Thermal instabilities in melt spinning of viscoelastic fibers

Nonisothermal melt spinning of viscoelastic fibers for which the viscosity varies in a step-like manner with respect to temperature is studied in this work. A set of one-dimensional equations based on the slender-jet approximation and the upper convected Maxwell model is used to describe the melt spinning process. The process is characterized by the force required to pull the fiber, the strength of external heating, and the draw ratio, the square of the ratio of the fiber diameter at the spinneret to that at the take-up roller. For low levels of elasticity and sufficiently strong external heating, there can be three pulling forces consistent with the same draw ratio, similar to the Newtonian case studied by Wylie et al. [31]. For higher levels of elasticity, the process exhibits a draw ratio plateau where the draw ratio hardly changes with the pulling force, reflecting a competition between thermal and elastic effects. As in the Newtonian case, external heating introduces a new instability – termed thermal instability – that is absent in isothermal systems. Linear stability analysis reveals that external heating improves stability for low levels of elasticity, but can worsen stability for higher levels of elasticity, which is again a consequence of the interplay between thermal and elastic effects. Nonlinear simulations indicate that the predictions of linear stability analysis carry over to the nonlinear regime, and show that unstable systems exhibit limit-cycle behavior. The results of the present work demonstrate a possible mechanism through which external heating can stabilize the melt spinning of viscoelastic fibers.     

Effect of inertia and gravity on the draw resonance in high-speed film casting of Newtonian fluids

The interplay between inertia and gravity is examined for Newtonian film casting in this study. Both linear and nonlinear stability analyses are carried out. Linear stability analysis indicates that while both inertia and gravity enhance the stability in film casting, inertia plays a more dominant role regarding the critical draw ratio. In contrast, the disturbance frequency is more sensitive to the effect of gravity. The nonlinear results show that at the critical draw ratio, the system oscillates harmonically, indicating the onset of a Hopf bifurcation. For a draw ratio above criticality, finite-amplitude disturbances are amplified, and sustained oscillation is achieved. It is found that the growth rate increases with draw ratio, but decreases with inertia and gravity, which suggests that initial transients tend to take longer to die out for a fluid with inertia and gravity. Transient post-critical calculations show that the nonlinearity can be effectively halted by inertia and gravity. The oscillation frequency (film-thickness amplitude) decreases (increases) with draw ratio. However, the film oscillates more frequently but less fiercely with stronger inertia and gravity effects. The rupture of the film is also examined, and is found to be delayed by inertia and gravity. Interestingly, although the oscillation amplitude is found to be weakest at the chill roll, it is at this location that the film tends to rupture first.     

Nonisothermal model of glass fiber drawing stability

Draw resonance is caused by a constant speed winder that leads to non-constant axial forces (Schultz, 1984). The well studied isothermal Newtonian fiber drawing predicts very modest critical draw ratios (around 20, much less than the typical production draw ratios for glass fibers of 10³ – 10⁵). The nonisothermal fiber drawing model presented here shows that cooling along the spin line strongly stabilizes the process. However, we show that the conclusion of Shah and Pearson (1972a,b) that non-isothermal Newtonian fiber spinning is unconditionally stable is based on non-converged numerical results. The choice of viscosity-temperature correlation function has a strong influence in determining the stability of the process. While viscoelasticity generally has an adverse effect on the stability, low viscoelasticity in the presence of extensional thinning helps to slightly improve the maximum critical draw ratio.Dedicated to the memory of Professor Tasos C. Papanastasiou     

Extremely large-amplitude oscillation of soft pipes conveying fluid under gravity

In this work, the nonlinear behaviors of soft cantilevered pipes containing internal fluid flow are studied based on a geometrically exact model, with particular focus on the mechanism of large-amplitude oscillations of the pipe under gravity. Four key parameters, including the flow velocity, the mass ratio, the gravity parameter, and the inclination angle between the pipe length and the gravity direction, are considered to affect the static and dynamic behaviors of the soft pipe. The stability ...     

Heat Transport in an Anisotropic Porous Medium Saturated with Variable Viscosity Liquid Under G-jitter and Internal Heating Effects

Thermo-rheological effect of temperature-dependent viscous fluid saturating a porous medium has been studied in the presence of imposed time periodic gravity field and internal heat source. Weak nonlinear stability analysis has been performed by using the power series expansion in terms of the amplitude of gravity modulation, which is considered to be small. Nusselt number is calculated numerically using Ginzburg–Landau equation. The nonlinear effects of thermo-mechanical anisotropies, internal heat source parameter, Vadász number, thermo-rheological parameter and amplitude of gravity modulation have been obtained and depicted graphically. Streamlines and isotherms have been drawn for different times. Comparisons have been made between various physical systems.     

Numerical investigation of the influence of gravity on flutter of cantilevered pipes conveying fluid

We have carried out a numerical investigation of the three dimensional nonlinear dynamics of a cantilevered pipe conveying fluid in the presence of gravity. The pipe may be misaligned at the clamped end with respect to gravity, and the effects of this misalignment are the main objects of the present investigation. The problem has been formulated using the Cosserat rod model. First, we have computed the equilibrium solutions and used them to experimentally validate both the Cosserat model and the constitutive law. Then, we have analyzed the occurrence of flutter, via Hopf bifurcation, for critical values of the relevant parameters of the problem, such as fluid to total mass ratio, dimensionless flow rate, dimensionless gravity and misalignment angle. The influence of the equilibrium solution on flutter has been explored, and the results of the linear stability analysis show that the stabilizing or destabilizing effect of fluid flow, either in or out of the plane of the pipe, depend crucially on the misalignment. We have also computed the non-linear periodic behavior after flutter instability by two different methods: the first one is by solving the full nonlinear equations by direct integration in time and space, while the second one is by assuming the time dependence given by an appropriate ansatz. Circular periodic orbits have then been studied and found that its loss of stability via Hopf bifurcation gives rise to stable planar periodic orbits. Finally, we have also computed the multiply periodic and chaotic behaviors which take place for sufficiently large values of the flow rate.     

On the analysis of 2D nonlinear gravity waves with a fully nonlinear numerical model

A fully nonlinear numerical method, developed on the basis of Euler equations, is used to study the dynamics of nonlinear gravity waves, mainly in the aspects of the propagation of Stokes wave with disturbed sidebands, the evolution of one wave packet and the interaction of two wave groups. These cases have previously been studied with the higher order spectral method, which will be an approximately fully nonlinear scheme if the order of nonlinearity is not large enough, while the present method in the case of the 2D model has an integration scheme that is exact to the computer precision. As expected, in most cases the results are consistent between these two numerical models and it is confirmed again that this fully nonlinear numerical model is also capable of maintaining a high accuracy and good convergence, particularly in the long-term evolutionary process.     

Periodic orbits,basins of attraction and chaotic beats in two coupled Kerr oscillators

Kerr oscillators are model systems which have practical applications in nonlinear optics. Optical Kerr effect, i.e., interaction of optical waves with nonlinear medium with polarizability χ ⁽³⁾ is the basic phenomenon needed to explain, for example, the process of light transmission in fibers and optical couplers. In this paper, we analyze the two Kerr oscillators coupler and we show that there is a possibility to control the dynamics of this system, especially by switching its dynamics from periodic to chaotic motion and vice versa. Moreover, the switching between two different stable periodic states is investigated. The stability of the system is described by the so-called maps of Lyapunov exponents in parametric spaces. Comparison of basins of attractions between two Kerr couplers and a single Kerr system is also presented.     

On the stability of nonlinear vibrations of coupled pendulums

The motion of two identical pendulums connected by a linear elastic spring is studied. The pendulums move in a fixed vertical plane in a homogeneous gravity field. The nonlinear problem of orbital stability of such a periodic motion of the pendulums is considered under the assumption that they vibrate in the same direction with the same amplitude. (This is one of the two possible types of nonlinear normal vibrations.) An analytic investigation is performed in the cases of small vibration amplitude or small rigidity of the spring. In a special case where the spring rigidity and the vibration amplitude are arbitrary, the study is carried out numerically. Arbitrary linear and nonlinear vibrations in the case of small rigidity (the case of sympathetic pendulums) were studied earlier [1, 2].     

Three-Dimensional Simulations for Convection Problem in Anisotropic Porous Media with Nonhomogeneous Porosity,Thermal Diffusivity,and Variable Gravity Effects

A convection problem in anisotropic and inhomogeneous porous media has been analyzed. In particular, the effect of variable permeability, thermal diffusivity, and variable gravity with respect to the vertical direction, has been studied. A linear and nonlinear stability analysis of the conduction solution has been performed. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using a three- dimensional simulation. Our results show that the linear threshold accurately predicts on the onset of instability in the basic steady state. However, the required time to arrive at the basic steady state increases significantly as the Rayleigh number tends to the linear threshold.     

Plane instability problem for a composite reinforced with a periodic row of short parallel fibers

The instability of a composite material reinforced with a periodic row of parallel short fibers is studied considering the interaction of neighboring fibers. Emphasis is on the mutual influence of short fibers in the matrix during loss of stability, depending on the distance between them. A piecewise-homogeneous medium model and the three-dimensional linearized theory of stability of deformable bodies are used     

Electroviscous potential flow in nonlinear analysis of capillary instability

We study the nonlinear stability of electrohydrodynamic of a cylindrical interface separating two conducting fluids of circular cross section in the absence of gravity using electroviscous potential flow analysis. The analysis leads to an explicit nonlinear dispersion relation in which the effects of surface tension, viscosity and electricity on the normal stress are not neglected, but the effect of shear stresses is neglected. Formulas for the growth rates and neutral stability curve are given in general. In the nonlinear theory, it is shown that the evolution of the amplitude is governed by a Ginzburg–Landau equation. When the viscosities are neglected, the cubic nonlinear Schrödinger equation is obtained. Further, it is shown that, near the marginal state, a nonlinear diffusion equation is obtained in the presence of viscosities. The various stability criteria are discussed both analytically and numerically and stability diagrams are obtained. It is also shown that, the viscosity has effect on the nonlinear stability criterion of the system, contrary to previous belief.     

Gravity Effects on Fiber Dynamics in Wall Turbulence

The present study investigated gravity effects on the dynamical behavior of inertial fibers suspended in a vertical channel flow. Direct numerical simulations were performed to obtain the turbulent flow field and the fibers were modelled as prolate spheroidal point particles. For each of the four fiber classes, three different gravity configurations were considered: upward flow with gravity opposing, downward flow with aiding gravity, and channel flow in absence of gravity. Results for the fiber distribution and the translational and rotational fiber motion were reported. In the near-wall region, the presence of gravity resulted in an increased fiber density in the downward flow but a nearly uniform distribution of fibers in upward flow. However, the preferential clustering of fibers in near-wall low-speed streaks was unaffected by gravity. The mean wall-normal or drift velocity of the fibers was higher in the downward flow and lower in the upward flow as compared to the case with no gravity. The suppressed drift velocity in the upward flow resulted in a more uniform fiber distribution throughout the channel in contrast to the near-wall accumulation of fibers in the two other cases. Overall gravity turned out to have negligible effects on some of the statistics of the least inertial fibers whereas the inclusion of gravity had a strong impact for heavier fibers.     

Instability and waves during generalized Newtonian fluid film flow down a vertical wall

The problem of linear stability of a non-Newtonian fluid film flowing down a vertical plane under the action of gravity is considered. The linear stability of steady-state flow with a plane free boundary and the nonlinear waves that arise if this flow is unstable are investigated. The results obtained for two rheological models, the power-law and Eyring fluids, are compared.     

旋转机械系统多自由度非线性动力学数值分析

首先将转子系统的动力响应问题归结为2n维未知向量v的一阶非线性动力学方程dv／dt=Hv＋f（v,t）,并给出了求解这一方程的一次近似式法和三次多项式迫近法。在非稳态、非线性油膜力等作用下,以刚性Jeffcott转子与112个自由度的汽轮发电机组低压转子系统为例,用上述求解方法分析了它们的动力响应及非线性动力学特性;其问,还将计算结果与Runge—Kutta法、Newmark法的相应结果进行了比较,并深入讨论了数值稳定性问题。汽轮发电机组的算例表明对一些具有较复杂的非线性右端项,、同时规模又较大的问题,如果采用四阶Runge—Kutta法,才算几步就因数值骤然增大而失控;但若用同样步长的一次近似式,由于它是一种显式的无条件稳定算式,则计算过程迅速且结果合理可靠。     

On the influence of the gravity force on the stability of an inhomogeneous layer under biaxial extension-compression

In the present paper, in the framework of the three-dimensional nonlinear theory of elasticity, we study the stability of a heavy layer under biaxial extension-compression. The elastic properties of the layer are assumed to be inhomogeneous along thickness and are described by a semilinear material model. We study the stability by using the bifurcation approach. By solving the linearized equilibrium equations, we obtain the critical curves and the stability domain in the plane of the loading parameters, for which we take the material elongation ratios along the coordinate axes lying in the layer plane. We analyze the influence of the layer thickness, specific weight, and material parameters on buckling. In particular, we find that, when studying stability, it is expedient to take the gravity force into account only if the layer rigidity decreases with increasing depth.     

Draw resonance in film casting of viscoelastic fluids: A linear stability analysis

In order to understand the role of viscoelasticity on draw resonance in the isothermal film casting process, a steady state analysis and a linear stability analysis for three-dimensional flow disturbances have been conducted. The constitutive equation used is a modified convected Maxwell model, with shear-rate dependent viscosity and fluid characteristic time. The numerical results indicate that the flow is stable below a lower critical draw ratio and above an upper critical draw ratio. Shear thinning in viscosity reduces the lower critical draw ratio and somewhat increases the upper critical draw ratio—thereby enlarging the region of instability. Slower shear reduction in fluid characteristic time dramatically decreases the upper critical draw ratio but has no significant effect on the lower critical draw ratio; therefore, fluids with higher characteristic time are more stable.     

Water waves in an elastic vessel

Linear and nonlinear analyses of water waves in an elastic vessel are carried out to study the dramatic phenomena of Dragon Wash as well as related controllable experiments. It is proposed that the capillary edge waves are generated by parametric resonance, which is shown to be a possible mechanism for both rectangular an circular vessels. For circular vessel, the normal geometric resonance is also operating, thus greatly enhance the dramatic effect. The mechanism of nonlinear mode-mode interaction is proposed for the generation of axisymmetric low-frequency gravity waves by the high- frequency external excitation. A simple model system is studied numerically to demonstrate explicitly this interaction mechanism.     

Weakly Nonlinear Stability Analysis of Temperature/Gravity-Modulated Stationary Rayleigh–Bénard Convection in a Rotating Porous Medium

The effect of time-periodic temperature/gravity modulation on thermal instability in a fluid-saturated rotating porous layer has been investigated by performing a weakly nonlinear stability analysis. The disturbances are expanded in terms of power series of amplitude of convection. The Ginzburg–Landau equation for the stationary mode of convection is obtained and consequently the individual effect of temperature/gravity modulation on heat transport has been investigated. Further, the effect of various parameters on heat transport has been analyzed and depicted graphically.     

Nonlinear waves in a liquid film on a near-horizontal surface

Nonlinear waves in a liquid film on a slightly inclined rigid plane are studied. A mathematical model is reduced to a system of two evolutionary equations for the layer thickness and the local fluid mass flow. In addition to viscous forces, gravity, and surface tension, the pressure difference over the layer thickness, induced by the gravity force projection on the normal to the underlying surface, is also taken into account. Spatially periodic solutions developing with time from small initial disturbances into regular nonlinear waves are considered. A spectral representation of the solution, the Galerkin method with respect to the uniform coordinate, and subsequent numerical calculation of the corresponding dynamic system on large time intervals are employed. Different variants in the space of the three governing parameters are calculated and some basic mechanisms of nonlinear dynamics of the two-dimensional waves are detected. The calculation results are compared with the existing experimental data. It is shown that the theoretical conclusions can be used to interpret and predict experiments.