On Echo Chains in Landau damping: Traveling Wave-like Solutions and Gevrey 3 as a Linear Stability Threshold

Three-dimensional and time-dependent simulations of viscoelastic Taylor–Couette flow of dilute polymer solutions are performed using a fully implicit parallel spectral time-splitting algorithm to discover flow patterns with various spatio-temporal symmetries, namely rotating standing waves (RSWs), disordered oscillations (DOs) and solitary vortex structures referred to as oscillatory strips (OSs) and diwhirls (DWs). A detailed account of the impact of flow transitions on molecular conformation and viscoelastic stress, velocity profiles, hydrodynamic drag force and energy spectra of time-dependent flow states is presented. Overall, predicted pattern selection and flow features compare very favourably with experimental observations. For elasticity number E, that signifies the ratio of elastic to viscous forces, >0.1, and when the shear rate (cylinder rotation speed) is increased above the linear stability threshold, the circular Couette flow (CCF) becomes unstable to RSWs which are characterized by a checkerboard-like pattern in the space–time plot of radial velocity, implying symmetry between inflow/outflow (I/O) regions. As the shear rate is further increased, perturbations that break the I/O symmetry are amplified leading to DOs and/or flame-like patterns with spectral mechanical energy transfer reminiscent of elastically induced low-Reynolds-number turbulence. However, when the shear rate is decreased from those at which such chaotic states are observed, the radially inward acting polymer body force created by flow-induced molecular stretching causes the development of narrow inflow regions surrounded by much broader weak outflow domains. This promotes the formation of solitary vortex structures, which can be stationary and axisymmetric (DWs) or time-dependent (OSs). The dynamics of the formation of these structures by merging and coalescence of vortex pairs and the implication of such events on instantaneous hydrodynamic force are studied. For O(1) values of E, OSs and DWs appear approximately at constant values of the We, defined as the ratio of polymer relaxation time to the inverse shear rate in the gap. As shear rate is decreased further, DWs decay to CCF although at We values less than the linear stability threshold. The flow transitions are hysteretic with respect to We, as evidenced by a plot of drag force versus We.

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Electrohydrodynamic linear stability analysis of dielectric liquids subjected to unipolar injection in a rectangular enclosure with rigid sidewalls

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.537 ◽

2014 ◽

Vol 758 ◽

pp. 586-602 ◽

Cited By ~ 15

Author(s):

A. T. Pérez ◽

P. A. Vázquez ◽

Jian Wu ◽

P. Traoré

Keyword(s):

Aspect Ratio ◽

Linear Stability ◽

Element Formulation ◽

Nonlinear Behaviour ◽

Algebraic Equations ◽

Stability Threshold ◽

Rectangular Enclosure ◽

Linear Partial Differential Equations ◽

Symmetry Properties ◽

Unipolar Injection

AbstractWe investigate the linear stability threshold of a dielectric liquid subjected to unipolar injection in a two-dimensional rectangular enclosure with rigid boundaries. A finite element formulation transforms the set of linear partial differential equations that governs the system into a set of algebraic equations. The resulting system poses an eigenvalue problem. We calculate the linear stability threshold, as well as the velocity field and charge density distribution, as a function of the aspect ratio of the domain. The stability parameter as a function of the aspect ratio describes paths of symmetry-breaking bifurcation. The symmetry properties of the different linear modes determine whether these paths cross each other or not. The resulting structure has important consequences in the nonlinear behaviour of the system after the bifurcation points.

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Modal and non-modal stability analysis of electrohydrodynamic flow with and without cross-flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.134 ◽

2015 ◽

Vol 770 ◽

pp. 319-349 ◽

Cited By ~ 17

Author(s):

Mengqi Zhang ◽

Fulvio Martinelli ◽

Jian Wu ◽

Peter J. Schmid ◽

Maurizio Quadrio

Keyword(s):

Stability Analysis ◽

Electric Field ◽

Linear Stability ◽

Poiseuille Flow ◽

Cross Flow ◽

Transient Growth ◽

Unstable Flow ◽

Electrohydrodynamic Flow ◽

Stability Threshold ◽

Energy Growth

We report the results of a complete modal and non-modal linear stability analysis of the electrohydrodynamic flow for the problem of electroconvection in the strong-injection region. Convective cells are formed by the Coulomb force in an insulating liquid residing between two plane electrodes subject to unipolar injection. Besides pure electroconvection, we also consider the case where a cross-flow is present, generated by a streamwise pressure gradient, in the form of a laminar Poiseuille flow. The effect of charge diffusion, often neglected in previous linear stability analyses, is included in the present study and a transient growth analysis, rarely considered in electrohydrodynamics, is carried out. In the case without cross-flow, a non-zero charge diffusion leads to a lower linear stability threshold and thus to a more unstable flow. The transient growth, though enhanced by increasing charge diffusion, remains small and hence cannot fully account for the discrepancy of the linear stability threshold between theoretical and experimental results. When a cross-flow is present, increasing the strength of the electric field in the high-$\mathit{Re}$Poiseuille flow yields a more unstable flow in both modal and non-modal stability analyses. Even though the energy analysis and the input–output analysis both indicate that the energy growth directly related to the electric field is small, the electric effect enhances the lift-up mechanism. The symmetry of channel flow with respect to the centreline is broken due to the additional electric field acting in the wall-normal direction. As a result, the centres of the streamwise rolls are shifted towards the injector electrode, and the optimal spanwise wavenumber achieving maximum transient energy growth increases with the strength of the electric field.

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Linear Vibrations of Statically Indeterminate Rotors on Angular Misaligned Hydrodynamic Bearings

Volume 4: Manufacturing Materials and Metallurgy; Ceramics; Structures and Dynamics; Controls, Diagnostics and Instrumentation; Education; IGTI Scholar Award; General ◽

10.1115/99-gt-333 ◽

1999 ◽

Cited By ~ 2

Author(s):

Demetrio C. Zachariadis

Keyword(s):

Finite Elements ◽

Linear Stability ◽

Design Stage ◽

Angular Misalignment ◽

Hydrodynamic Bearings ◽

Early Design ◽

Rotor Systems ◽

Early Design Stage ◽

Stability Threshold ◽

Linear Vibrations

The influence of journal’s static and dynamic angular misalignment on the synchronous unbalance response and linear stability threshold of statically indeterminate rotors is analysed. Both short and finite bearing models are considered in order to calculate hydrodynamic reactions, and rotating parts are modeled using beam finite elements. The results provide descriptions of the effects of the consideration of the 32 coefficient bearing model on linear vibrations analyses, together with guidelines for early design stage identification of rotor systems sensitive to angular misalignments.

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Competition between the centrifugal and strato-rotational instabilities in the stratified Taylor–Couette flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.15 ◽

2018 ◽

Vol 840 ◽

pp. 5-24 ◽

Cited By ~ 6

Author(s):

Junho Park ◽

Paul Billant ◽

Jong-Jin Baik ◽

Jaemyeong Mango Seo

Keyword(s):

Stability Analysis ◽

Linear Stability ◽

Couette Flow ◽

Linear Stability Analysis ◽

Reynolds Numbers ◽

Axisymmetric Mode ◽

Stability Threshold ◽

Taylor Couette ◽

The Stability ◽

Taylor Couette Flow

The stably stratified Taylor–Couette flow is investigated experimentally and numerically through linear stability analysis. In the experiments, the stability threshold and flow regimes have been mapped over the ranges of outer and inner Reynolds numbers: $-2000<Re_{o}<2000$ and $0<Re_{i}<3000$, for the radius ratio $r_{i}/r_{o}=0.9$ and the Brunt–Väisälä frequency $N\approx 3.2~\text{rad}~\text{s}^{-1}$. The corresponding Froude numbers $F_{o}$ and $F_{i}$ are always much smaller than unity. Depending on $Re_{o}$ (or equivalently on the angular velocity ratio $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D6FA}_{o}/\unicode[STIX]{x1D6FA}_{i}$), three different regimes have been identified above instability onset: a weakly non-axisymmetric mode with low azimuthal wavenumber $m=O(1)$ is observed for $Re_{o}<0$ ($\unicode[STIX]{x1D707}<0$), a highly non-axisymmetric mode with $m\sim 12$ occurs for $Re_{o}>840$ ($\unicode[STIX]{x1D707}>0.57$) while both modes are present simultaneously in the lower and upper parts of the flow for $0\leqslant Re_{o}\leqslant 840$ ($0\leqslant \unicode[STIX]{x1D707}\leqslant 0.57$). The destabilization of these primary modes and the transition to turbulence as $Re_{i}$ increases have been also studied. The linear stability analysis proves that the weakly non-axisymmetric mode is due to the centrifugal instability while the highly non-axisymmetric mode comes from the strato-rotational instability. These two instabilities can be clearly distinguished because of their distinct dominant azimuthal wavenumber and frequency, in agreement with the recent results of Park et al. (J. Fluid Mech., vol. 822, 2017, pp. 80–108). The stability threshold and the characteristics of the primary modes observed in the experiments are in very good agreement with the numerical predictions. Moreover, we show that the centrifugal and strato-rotational instabilities are observed simultaneously for $0\leqslant Re_{o}\leqslant 840$ in the lower and upper parts of the flow, respectively, because of the variations of the local Reynolds numbers along the vertical due to the salinity gradient.

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Stability of the Landau resonance for drift modes in rotating tokamak plasma

Journal of Plasma Physics ◽

10.1017/s0022377803002253 ◽

2003 ◽

Vol 69 (5) ◽

pp. 371-382

Author(s):

E. ASP ◽

V. P. PAVLENKO ◽

S. M. REVENCHUK

Keyword(s):

Dispersion Relation ◽

Linear Stability ◽

High Velocity ◽

Velocity Shear ◽

Landau Damping ◽

Tokamak Plasma ◽

Drift Waves ◽

Two Dimensional

The linear stability of drift waves in a poloidally rotating tokamak plasma is considered. The derived dispersion relation features a peaking of the diamagnetic frequency which gives the drift modes an irreducible two-dimensional character. We then show that inverse Landau damping can be suppressed and even stabilized, if the flow's shear is strong. Even though the instability, excited by the Landau resonance, is stronger at a high velocity shear for positive rotation velocities, effects due to the rotation of the plasma can reverse the sign and induce damping of the two-dimensional drift modes. This stabilizing mechanism works only for positive rotation velocities. For negative rotation velocities, we show that only modes with high poloidal mode numbers are unstable.

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Linear Stability Analysis of Short Porous Journal Bearings—Use of the Brinkman Model

Journal of Tribology ◽

10.1115/1.2830600 ◽

1995 ◽

Vol 117 (1) ◽

pp. 199-202 ◽

Cited By ~ 2

Author(s):

Jaw-Ren Lin ◽

Chi-Chuan Hwang

Keyword(s):

Linear Stability ◽

Flow Model ◽

Slip Flow ◽

Journal Bearings ◽

Shear Stresses ◽

Brinkman Model ◽

Stability Threshold ◽

Viscous Shear ◽

Shear Effects ◽

The Stability

On the basis of a Brinkman model (BM), this paper predicts that the effects of viscous shear stresses on the linear stability of short porous journal bearings are apparent and not negligible. Compared with those of the slip-flow model (SFM) and the Darcy model (DM), the viscous shear effects provide a significant increase in the stability threshold speeds of short porous journal bearings.

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Nonlinear evolution of two fast-particle-driven modes near the linear stability threshold

Physics of Plasmas ◽

10.1063/1.3601136 ◽

2011 ◽

Vol 18 (6) ◽

pp. 062109 ◽

Cited By ~ 5

Author(s):

Jarosław Zaleśny ◽

Grzegorz Galant ◽

Mietek Lisak ◽

Sławomir Marczyński ◽

Paweł Berczyński ◽

...

Keyword(s):

Linear Stability ◽

Nonlinear Evolution ◽

Fast Particle ◽

Stability Threshold

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Fingering Instability of a Gravity-Driven Thin Film Flowing Down a Vertical Tube with Wall Slippage

Applied Sciences ◽

10.3390/app10010076 ◽

2019 ◽

Vol 10 (1) ◽

pp. 76

Author(s):

Chicheng Ma ◽

Shuaizhao Hu ◽

Guangxu Dong ◽

Bo Li

Keyword(s):

Thin Film ◽

Stability Analysis ◽

Linear Stability ◽

Linear Stability Analysis ◽

Traveling Wave ◽

Flow Instability ◽

Vertical Tube ◽

Petroleum Engineering ◽

Gravity Driven ◽

Wall Slippage

Inspired by the antiwetting property of pitcher plants, specialists have designed different functional material with slippery surfaces, and a directional slippery surface has been fabricated. This paper considers a gravity-driven liquid film coating the interior surface of a vertical tube, and different slippery lengths in the azimuthal direction and the axial direction are taken into account. The evolution equation of coating flow is derived using the thin film model, and time responses for two dimensional flow are calculated. Linear stability analysis (LSA) is given based on the traveling wave solutions, demonstrating that the axial slippery effect suppresses the flow instability and causes a larger traveling wave speed. Simultaneously, the azimuthal slippery effect plays a destabilizing role for perturbations with small wavenumbers and it is stabilizing for large wavenumbers. Direct simulations of the fingering flow patterns agree well with the linear stability analysis. Our results offer insight into the influence of wall slippage on the flow stability of liquid in petroleum engineering.

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