Experimental evidence for an intrinsic route to polymer melt fracture phenomena: a nonlinear instability of viscoelastic Poiseuille flow

The production rate of polymer fibers by extrusion is usually limited by the appearance of a series of instabilities ("melt fracture") that lead to unwanted undulations of the surface. We present both qualitative and quantitative experimental evidence that-in addition to previously known polymer-specific scenarios-there is an intrinsic route towards melt fracture type phenomena: a nonlinear ("subcritical") instability of viscoelastic Poiseuille flow.     

Melt fracture of entangled polymers*

We construct a model for a slippage plane in a sheared melt, based on a balance between reptation bridging and shear debonding. The resulting state could show up at rather low shear rates and be locally stable. But it is not easy to nucleate: the conventional entangled state is also locally stable. We propose that slippage occurs on solid walls: either at the container surface, or on dust particles floating in the melt. Slippage at solid/melt interfaces was studied (experimentally and theoretically) long ago. There is a critical stress for slippage: our estimate (for strong adsorption of melt chains on the solid) gives (plateau modulus) for typical cases. Thus, melt fracture is expected at moderate stresses, in agreement with observations by S.Q. Wang and coworkers.     

Structure-driven nonlinear instability as the origin of the explosive reconnection dynamics in resistive double tearing modes

The onset of abrupt magnetic reconnection events, observed in the nonlinear evolution of double tearing modes (DTM), is investigated via reduced resistive magnetohydrodynamic simulations. We have identified the critical threshold for the parameters characterizing the linear DTM stability leading to the bifurcation to the explosive dynamics. A new type of secondary instability is discovered that is excited once the magnetic islands on each rational surface reach a critical structure characterized here by the width and the angle rating their triangularization. This new instability is an island structure-driven nonlinear instability, identified as the trigger of the subsequent nonlinear dynamics which couples flow and flux perturbations. This instability only weakly depends on resistivity.     

Long waves in streamwise varying shear flows: new mechanisms for a weakly nonlinear instability

The weakly nonlinear dynamics of long waves embedded in marginally stable shear flows that vary in the streamwise direction are shown to be governed by a variable-coefficient Boussinesq equation. Depending on the local stability characteristics of the flow, new nonmodal or modal instabilities may emerge that serve as natural mechanisms for achieving amplitude thresholds necessary for weakly nonlinear instability and the transition to fully finite-amplitude states.     

二维不可压流体Kelvin-Helmholtz不稳定性的弱非线性研究

通过将扰动速度势展至三阶,提出了Kelvin-Helmholtz（KH）不稳定性的弱非线性理论.在模耦合过程中观察到一个重要的共振现象,共振使得模耦合过程变得相当复杂,单模扰动很快进入非线性区,产生大量高次谐波,共振加强了非线性作用.分析了单模扰动中二次和三次谐波产生效应,以及对基模指数增长的非线性校正.模拟结果支持了解析理论.利用该理论,分析了KH不稳定的非线性阈值问题. 关键词： Kelvin-Helmholtz不稳定性 弱非线性理论 非线性阈值     

表面张力对Rayleigh-Taylor不稳定性谐波的影响

Using the method of the parameter expansion up to the third order, explicitly investigates surface tension effect on harmonics at weakly nonlinear stage in Rayleigh-Taylor instability (RTI) for arbitrary Atwood numbers and compares the results with those of classical RTI within the framework of the third-order weakly nonlinear theory. It is found that surface tension strongly reduces the linear growth rate of time, resulting in mild growth of the amplitude of the fundamental mode, and changes amplitudes of the second and third harmonics, as is expressed as a tension factor coupling in amplitudes of the harmonics. On the one hand, surface tension can either decrease or increase the space amplitude; on the other hand, surface tension can also change their phases for some conditions which are explicitly determined.     

On the modulation instability of surface waves on a large-scale shear flow

The problem of the modulation instability of a weakly nonlinear quasi-monochromatic wave on the surface of deep water in the presence of a steady-state collinear large-scale inhomogeneous flow (e.g., of a jet type) has been considered. In a certain range of (obtuse) angles of incidence of the wave with respect to the flow direction, the steepness of the refracted wave increases considerably, which contributes to the enhancement of nonlinear effects, including the formation of so-called rogue waves. The corresponding nonlinear Schrödinger equation with variable coefficients suitable for the analysis of the modulation instability has been derived.     

Instability of moving gratings in photorefractive crystals

It is shown that a moving intensity grating in a photorefractive crystal with a sufficiently large lifetimemobility product can give rise to an instability against excitation of weakly damped space-charge waves. Increments and threshold of the instability are investigated for the threedimensional case. Subharmonics and other nonlinear scenarios of saturation are discussed.     

Melt fracture of entangled polymers ( small star, filled)

We construct a model for a slippage plane in a sheared melt, based on a balance between reptation bridging and shear debonding. The resulting state could show up at rather low shear rates and be locally stable. But it is not easy to nucleate: the conventional entangled state is also locally stable. We propose that slippage occurs on solid walls: either at the container surface, or on dust particles floating in the melt. Slippage at solid/melt interfaces was studied (experimentally and theoretically) long ago. There is a critical stress sigma* for slippage: our estimate (for strong adsorption of melt chains on the solid) gives sigma* approximately 1/3 (plateau modulus) for typical cases. Thus, melt fracture is expected at moderate stresses, in agreement with observations by S.Q. Wang and coworkers.     

Numerical investigation of hydrodynamics behaviour of melt layer during laser cutting of steel

Understanding of the melt layer hydrodynamic behaviour during laser-cutting process under gas jet assistance is of high importance for cut quality control. In the present work, a numerical model is developed to calculate the three-dimensional behaviour of the melt flow on the kerf front, while an inert gas jet interacts with the melt film. Fluent CFD code is used to solve the governing hydrodynamic equations by finite volume method. The results show that the melt flow on the kerf front reveals a strong instability, which depends on the cutting speed and on the gas jet velocity. Global flow behaviour (gas and molten metal flows) computed using a laminar model, reveals oscillations of the gas–metal liquid interface, which is assimilated to Kelvin–Helmholtz instability. The origin of this instability is discussed in terms of instabilities in thermal dynamics and hydrodynamics. Instability in thermal dynamics is related to the localized melting, while the instability in hydrodynamics is governed by forces balance between gas and resistant surface tension.     

Rogue edge waves in the ocean

Theoretically possible rogue edge wave are studied over cylindrical bottom in the framework of nonlinear shallow water equations in a weakly nonlinear limit. The nonlinear mechanisms (nonlinear dispersion enhancement, modulation instability and multimodal interactions) of possible anomalous edge wave appearance are analyzed.     

Modulational instability and localized breather in discrete Schrödinger equation with helicoidal hopping and a power-law nonlinearity

We investigate the discrete nonlinear Schrödinger model with helicoidal hopping and a power-law nonlinearity, motivated by the tunable nonlinearity in the model of DNA chain and ultra-cold atoms trapped in a helix-shaped optical trap. In the study of modulational instability, we find a successive destabilization along with increasing nonlinear-power. In particular, the critical amplitudes of second-stage instability decrease as nonlinear-power increases. Furthermore, it is shown that information on the stability properties of weakly localized solutions can be inferred from the plane-wave modulational instability results. This link enable us to analytically estimate the critical parameters at which the breather solutions turn unstable, and find these parameters are dramatically influenced by the nonlinear-power. The stability properties of localized breathers perform an obvious change when the nonlinear power crosses a critical value ${\gamma }_{cr}$. It is reflected that at weak nonlinearity the breathers exhibit monostability, while exceeding ${\gamma }_{cr}$ the bistability and instability will set in. The interplay between nonlinear-power and long-range hopping on the stability properties of breathers is also discussed in detail.     

Spatial Unfolding of Elementary Bifurcations

We consider solutions of a partial differential equation which are homogeneous in space and stationary or periodic in time. We study the stability with respect to large wavelength perturbations and the weakly nonlinear behavior around these solutions, especially when they are close to bifurcations for the ordinary differential equation governing the homogeneous solutions of the PDE. We distinguish cases where a spatial parity symmetry holds. All bifurcations occurring generically for two-dimensional ODES are treated. Our main result is that for almost homoclinic periodic solutions instability is generic.     

Convective Brillouin amplification in a homogeneous plasma slab

The coupled-mode Brillouin backscatter equations, describing weakly nonlinear decay of a damped pump into two damped daughter waves, are solved analytically in the convective instability regime, yielding the convective instability threshold and backward amplification of a noise source.     

Relativistic degenerate effects of electrons and positrons on modulational instability of quantum ion acoustic waves in dense plasmas with two polarity ions

The nonlinear propagation of quantum ion acoustic wave(QIAW) is investigated in a four-component plasma composed of warm classical positive ions and negative ions,as well as inertialess relativistically degenerate electrons and positrons.A nonlinear Schrodinger equation is derived by using the reductive perturbation method,which governs the dynamics of QIAW packets.The modulation instability analysis of QIAWs is considered based on the typical parameters of the white dwarf.The results exhibit that both in the weakly relativistic limit and in the ultrarelativistic limit,the modulational instability regions are sensitively dependent on the ratios of temperature and number density of negative ions to those of positive ions respectively,and on the relativistically degenerate effect as well.     

Nonlinear competition between asters and stripes in filament-motor systems

A model for polar filaments interacting via molecular motor complexes is investigated which exhibits bifurcations to spatial patterns. It is shown that the homogeneous distribution of filaments, such as actin or microtubules, may become either unstable with respect to an orientational instability of a finite wave number or with respect to modulations of the filament density, where long-wavelength modes are amplified as well. Above threshold nonlinear interactions select either stripe patterns or periodic asters. The existence and stability ranges of each pattern close to threshold are predicted in terms of a weakly nonlinear perturbation analysis, which is confirmed by numerical simulations of the basic model equations. The two relevant parameters determining the bifurcation scenario of the model can be related to the concentrations of the active molecular motors and of the filaments, respectively, which both could be easily regulated by the cell.     

Weakly turbulent instability of anti-de Sitter spacetime

We study the nonlinear evolution of a weakly perturbed anti-de?Sitter (AdS) space by solving numerically the four-dimensional spherically symmetric Einstein-massless-scalar field equations with negative cosmological constant. Our results suggest that AdS space is unstable under arbitrarily small generic perturbations. We conjecture that this instability is triggered by a resonant mode mixing which gives rise to diffusion of energy from low to high frequencies.     

The role of linear damping for explosive instabilities

It is shown that conservation laws of energy and momentum contain often sufficient information on various weakly nonlinear interactions. The changes in the temporal development of an explosive instability or its stabilization due to the linear damping of the interacting waves are discussed in more detail.Nademlýnská 600, Praha 9, Czechoslovakia.     

Instabilities of waves in nonlinear disordered media

We develop a self-consistent theory of temporal fluctuations of a speckle pattern resulting from the multiple scattering of a coherent wave in a weakly nonlinear disordered medium. The speckle pattern is shown to become unstable if the nonlinearity exceeds a threshold value. The instability is due to a feedback provided by the multiple scattering and manifests itself in spontaneous fluctuations of the scattered intensity. The development of instability is independent of the sign of nonlinearity.     

Nonlinear saturation of the thermoacoustic instability

A weakly nonlinear theory of the thermoacoustic instability in gas-filled tubes is developed in the time domain by exploiting the difference between the instability time scale and the period of standing waves. By carrying the expansion to fourth order in the perturbation parameter, explicit results for the initial growth, nonlinear evolution, and final saturation are obtained. The dependence of the saturation amplitude upon the temperature difference in the stack, the tube geometry, stack plate spacing, Prandtl number, and other parameters is illustrated.