Surfactant- and gravity-dependent instability of two-layer channel flows: linear theory covering all wavelengths. Part 1. ‘Long-wave’ regimes

2019 ◽  
Vol 863 ◽  
pp. 150-184 ◽  
Author(s):  
Alexander L. Frenkel ◽  
David Halpern ◽  
Adam J. Schweiger
Keyword(s):  
Growth Rate ◽  
Dispersion Equation ◽  
Normal Modes ◽  
Bond Number ◽  
Velocity Shear ◽  
Immiscible Fluid ◽  
Parallel Plates ◽  
Linear Eigenvalue Problem ◽  
Insoluble Surfactant ◽  
Flow Disturbances

A linear stability analysis of a two-layer plane Couette flow of two immiscible fluid layers with different densities, viscosities and thicknesses, bounded by two infinite parallel plates moving at a constant relative velocity to each other, with an insoluble surfactant monolayer along the interface and in the presence of gravity is carried out. The normal modes approach is applied to the equations governing flow disturbances in the two layers. These equations, together with boundary conditions at the plates and the interface, yield a linear eigenvalue problem. When inertia is neglected the velocity amplitudes are the linear combinations of certain hyperbolic functions, and a quadratic dispersion equation for the increment, that is the complex growth rate, is obtained, where coefficients depend on the aspect ratio, the viscosity ratio, the basic velocity shear, the Marangoni number $Ma$ that measures the effects of surfactant and the Bond number $Bo$ that measures the influence of gravity. An extensive investigation is carried out that examines the stabilizing or destabilizing influences of these parameters. Since the dispersion equation is quadratic in the growth rate, there are two continuous branches of the normal modes: a robust branch that exists even with no surfactant, and a surfactant branch that, to the contrary, vanishes when $Ma\downarrow 0$. Regimes have been uncovered with crossings of the two dispersion curves, their reconnections at the point of crossing and separations as $Bo$ changes. Due to the availability of the explicit forms for the growth rates, in many instances the numerical results are corroborated with analytical asymptotics.

scholarly journals Surfactant and gravity dependent instability of two-layer Couette flows and its nonlinear saturation

2017 ◽  
Vol 826 ◽  
pp. 158-204 ◽  
Author(s):  
Alexander L. Frenkel ◽  
David Halpern
Keyword(s):  
Evolution Equations ◽  
Normal Modes ◽  
Gravitational Effect ◽  
Bond Number ◽  
Nonlinear Evolution Equations ◽  
Immiscible Fluid ◽  
Base Shear ◽  
Diffusion Term ◽  
Insoluble Surfactant ◽  
Infinite Case

A horizontal channel flow of two immiscible fluid layers with different densities, viscosities and thicknesses, subject to vertical gravitational forces and with an insoluble surfactant monolayer present at the interface, is investigated. The base Couette flow is driven by the uniform horizontal motion of the channel walls. Linear and nonlinear stages of the (inertialess) surfactant and gravity dependent long-wave instability are studied using the lubrication approximation, which leads to a system of coupled nonlinear evolution equations for the interface and surfactant disturbances. The (inertialess) instability is a combined result of the surfactant action characterized by the Marangoni number $Ma$ and the gravitational effect corresponding to the Bond number $Bo$ that ranges from $-\infty$ to $\infty$. The other parameters are the top-to-bottom thickness ratio $n$, which is restricted to $n\geqslant 1$ by a reference frame choice, the top-to-bottom viscosity ratio $m$ and the base shear rate $s$. The linear stability is determined by an eigenvalue problem for the normal modes, where the complex eigenvalues (determining growth rates and phase velocities) and eigenfunctions (the amplitudes of disturbances of the interface, surfactant, velocities and pressures) are found analytically by using the smallness of the wavenumber. For each wavenumber, there are two active normal modes, called the surfactant and the robust modes. The robust mode is unstable when $Bo/Ma$ falls below a certain value dependent on $m$ and $n$. The surfactant branch has instability for $m<1$, and any $Bo$, although the range of unstable wavenumbers decreases as the stabilizing effect of gravity represented by $Bo$ increases. Thus, for certain parametric ranges, even arbitrarily strong gravity cannot completely stabilize the flow. The correlations of vorticity-thickness phase differences with instability, present when gravitational effects are neglected, are found to break down when gravity is important. The physical mechanisms of instability for the two modes are explained with vorticity playing no role in them. This is in marked contrast to the dynamical role of vorticity in the mechanism of the well-known Yih instability due to effects of inertia, and is contrary to some earlier literature. Unlike the semi-infinite case that we previously studied, a small-amplitude saturation of the surfactant instability is possible in the absence of gravity. For certain $(m,n)$-ranges, the interface deflection is governed by a decoupled Kuramoto–Sivashinsky equation, which provides a source term for a linear convection–diffusion equation governing the surfactant concentration. When the diffusion term is negligible, this surfactant equation has an analytic solution which is consistent with the full numerics. Just like the interface, the surfactant wave is chaotic, but the ratio of the two waves turns out to be constant.

scholarly journals Surfactant- and gravity-dependent instability of two-layer channel flows: linear theory covering all wavelengths. Part 2. Mid-wave regimes

2019 ◽  
Vol 863 ◽  
pp. 185-214 ◽  
Author(s):  
Alexander L. Frenkel ◽  
David Halpern ◽  
Adam J. Schweiger
Keyword(s):  
Normal Modes ◽  
Bond Number ◽  
Base Shear ◽  
System Parameters ◽  
Critical Curves ◽  
Extremal Points ◽  
Insoluble Surfactant ◽  
Fluid Layers ◽  
Instability Regions ◽  
Unstable Branch

The joint effects of an insoluble surfactant and gravity on the linear stability of a two-layer Couette flow in a horizontal channel are investigated. The inertialess instability regimes are studied for arbitrary wavelengths and with no simplifying requirements on the system parameters: the ratio of thicknesses of the two fluid layers; the viscosity ratio; the base shear rate; the Marangoni number $Ma$; and the Bond number $Bo$. As was established in the first part of this investigation (Frenkel, Halpern & Schweiger, J. Fluid Mech., vol. 863, 2019, pp. 150–184), a quadratic dispersion equation for the complex growth rate yields two, largely continuous, branches of the normal modes, which are responsible for the flow stability properties. This is consistent with the surfactant instability case of zero gravity studied in Halpern & Frenkel (J. Fluid Mech., vol. 485, 2003, pp. 191–220). The present paper focuses on the mid-wave regimes of instability, defined as those having a finite interval of unstable wavenumbers bounded away from zero. In particular, the location of the mid-wave instability regions in the ($Ma$, $Bo$)-plane, bounded by their critical curves, depending on the other system parameters, is considered. The changes of the extremal points of these critical curves with the variation of external parameters are investigated, including the bifurcation points at which new extrema emerge. Also, it is found that for the less unstable branch of normal modes, a mid-wave interval of unstable wavenumbers may sometimes coexist with a long-wave one, defined as an interval having a zero-wavenumber endpoint.

scholarly journals Magnetic field generation in a jet-sheath plasma via the kinetic Kelvin-Helmholtz instability

2013 ◽  
Vol 31 (9) ◽  
pp. 1535-1541 ◽  
Author(s):  
K.-I. Nishikawa ◽  
P. Hardee ◽  
B. Zhang ◽  
I. Duţan ◽  
M. Medvedev ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Growth Rate ◽  
Magnetic Fields ◽  
Electric Field ◽  
Flow Direction ◽  
Transverse Magnetic ◽  
Velocity Shear ◽  
Helmholtz Instability ◽  
Magnetic Field Generation ◽  
Relativistic Jet

Abstract. We have investigated the generation of magnetic fields associated with velocity shear between an unmagnetized relativistic jet and an unmagnetized sheath plasma. We have examined the strong magnetic fields generated by kinetic shear (Kelvin–Helmholtz) instabilities. Compared to the previous studies using counter-streaming performed by Alves et al. (2012), the structure of the kinetic Kelvin–Helmholtz instability (KKHI) of our jet-sheath configuration is slightly different, even for the global evolution of the strong transverse magnetic field. In our simulations the major components of growing modes are the electric field Ez, perpendicular to the flow boundary, and the magnetic field By, transverse to the flow direction. After the By component is excited, an induced electric field Ex, parallel to the flow direction, becomes significant. However, other field components remain small. We find that the structure and growth rate of KKHI with mass ratios mi/me = 1836 and mi/me = 20 are similar. In our simulations saturation in the nonlinear stage is not as clear as in counter-streaming cases. The growth rate for a mildly-relativistic jet case (γj = 1.5) is larger than for a relativistic jet case (γj = 15).

scholarly journals Velocity Shear Instabilities in the Anisotropic Solar Wind

1985 ◽  
Vol 107 ◽  
pp. 371-374
Author(s):  
Stefano Migliuolo
Keyword(s):  
Magnetic Field ◽  
Growth Rate ◽  
Solar Wind ◽  
Magnetic Energy ◽  
Equilibrium Temperature ◽  
Acoustic Mode ◽  
Plasma Pressure ◽  
Velocity Shear ◽  
Linear Growth Rate ◽  
Quasilinear Theory

The linear and quasilinear theory of perturbations in finite-β (β is the ratio of plasma pressure to magnetic energy density), collisionless plasmas, that have sheared (velocity) flows, is developed. A simple, one-dimensional magnetic field geometry is assumed to adequately represent solar wind conditions near the sun (i.e., at R ≃ 0.3 AU). Two modes are examined in detail: an ion-acoustic mode (finite-β stabilized) and a compressional Alfven mode (finite-β threshold, high-β stabilization). The role played by equilibrium temperature anisotropies, in the linear stability of these modes, is also presented. From the quasilinear theory, two results are obtained. First, the feedback of these waves on the state of the wind is such as to heat (cool) the ions in the direction perpendicular (parallel) to the equilibrium magnetic field. The opposite effect is found for the electrons. This is in qualitative agreement with the observed anisotropies of ions and electrons, in fast solar wind streams. Second, these quasilinear temperature changes are shown to result in a quasilinear growth rate that is lower than the linear growth rate, suggesting saturation of these instabilities.

Notes on the stability problem for inhomogeneous equilibria

1983 ◽  
Vol 29 (2) ◽  
pp. 275-286 ◽  
Author(s):  
K. R. Symon
Keyword(s):  
Growth Rate ◽  
Normal Mode ◽  
Stability Problem ◽  
Normal Modes ◽  
Electromagnetic Mode ◽  
Spectral Matrix ◽  
First Order ◽  
Real Frequency ◽  
Special Cases ◽  
The Stability

Several conclusions regarding the stability of inhomogeneous Vlasov equilibria are drawn from earlier work. A technique is presented for generating first-order formulae for the change δω in the frequency of any normal mode, when a parameter λ characterizing the equilibrium is changed slightly. Several applications are given, including a first-order calculation of the growth rate or damping of an electromagnetic mode due to the presence of plasma. A condition is derived for the existence of a normal mode with real frequency. When there are ignorable co-ordinates, the normal modes can be written in the form of waves propagating in the ignorable directions. The character of the modes depends on certain symmetries in the dynamic spectral matrix. Special cases arise when the orbits can be approximated in certain ways.

scholarly journals Kelvin-Helmholtz Instabilities in a Magnetised Compressible Plasma (Sheared Flow)

1985 ◽  
Vol 107 ◽  
pp. 375-377
Author(s):  
T. P. Ray ◽  
A. I. Ershkovich
Keyword(s):  
Magnetic Field ◽  
Dispersion Equation ◽  
Uniform Magnetic Field ◽  
Linear Velocity ◽  
Velocity Shear ◽  
Compressible Plasma ◽  
Sheared Flow ◽  
Method Of Solution

We have investigated Kelvin-Helmholtz (K-H) instabilities for a homogeneous compressible plasma containing a uniform magnetic field and a linear velocity shear. A derivation of the relevant K-H dispersion equation and details regarding method of solution are given elsewhere (submitted to Mon. Not. R. Astr. Soc.). We present here an outline of our results.

Linear Instability Analysis of an Annular Liquid Sheet With Inner and Outer Gas Flow

2009 ◽  
Author(s):  
Fathollah Ommi ◽  
Seid Askari Mahdavi ◽  
S. Mostafa Hosseinalipour ◽  
Ehsan Movahednejad
Keyword(s):  
Growth Rate ◽  
Dispersion Equation ◽  
Liquid Sheet ◽  
Linear Instability ◽  
Wave Number ◽  
Unstable Wave ◽  
Instability Analysis ◽  
Air Stream ◽  
Linear Instability Analysis ◽  
Air Streams

A linear instability analysis of an inviscid annular liquid sheet emanating from an atomizer subjected to inner and outer swirling air streams has been carried out. The dimensionless dispersion equation that governs the instability is derived. The dispersion equation solved by Numerical method to investigate the effects of the liquid-gas swirl orientation on the maximum growth rate and its corresponding unstable wave number that it produces the finest droplets. To understand the effect of air swirl orientation with respect to liquid swirl direction, four possible combinations with both swirling air streams with respect to the liquid swirl direction have been considered. Results show that at low liquid swirl Weber number a combination of co-inner air stream and counter-outer air stream has the largest most unstable wave number and shortest breakup length. The combination of inner and the outer air stream co-rotating with the liquid has the highest growth rate.

A conjecture on the least stable mode for the energy stability of plane parallel flows

2019 ◽  
Vol 881 ◽  
pp. 794-814 ◽  
Author(s):  
Xiangming Xiong ◽  
Zhi-Min Chen
Keyword(s):  
Critical Reynolds Number ◽  
Normal Modes ◽  
Slip Boundary Condition ◽  
Energy Stability ◽  
Parallel Plates ◽  
Stable Mode ◽  
Slip Boundary ◽  
Parallel Flows ◽  
Parallel Shear Flows ◽  
Decreasing Functions

In the energy stability theory, the critical Reynolds number is usually defined as the minimum of the first positive eigenvalue $R_{1}$ of an eigenvalue equation for all wavenumber pairs $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$, where $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ are the streamwise and spanwise wavenumbers of the normal mode. We prove that $(\cos \unicode[STIX]{x1D703}\pm 1)R_{1}$ are decreasing functions of $\unicode[STIX]{x1D703}=\arctan (\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D6FC})$ for the parallel flows between no-slip or slip parallel plates with or without variations in temperature. Numerical results inspire us to conjecture that $R_{1}$ is also a decreasing function of $\unicode[STIX]{x1D703}$ for the parallel shear flows under the no-slip boundary condition and without variations in temperature. If the conjecture is correct, the least stable normal modes for the energy stability will be streamwise vortices for these base flows.

scholarly journals Benchmark solution for the stability of plane Couette flow with net throughflow

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
B. M. Shankar ◽  
I. S. Shivakumara
Keyword(s):  
Reynolds Number ◽  
Fluid Flow ◽  
Couette Flow ◽  
Normal Modes ◽  
Base Flow ◽  
Parallel Plates ◽  
Plane Couette Flow ◽  
Benchmark Solution ◽  
Vertical Throughflow ◽  
The Stability

AbstractThis paper investigates the stability of an incompressible viscous fluid flow between relatively moving horizontal parallel plates in the presence of a uniform vertical throughflow. A linear stability analysis has been performed by employing the method of normal modes and the resulting stability equation is solved numerically using the Chebyshev collocation method. Contrary to the stability of plane Couette flow (PCF) to small disturbances for all values of the Reynolds number in the absence of vertical throughflow, it is found that PCF becomes unstable owing to the change in the sign of growth rate depending on the magnitude of throughflow. The critical Reynolds number triggering the instability is computed for different values of throughflow dependent Reynolds number and it is shown that throughflow instills both stabilizing and destabilizing effect on the base flow. It is seen that the direction of throughflow has no influence on the stability of fluid flow. A comparative study between plane Poiseuille flow and PCF has also been carried out and the similarities and differences are highlighted.

Strato-rotational instability without resonance

2018 ◽  
Vol 846 ◽  
pp. 815-833 ◽  
Author(s):  
Chen Wang ◽  
Neil J. Balmforth
Keyword(s):  
Three Dimensional ◽  
Normal Modes ◽  
Phase Speed ◽  
Flow Speed ◽  
Rotational Instability ◽  
Buoyancy Frequency ◽  
Critical Levels ◽  
Mean Flow ◽  
Linear Eigenvalue Problem ◽  
Resonant Interactions

Strato-rotational instability (SRI) is normally interpreted as the resonant interactions between normal modes of the internal or Kelvin variety in three-dimensional settings in which the stratification and rotation are orthogonal to both the background flow and its shear. Using a combination of asymptotic analysis and numerical solution of the linear eigenvalue problem for plane Couette flow, it is shown that such resonant interactions can be destroyed by certain singular critical levels. These levels are not classical critical levels, where the phase speed $c$ of a normal mode matches the mean flow speed $U$, but are a different type of singularity where $(c-U)$ matches a characteristic gravity-wave speed $\pm N/k$, based on the buoyancy frequency $N$ and streamwise horizontal wavenumber $k$. Instead, it is shown that a variant of SRI can occur due to the coupling of a Kelvin or internal wave to such ‘baroclinic’ critical levels. Two characteristic situations are identified and explored, and the conservation law for pseudo-momentum is used to rationalize the physical mechanism of instability. The critical level coupling removes the requirement for resonance near specific wavenumbers $k$, resulting in an extensive continuous band of unstable modes.

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