Azimuthal instability of the interface in a shear banded flow by direct visual observation

The stability of the shear banded flow of a Maxwellian fluid is studied from an experimental point of view using rheology and flow visualization with polarized light. We show that the one-layer homogeneous flow cannot sustain shear rates corresponding to the end of the stress plateau. The high shear rate branch is not found and the shear stress oscillates at the end of the plateau. An azimuthal instability appears: the shear induced band becomes unstable and the interface between the two bands undulates in time and space with a period τ, a wavelength λ and a wave vector k parallel to the direction of the tangential velocity.     

Unstable Surface Waves in Running Water

We consider the stability of periodic gravity free-surface water waves traveling downstream at a constant speed over a shear flow of finite depth. In case the free surface is flat, a sharp criterion of linear instability is established for a general class of shear flows with inflection points and the maximal unstable wave number is found. Comparison to the rigid-wall setting testifies that the free surface has a destabilizing effect. For a class of unstable shear flows, the bifurcation of nontrivial periodic traveling waves is demonstrated at all wave numbers. We show the linear instability of small nontrivial waves that appear after bifurcation at an unstable wave number of the background shear flow. The proof uses a new formulation of the linearized water-wave problem and a perturbation argument. An example of the background shear flow of unstable small-amplitude periodic traveling waves is constructed for an arbitrary vorticity strength and for an arbitrary depth, illustrating that vorticity has a subtle influence on the stability of free-surface water waves.     

The effect of two-dimensional shear flow on the stability of a crystal interface in a supercooled melt

A model is developed based on the time-related thermal diffusion equations to investigate the effects of twodimensional shear flow on the stability of a crystal interface in the supercooled melt of a pure substance.Similar to the three-dimensional shear flow as described in our previous paper,the two-dimensional shear flow can also be found to reduce the growth rate of perturbation amplitude.However,compared with the case of the Laplace equation for a steady-state thermal diffusion field,due to the existence of time partial derivatives of the temperature fields in the diffusion equation the absolute value of the gradients of the temperature fields increases,therefore destabilizing the interface.The circular interface is more unstable than in the case of Laplace equation without time partial derivatives.The critical stability radius of the crystal interface increases with shearing rate increasing.The stability effect of shear flow decreases remarkably with the increase of melt undercooling.     

The effect of two-dimensional shear flow on the stability of a crystal interface in the supercooled melt

A model is developed based on the time-related thermal diffusion equations to investigate the effects of two-dimensional shear flow on the stability of a crystal interface in the supercooled melt of pure substance. Similar to the three-dimensional shear flow as described in our previous paper, the two-dimensional shear flow can also be found to reduce the growth rate of perturbation amplitude. However, compared with the case of Laplace equation for steady state thermal diffusion field, due to the existence of time partial derivatives of the temperature fields in diffusion equation the absolute value of the gradients of the temperature fields increases, therefore destabilizing the interface. The circular interface is more unstable than in the case of Laplace equation without time partial derivatives. The critical stability radius of the crystal interface increases with shearing rate increasing. The stability effect of shear flow decreases remarkably with the increase of melt undercooling.     

Secondary instabilities of interfacial waves due to coupling with a long wave mode in a two-layer Couette flow

We report experiments on the stability of interfacial waves in a two-layer Couette flow. As the shear rate is increased, the periodic wave train arising from the primary instability undergoes a secondary instability which results in wave coalescence or nucleation, after a long transient. This secondary instability crucially involves the coupling with a long wave mode, which corresponds to variations of the mean interface level. These observations are favourably compared to stability results on travelling wave solutions for a set of two coupled equations, one for the envelope of a weakly unstable wave packet, and the other for the marginal long wave mode with zero wave number. A physical mechanism for this instability is proposed, as well as an interpretation for the onset of chaos.     

Two-dimensional perturbations in a scalar model for shear banding

We present an analytical study of a toy model for shear banding, without normal stresses, which uses a piecewise linear approximation to the flow curve (shear stress as a function of shear rate). This model exhibits multiple stationary states, one of which is linearly stable against general two-dimensional perturbations. This is in contrast to analogous results for the Johnson-Segalman model, which includes normal stresses, and which has been reported to be linearly unstable for general two-dimensional perturbations. This strongly suggests that the linear instabilities found in the Johnson-Segalman can be attributed to normal stress effects.     

Nonlinear dynamics of an interface between shear bands

We study numerically the nonlinear dynamics of a shear banding interface in two-dimensional planar shear flow, within the nonlocal Johnson-Segalman model. Consistent with a recent linear stability analysis, we find that an initially flat interface is unstable with respect to small undulations for a sufficiently small ratio of the interfacial width l to cell length L(x). The instability saturates in finite amplitude interfacial fluctuations. For decreasing l/L(x) these undergo a nonequilibrium transition from simple traveling interfacial waves with constant average wall stress, to periodically rippling waves with a periodic stress response. When multiple shear bands are present we find erratic interfacial dynamics and a stress response suggesting low dimensional chaos.     

Non-linear wave and Schrödinger equations

We investigate stability of periodic and quasiperiodic solutions of linear wave and Schrödinger equations under non-linear perturbations. We show in the case of the wave equations that such solutions are unstable for generic perturbations. For the Schrödinger equations periodic solutions are stable while the quasiperiodic ones are not. We extend these results to periodic solutions of non-linear equations.Partially supported by NSERC under Grant NA7901     

Transition to turbulence in a shear flow

We analyze the properties of a 19-dimensional Galerkin approximation to a parallel shear flow. The laminar flow with a sinusoidal shape is stable for all Reynolds numbers Re. For sufficiently large Re additional stationary flows occur; they are all unstable. The lifetimes of finite amplitude perturbations shows a fractal dependence on amplitude and Reynolds number. These findings are in accord with observations on plane Couette flow and suggest a universality of this transition scenario in shear flows.     

Evolution of local ideal helical perturbations in cylindrical plasma

The evolution of a local helical perturbation and its stability property for arbitrary magnetic shear configurations are investigated for the case of in cylindrical geometry. An analytic stability criterion has been obtained which predicts that a strong magnetic shear will enhance the instability in the positive shear region but enhance the stability in the negative shear region. The perturbations with the poloidal and toroidal perturbation mode numbers m/n=1/1 is most unstable due to the stabilizing terms increasing with m. For m/n=1/1 local perturbations in the conventional positive magnetic shear (PMS) configurations, a larger q_{min} exhibits a weaker shear in the core and is favourable to the stability, while in the reversed magnetic shear (RMS) configurations, a larger q_0 corresponds to a stronger positive shear in the middle region, which enhances the instability. No instabilities are found for m≥2 local perturbations. The stability for RMS configuration is not better than that for PMS configuration.     

Surface gravity waves in deep fluid at vertical shear flows

Special features of surface gravity waves in a deep fluid flow with a constant vertical shear of velocity is studied. It is found that the mean flow velocity shear leads to a nontrivial modification of the dispersive characteristics of surface gravity wave modes. Moreover, the shear induces generation of surface gravity waves by internal vortex mode perturbations. The performed analytical and numerical study show that surface gravity waves are effectively generated by the internal perturbations at high shear rates. The generation is different for the waves propagating in the different directions. The generation of surface gravity waves propagating along the main flow considerably exceeds the generation of surface gravity waves in the opposite direction for relatively small shear rates, whereas the latter wave is generated more effectively for high shear rates. From the mathematical standpoint, the wave generation is caused by non-self-adjointness of the linear operators that describe the shear flow.     

Instability and Stability of Rolls in the Swift–Hohenberg Equation

We develop a method for the stability analysis of bifurcating spatially periodic patterns under general nonperiodic perturbations. In particular, it enables us to detect sideband instabilities. We treat in all detail the stability question of roll solutions in the two–dimensional Swift–Hohenberg equation and derive a condition on the amplitude and the wave number of the rolls which is necessary and sufficent for stability. Moreover, we characterize the set of those wave vectors which give rise to unstable perturbations. Dedicated to Professor K. Kirchg?ssner on the occasion of his sixty-fifth birthday Received: 25 October 1996 / Accepted: 24 March 1997     

On the stability of periodic waves in a resonant medium

We study the stability of two periodic waves existing in two-level systems. It is shown that one periodic wave is unstable, while the other is stable up to one-dimensional perturbations. The results are obtained using the formalism of supersymmetric quantum mechanics for one-dimensional periodic potentials.     

Growth rates of the buoyancy-driven instability of an autocatalytic reaction front in a narrow cell

Experimental studies were performed on the buoyancy-driven instability of an autocatalytic reaction front in a quasi-2D cell. The unstable density stratification at an ascending front leads to convection that results in a fingerlike front deformation. The growth rates of the spatial modes of the instability are determined at the initial stage. A stabilization is found at higher wave numbers, while the system is unstable against low wave number perturbations. Whereas comparison with a reported model governed by Hele-Shaw flow fails, a two-dimensional Navier-Stokes model yields more satisfactory results. Still, present deviations suggest the presence of an additional mechanism that suppresses the growth.     

Stability of magnetohydrodynamic stratified shear flows

Summary The linear stability of a stratified shear flow of a perfectly conducting bounded fluid in the presence of a magnetic field aligned with the flow and buoyancy forces has been studied under Boussinesq approximation. A new upper bound has been obtained for the range of real and imaginary parts of the complex wave velocity for growing perturbations. The upper bound depends on minimum Richardson number, wave number, Alfvén velocity and basic flow velocity. H?iland's necessary criterion for instability of hydrodynamic stratified homogeneous shear flow is modified and its analog for nonhomogeneous magnetohydrodynamic cases is derived. Finally the upper bound for the growth rate ofKC _i and its variants, whereK is the wave number andC _i the imaginary part of complex wave velocity, is derived as the necessary condition of instability. All estimates remain valid even when the minimum richardson numberJ ₁, for some practical problems, exceeds 1/4 for growing perturbations. The authors of this paper have agreed to not receive the proofs for correction.     

Linear stability of particle-laden thin films

Recent experiments [Zhou, Dupuy, Bertozzi, and Hosoi, Phys. Rev. Lett., 94 (2005)] of particle-laden film flow on an incline demonstrate new behavior distinct from the well-known clear fluid case, including the formation of a particle-rich ridge which can stabilize the advancing contact line with respect to “fingering" perturbations. We consider a model similar to that of Zhou et al. with the additional regularizing effect of shear-induced diffusion. A linear stability analysis demonstrates that particle settling moderately reduces the growth rate of unstable modes, while increasing the most unstable wavelength.     

Resonant generation of internal waves on a model continental slope

We study internal wave generation in a laboratory model of oscillating tidal flow on a continental margin. Waves are found to be generated only in a near-critical region where the slope of the bottom topography matches that of internal waves. Fluid motion with a velocity an order of magnitude larger than that of the forcing occurs within a thin boundary layer above the bottom surface. The resonant wave is unstable because of strong shear; Kelvin-Helmholtz billows precede wave breaking. This work provides a new explanation for the intense boundary flows on continental slopes.     

Kelvin Helmholtz Instability in a Magnetoplasma

Two dimensional transverse Kelvin-Helmholtz (K-H) instability has been studied at the interface between the two fluids (plasma medium) of finite thickness in relative motion to each other. The perturbations on the interface are assumed to be electromagnetic and a dispersion relation is obtained. The interface (boundary) has been found to be unstable for a wide range of perturbation wavelengths (wave numbers kx, ky). It is shown that the modification introduced by electromagnetic (quasi-electrostatic) perturbations in comparison to electrostatic one is to reduce the growth rate of perturbations. The growth rate maximizes when kx = ky. The applications of this study have been discussed to explain some of the observed ionospheric (auroral arc formation) and magnetospheric (unstable magnetopause boundary, hydromagnetic pulsations) phenomena.     

Rayleigh-Taylor instability in a spherical configuration: A viscous potential flow approach

In this paper, Rayleigh-Taylor (RT) instability at the spherical interface is analyzed through the potential flow theory of viscous fluids. The linear stability theory is utilized including normal mode procedure. The viscous, incompressible fluids are confined in the spherical shell and the interface distortion is expressed in terms of spherical harmonic functions. We achieve a quadratic equation for the illustration of instability/stability criterion. We also compute the highest growth of perturbations and corresponding mode. The Plane configuration results are recovered when the mode of perturbations is very large. Viscosity is found to stabilize the interface while the Weber number and Reynolds number have destabilizing nature.     

Inhomogeneous oscillatory structures in fractional reaction-diffusion systems

We perform a study of the two component fractional reaction-diffusion system with cubic nonlinearity. The linear stage of the system stability is studied for different values of the system parameters. It is shown that for a certain value of the fractional derivatives index, a new type of instability takes place with respect to perturbations of finite wave number. As a result, inhomogeneous oscillations with this wave number become unstable and lead to nonlinear oscillations which result in spatial oscillatory structure formation.