ 0·26ω0); and highly unstable electrically transverse modes (with γ ≃ l·5ω0).

scholarly journals Stability of an electrified liquid jet approaching to critical point

2020 ◽  
Vol 12 (7) ◽  
pp. 168781402093637
Author(s):  
Zixuan Fang ◽  
Ping Wang
Keyword(s):  
Surface Tension ◽  
Critical Point ◽  
Weber Number ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Wave Number ◽  
Electrical Field ◽  
Electric Stress ◽  
Electrical Field Intensity ◽  
The Stability

This article reports the linear stability analysis of a thermodynamic-transcritical jet sprayed to a radial electrical field. An asymptotic approach was used to obtain the stability solution of a supercritical jet subjected to electrical field. In order to obtain the solutions for the electrified supercritical jet, the surface tension was decreased and consequently led the increase in Weber number in the linear governing equation of subcritical charged jet. To investigate the role of surface tension and electric stress playing in the destabilizing process when approaching the critical point, the energy budget is performed by tracing the energy sources. It was found that, when the Weber number is increased to a sufficiently large value, the solution will become an asymptotic value, which can be considered as a solution under the supercritical conditions. The electric stress can increase both the maximum growth rate and the dominant wave number of electrified supercritical jet, that is, higher electrical field intensity would enhance the instability of the electrified supercritical jet and decrease the wavelength of the disturbances.

Stability of the solar wind against the whistler mode

1978 ◽  
Vol 20 (2) ◽  
pp. 225-230 ◽  
Author(s):  
P. Revathy
Keyword(s):  
Growth Rate ◽  
Solar Wind ◽  
Relative Velocity ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
The Sun ◽  
Thermal Velocity ◽  
Whistler Mode ◽  
Minimum Value ◽  
The Stability

The stability of the solar wind against the whistler mode is analyzed. It is shown that the solar wind can become unstable owing to this mode after a distance of 100 R from the sun. The minimum value of the relative velocity (Ur) between ∝-particles and protons for the excitation of this instability is the proton thermal velocity αi. The maximum growth rate at 200 R occurs for the values of the parameters kρi = 0·3,(T┤/T∥)i = 0·3 and Ur/αi = 0·5.

Oblique wind waves generated by the instability of wind blowing over water

1996 ◽  
Vol 316 ◽  
pp. 163-172 ◽  
Author(s):  
L. C. Morland
Keyword(s):  
Gravity Waves ◽  
Wind Waves ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Mean Flow ◽  
Oblique Angle ◽  
Wind Speeds ◽  
The Mean ◽  
Directional Distributions ◽  
The Waves

The growth rates of gravity waves are computed from linear, inviscid stability theory for wind velocity profiles that are representative of the mean flow in a turbulent boundary layer. The energy transfer to the waves is largely concentrated in an angle (to the wind) interval that broadens with increasing wind speed and narrows with increasing wavelength. At sufficiently high wind speeds and sufficiently short wavelengths, the waves of maximum growth rate propagate at an oblique angle to the wind. The connection with bimodal directional distributions of observed spectra is discussed.

Relativistic transverse modulational instability of two electron cyclotron waves

1996 ◽  
Vol 55 (3) ◽  
pp. 327-338 ◽  
Author(s):  
Kwang-Sup Yang
Keyword(s):  
Ponderomotive Force ◽  
Modulational Instability ◽  
Maximum Growth ◽  
Finite Amplitude ◽  
Equation Model ◽  
Maximum Growth Rate ◽  
Electron Cyclotron ◽  
Mass Variation ◽  
Single Wave ◽  
Electron Cyclotron Waves

The transverse modulational instabilities of two finite-amplitude electron-cyclotron waves due to the ponderomotive force and relativistic mass variation of electrons are considered using the coupled nonlinear Schrödinger equation model. The waves are modulationally unstable, with maximum growth rate larger than that of a single wave. The stable waves can be unstable by the effect of coupling. The instability is caused only by the mass variation of electrons, and the contribution of the ponderomotive force is negligibly small. The instability of copropagating waves has a convective nature and that due to the counterpropagating waves has an absolute nature, no matter how large the pump intensities are. The threshold of modulational instability is also considered for finite-length plasmas.

Magnetohydrodynamic parametric instabilities driven by a standing Alfvén wave in a low-β plasma

1996 ◽  
Vol 56 (2) ◽  
pp. 251-264 ◽  
Author(s):  
L. P. L. Oliveira ◽  
A. C.-L. Chian
Keyword(s):  
Mode Coupling ◽  
Wave Equations ◽  
Maximum Growth ◽  
Finite Amplitude ◽  
Maximum Growth Rate ◽  
Density Fluctuations ◽  
Alfven Wave ◽  
Magnetic Fluctuations ◽  
Parametric Instabilities ◽  
The Stability

The stability of a finite-amplitude standing Alfvén wave of circular polarization in a low-β plasma is studied using a set of nonlinearly coupled MHD wave equations. In the presence of a standing Alfvén pump, two distinct gratings associated with the density fluctuations are excited: those due to the ponderomotive beating of the pump magnetic field, and those due the induced magnetic fluctuations. The roles played by the two gratings in the mode coupling are analysed. Both convective and purely growing regimes of the MHD parametric instabilities can be produced by a standing Alfvén wave. In both regimes, the maximum growth rate increases as the pump amplitude increases, and decreases as increases. Tn the presence of the second grating, a new unstable convective regime appears that widens the overall instability bandwidth.

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