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 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.

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.

Beating in Three-Mass Systems Under Step Velocity Input. Part 2—Experimental

1967 ◽  
Vol 89 (2) ◽  
pp. 290-295
Author(s):  
W. F. Stokey ◽  
R. A. Wunder
Keyword(s):  
Experimental Study ◽  
Normal Modes ◽  
Experimental Results ◽  
System Parameters ◽  
Step Velocity ◽  
Velocity Input

An experimental study of the response of three-mass systems to step changes of the base velocity is described. Recordings that show the beating that can occur between the vibrations in two normal modes of such systems are included. The excitation of these modes for various values of the system parameters is shown by calculated curves and by experimental results.

Chaotic dynamics for a class of single-machine-infinite bus power system

2016 ◽  
Vol 24 (3) ◽  
pp. 582-587 ◽  
Author(s):  
Liangqiang Zhou ◽  
Fangqi Chen
Keyword(s):  
Power Systems ◽  
Power System ◽  
Single Machine ◽  
Chaotic Dynamics ◽  
Numerical Results ◽  
Chaotic Motions ◽  
System Parameters ◽  
Critical Curves ◽  
Single Machine Infinite Bus

The chaotic motions are investigated both analytically and numerically for a class of single-machine-infinite bus power systems. The mechanism and parametric conditions for chaotic motions of this system are obtained rigorously. The critical curves separating the chaotic and non-chaotic regions are presented. The chaotic feature of the system parameters is discussed in detail. It is shown that there exist chaotic bands for this system, and the bands vary with the system parameters. Some new dynamical phenomena are presented. Numerical results are given, which verify the analytical ones.

Parametric Instability of a Rotating Circular Ring With Moving, Time-Varying Springs

2005 ◽  
Vol 128 (2) ◽  
pp. 231-243 ◽  
Author(s):  
Sripathi Vangipuram Canchi ◽  
Robert G. Parker
Keyword(s):  
Parametric Instability ◽  
Circular Ring ◽  
Time Varying ◽  
Ring Gear ◽  
Geometric Description ◽  
Bending Vibrations ◽  
System Parameters ◽  
Ring Rotation ◽  
Special Cases ◽  
Instability Regions

Parametric instabilities of in-plane bending vibrations of a rotating ring coupled to multiple, discrete, rotating, time-varying stiffness spring-sets of general geometric description are investigated in this work. Instability boundaries are identified analytically using perturbation analysis and given as closed-form expressions in the system parameters. Ring rotation and time-varying stiffness significantly affect instability regions. Different configurations with a rotating and nonrotating ring, and rotating spring-sets are examined. Simple relations governing the occurrence and suppression of instabilities are discussed for special cases with symmetric circumferential spacing of spring-sets. These results are applied to identify possible conditions of ring gear instability in example planetary gears.

Stability of surfactant-laden core–annular flow and rod–annular flow to non-axisymmetric modes

2013 ◽  
Vol 716 ◽  
Author(s):  
M. G. Blyth ◽  
Andrew P. Bassom
Keyword(s):  
Linear Stability ◽  
Annular Flow ◽  
Convective Motion ◽  
Straight Pipe ◽  
Pipe Axis ◽  
Insoluble Surfactant ◽  
Fluid Layers ◽  
Axisymmetric Disturbances ◽  
Quiescent Fluid ◽  
Fluid Configuration

AbstractThe linear stability of core–annular fluid arrangements are considered in which two concentric viscous fluid layers occupy the annular region within a straight pipe with a solid rod mounted on its axis when the interface between the fluids is coated with an insoluble surfactant. The linear stability of this arrangement is studied in two scenarios: one for core–annular flow in the absence of the rod and the second for rod–annular flow when the rod moves parallel to itself along the pipe axis at a prescribed velocity. In the latter case the effect of convective motion on a quiescent fluid configuration is also considered. For both flows the emphasis is placed on non-axisymmetric modes; in particular their impact on the recent stabilization to axisymmetric modes at zero Reynolds number discovered by Bassom, Blyth and Papageorgiou (J. Fluid. Mech., vol. 704, 2012, pp. 333–359) is assessed. It is found that in general non-axisymmetric disturbances do not undermine this stabilization, but under certain conditions the flow may be linearly stable to axisymmetric disturbances but linearly unstable to non-axisymmetric disturbances.

Buoyancy-driven interactions of viscous drops with deforming interfaces

2001 ◽  
Vol 446 ◽  
pp. 253-269 ◽  
Author(s):  
JOSEPH KUSHNER ◽  
MICHAEL A. ROTHER ◽  
ROBERT H. DAVIS
Keyword(s):  
Castor Oil ◽  
Viscosity Ratio ◽  
Bond Number ◽  
Glycerol Water ◽  
System Parameters ◽  
Viscous Drops ◽  
Water Drops ◽  
Pass Through ◽  
Theoretical Results ◽  
Good Agreement

Experiments were conducted on the interactions of two different-sized deformable drops moving due to gravity in an immiscible viscous fluid at low Reynolds number. As the drops come close to each other, several interactions are possible: (i) separation of the drops, (ii) capture of the smaller drop behind the larger drop, (iii) breakup of the smaller drop into two or more drops, and (iv) pass-through of one drop through the other, with possible cycle interaction or leap-frogging. The interactions depend on several system parameters, including the drop-to-medium viscosity ratio, the radius ratio of the two drops, the initial horizontal offset of the two drops at large vertical separation, and the gravitational Bond number (which represents the ratio of buoyant forces to interfacial tension forces for the larger drop and describes how much the drops will deform). Experimental analysis was conducted by videotaping trajectories of glycerol–water drops of various compositions falling in castor oil. The results show good agreement with available theoretical results, both for interaction maps and individual trajectories. The results also provide data beyond the present limitations of theoretical algorithms and reveal the new pass-through phenomenon.

scholarly journals Chaotic Motions of the Duffing-Van der Pol Oscillator with External and Parametric Excitations

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Liangqiang Zhou ◽  
Fangqi Chen
Keyword(s):  
Numerical Results ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Chaotic Motions ◽  
System Parameters ◽  
Critical Curves

The chaotic motions of the Duffing-Van der Pol oscillator with external and parametric excitations are investigated both analytically and numerically in this paper. The critical curves separating the chaotic and nonchaotic regions are obtained. The chaotic feature on the system parameters is discussed in detail. Some new dynamical phenomena including the controllable frequency are presented for this system. Numerical results are given, which verify the analytical ones.

Beating in Three-Mass Systems Under Step Velocity Input. Part 1—Analytical

1967 ◽  
Vol 89 (2) ◽  
pp. 285-289
Author(s):  
W. F. Stokey ◽  
R. A. Wunder
Keyword(s):  
Beat Frequency ◽  
Normal Modes ◽  
Static Force ◽  
Maximum Acceleration ◽  
System Parameters ◽  
Step Velocity ◽  
Wide Range ◽  
The Masses ◽  
Velocity Input

The response of three-mass systems to step changes of the base velocity is discussed. It is shown that beating, which may occur between the vibrations in two normal modes, can result in a large increase in the maximum acceleration, or effective static force, undergone by one of the masses. Included are curves which show the maximum effective static forces on the system masses, and the beat frequency, for a wide range of system parameters.

Parametric Instability of a Rotating Circular Ring With Moving, Time-Varying Springs

2007 ◽  
Author(s):  
Sripathi Vangipuram Canchi ◽  
Robert G. Parker
Keyword(s):  
Parametric Instability ◽  
Circular Ring ◽  
Time Varying ◽  
Ring Gear ◽  
Geometric Description ◽  
Bending Vibrations ◽  
System Parameters ◽  
Ring Rotation ◽  
Special Cases ◽  
Instability Regions

Parametric instabilities of in-plane bending vibrations of a rotating ring coupled to multiple, discrete, rotating, time-varying stiffness spring-sets of general geometric description are investigated in this work. Instability boundaries are identified analytically using perturbation analysis and given as closed-form expressions in the system parameters. Ring rotation and time-varying stiffness significantly affect instability regions. Different configurations with a rotating and non-rotating ring, and rotating spring-sets are examined. Simple relations governing the occurrence and suppression of instabilities are discussed for special cases with symmetric circumferential spacing of spring-sets. These results are applied to identify possible conditions of ring gear instability in example planetary gears.

Sign in / Sign up

Export Citation Format

Share Document