Phase slip solutions in magnetically modulated Taylor–Couette flow

We numerically investigate Taylor–Couette flow in a wide-gap configuration, with \({r_i/r_o=1/2}\), the inner cylinder rotating, and the outer cylinder stationary. The fluid is taken to be electrically conducting, and a magnetic field of the form \({B_z\approx(1 + \cos(2\pi z/z_0))/2}\) is externally imposed, where the wavelength \({z_0=50(r_o-r_i)}\). Taylor vortices form where the field is weak, but not where it is strong. As the Reynolds number measuring the rotation rate is increased, the initial onset of vortices involves phase slip events, whereby pairs of Taylor vortices are periodically formed and then drift outward, away from the midplane where \({B_z=0}\). Subsequent bifurcations lead to a variety of other solutions, including ones both symmetric and asymmetric about the midplane. For even larger Reynolds numbers, a different type of phase slip arises, in which vortices form at the outer edges of the pattern and drift inward, disappearing abruptly at a certain point. These solutions can also be symmetric or asymmetric about the midplane and co-exist at the same Reynolds number. Many of the dynamics of these phase slip solutions are qualitatively similar to previous results in geometrically ramped Taylor–Couette flows.     

Onset of Taylor-Görtler instability in accelerated circular Couette flow

Summary The onset of Taylor-G?rtler vortices in a developing Couette flow induced by the inner cylinder rotating with time-dependent manner is analyzed using linear theory. It is well known that there is a critical Taylor number Ta _c at which Taylor vortices set in between two concentric cylinders. For Ta > Ta _c Taylor-G?rtler vortices are detected experimentally at a certain elapsed time. In the present study the critical time t _c to represent the onset of a fastest growing instability, which then grows as toroidal vortices, is analyzed using the propagation theory. Available experimental data indicate that for large Ta secondary motion is detected starting from a certain time t _m ≈ 4t _c. This means that the growth period of initiated instabilities is needed for secondary motion to be detected experimentally. The new measures to represent the onset of a fastest growing instability in the primary time-dependent Couette flow are suggested.     

Numerical and experimental study on the flow history effects of axial flow on the Couette–Taylor flow

In this research, experimental and numerical techniques are used to study the flow history effects of axial flow on the Couette–Taylor flow. For the experimental investigation, the flow is visualized using the PIV technique with reflective particles with a density of 1.62 g/cm³. Dispersed in a solution, the particles have a strong refraction index equal to 1.85. In this study, two protocols are adopted to study the effect of an axial flow superimposed on a Couette–Taylor flow, and of the history of the flow. The first one, the direct protocol, consists of imposing an azimuthal flow to the inner cylinder. In this case, when the regime is established, the axial flow is superimposed. The second protocol, the inverse protocol, consists of imposing first the axial flow in the gap of the system, after which an azimuthal flow is conveyed. The Couette–Taylor flow with axial flow is strongly dependent on the flow history (the protocol). Thus, the flow structures and development for different protocols are studied and analyzed here experimentally and numerically. In addition, from the numerical results, mathematical models for the two protocols are presented. For the direct protocol, a new relation between the axial Reynolds number, which stabilizes the Couette–Taylor flow, and the Taylor number is presented; for the inverse protocol, a new mathematical model for the critical Taylor number is developed as a function of the axial Reynolds number and also the first critical Taylor number without axial flow.     

Experiments on two immiscible fluids in spherical Couette flow

This paper deals with the experimental instabilities analysis of spherical Couette flow. We consider the flow of two immiscible fluids superimposed between concentric spheres when the outer sphere is fixed and the inner one rotates. The working fluids have rather different viscosities and thus different Reynolds numbers. The obtained results are compared with a reference case of filled gap using one fluid (Γ _max = 20). Experiments are performed for different aspect ratio values, and Laser photometric technique is used for visualization. Our analysis is mainly focused on the type of instabilities and their relationship with the laminar-turbulent transition regime. We intend to explore the combined effects of the aspect ratio and the interaction between the two superposed fluids on the appearance of different instability evolutions. The evolution of the phase velocity for different aspect ratio of heavy fluid Γ _HF = H _HF/d is presented. The immiscible fluids are separated by a liquid–liquid interface (water–oil). In order to control instability occurrence, Taylor number variation is presented versus aspect ratio. Instability phenomena are found to be the same as for the nominal case for large heavy fluid aspect ratios. The first equatorial symmetry breaking of the flow is observed for a critical value Γ _c  = 13 where the Taylor vortex flow is introduced with three stationary cells. For the same aspect ratio, the interaction of the immiscible fluids leads to the appearance of gravitational waves near the equatorial zone. A surface cell, starting before the appearance of Taylor vortices, is detected in the light fluid for low aspect ratios. This cell of Ekman type has not been observed before, to our best knowledge, in spherical Couette flow.     

Modulated rotating waves in O(2) mode interactions

The interaction of steady-state and Hopf bifurcations in the presence of O(2) symmetry generically gives a secondary Hopf bifurcation to a family of 2-tori, from the primary rotating wave branch. We present explicit formulas for the coefficients which determine the direction of bifurcation and the stability of the 2-tori. These formulas show that the tori are determined by third-degree terms in the normal-form equations, evaluated at the origin. The flow on the torus near criticality has a small second frequency, and is close to linear flow, without resonances. Existence of an additional SO(2) symmetry, as in the Taylor-Couette problem, forces the flow to be exactly linear; however, the tori are unstable at bifurcation in the Taylor-Couette case. More generally, these tori may reveal themselves physically as slowly modulated rotating waves, for example in reaction-diffusion problems.     

Modulated rotating waves in O(2) mode interactions

The interaction of steady-state and Hopf bifurcations in the presence of O(2) symmetry generically gives a secondary Hopf bifurcation to a family of 2-tori, from the primary rotating wave branch. We present explicit formulas for the coefficients which determine the direction of bifurcation and the stability of the 2-tori. These formulas show that the tori are determined by third-degree terms in the normal-form equations, evaluated at the origin. The flow on the torus near criticality has a small second frequency, and is close to linear flow, without resonances. Existence of an additional SO(2) symmetry, as in the Taylor-Couette problem, forces the flow to be exactly linear; however, the tori are unstable at bifurcation in the Taylor-Couette case. More generally, these tori may reveal themselves physically as slowly modulated rotating waves, for example in reaction-diffusion problems.     

A tertiary hopf bifurcation with applications to problems with symmetries

We develop explicit criteria for the occurrence of a tertiary Hopf bifurcation, and stability of the bifurcating orbits, in a special class of two-parameter systems of ordinary differential equations. We use these results to discuss tertiary Hopf and torus bifurcations in some bifurcation problems with symmetries such as steady-state-Hopf and Hopf-Hopf interaction problems. To analyse (and even detect) these bifurcations we use invariant coordinates and rescaling techniques     

Symmetric and asymmetric Taylor vortex flow in spherical gaps

Summary The rotationally symmetric flow between two concentric rotating spheres is investigated both theoretically and experimentally. The non-uniqueness of the supercritical flow exhibits three different modes with zero, one and two Taylor vortices in each hemisphere. These modes are realized by different accelerations of the inner sphere from the state at rest. A initial value code, based on a finite difference method, is used for the numerical simulation. The existence regions of the different supercritical flows are connected with symmetric and asymmetric transitions. It is found, that a steady state can exist asymmetric with respect to the equator. The flow is analyzed by plotting the size of the Taylor vortices, the depending variables , ,V, the velocity distributions and the torque. A comparison between theory and experiments for the observed modes of flow is given.     

Generator of quasi-periodic oscillations featuring two-dimensional torus doubling bifurcations

A new autonomous differential dynamical system with dimension N = 4 is introduced, which has solutions in the form of stable two-frequency oscillations and features a sequence of period-doubling bifurcations of two-dimensional ergodic tori. At the points of period-doubling bifurcations, no resonances are observed on a torus and only ergodic tori exhibit doubling.     

Chaos in the Takens–Bogdanov bifurcation with O(2) symmetry

The Takens–Bogdanov bifurcation is a codimension-two bifurcation that provides a key to the presence of complex dynamics in many systems of physical interest. When the system is translation-invariant in one spatial dimension with no left-right preference the imposition of periodic boundary conditions leads to the Takens–Bogdanov bifurcation with O(2) symmetry. This bifurcation, analyzed by G. Dangelmayr and E. Knobloch, Phil. Trans. R. Soc. London A 322, 243 (1987), describes the interaction between steady states and travelling and standing waves in the nonlinear regime and predicts the presence of modulated travelling waves as well. The analysis reveals the presence of several global bifurcations near which the averaging method (used in the original analysis) fails. We show here, using a combination of numerical continuation and the construction of appropriate return maps, that near the global bifurcation that terminates the branch of modulated travelling waves, the normal form for the Takens–Bogdanov bifurcation admits cascades of period-doubling bifurcations as well as chaotic dynamics of Shil'nikov type. Thus chaos is present arbitrarily close to the codimension-two point.     

Taylor vortices between almost cylindrical boundaries

Summary We consider a modified Taylor problem, with the fluid flowing between a rotating inner circular cylinder and an outer stationary surface whose radius is a constant plus a small and slowly varying function of the axial co-ordinate z. This variation is chosen in such a way that the flow is locally more unstable near z=0 than near z=±, so that Taylor vortices appear more readily near z=0. The theory is developed to show how vortices of strength varying with z develop as the speed of rotation is increased through a critical value which is a perturbation of the classical value. Wave number changes in the axial direction are also calculated.     

Stability of power law fluid flow between rotating cylinders

The centrifugal instability of power law fluid flow when the fluid fills the gap between two rotating concentric cylinders of infinite length is addressed. The instability of the circular Couette flow is determined via linear stability analysis. In the narrow gap case, the critical Taylor number is determined analytically, while in the general gap case, a finite-difference numerical method is employed. The results are shown to be in agreement with existing theoretical and experimental findings. The super-critical flow is investigated by means of a weakly non-linear analysis method. The Taylor-vortex flow (the secondary stable flow) is obtained. The instability of this flow is determined as we present the critical Taylor number when the Taylor vortices begin to exhibit a waviness in the azimuth     

直线运动柔性梁非线性动力学--组合参数共振与内共振联合激励

针对基础直线运动柔性梁，基于Kane方程建立了相应的非线性动力学方程。采用多尺度法并结合笛卡尔坐标变换，导出了系统受前两阶模态间3：1内共振及其组合参激共振时的非线性调制方程组，数值求解了该方程组的定常解及相应的稳定性问题。研究表明，系统的平凡响应与双模态非平凡响应共存，由内共振所产生的非平凡响应皆为不稳定的鞍点，平凡及非平凡解分支都存在Hopf分岔现象，一些稳定的极限环随参数变化最终经倍周期分岔后产生混沌运动。     

高阶扭转模态耦合下覆冰导线的稳定性和影响因素分析

利用Galerkin法建立面内前四阶和扭转前四阶模态耦合的覆冰导线动力学模型。借助分岔理论分析各阶模态的失稳临界条件,研究导线系统在不同风速、扭转阻尼比、档距及初始拉力下各阶模态的失稳规律,并利用数值模拟对理论分析结果进行验证。研究结果表明:考虑了扭转前四阶模态的导线模型,其面内前四阶模态特征值实部随风速变化的响应曲线先后经历2次Hopf分岔,呈限幅振动;扭转阻尼比的增大扩大了面内模态的失稳风速区域;随着档距增大,面内模态的2个Hopf分岔点和扭转模态的一个Hopf分岔点分别左移,表明大档距时,扭转模态逐渐代替面内模态的舞动;初始拉力对面内模态的失稳区域影响显著,而对扭转模态的影响很小。以上结论可为工程中导线的优化设计提供理论依据。     

Amplitude modulated dynamics and bifurcations in the resonant response of a structure with cyclic symmetry

Summary Periodic structures with cyclic symmetry are often used as idealized models of physical systems and one such model structure is considered. It consists ofn identical particles, arranged in a ring, interconnected by extensional springs with nonlinear stiffness characteristics, and hinged to the ground individually by nonlinear torsional springs. These cyclic structures that, in their linear approximations, are known to possess pairwise double degenerate natural frequencies with orthogonal normal modes, are studied for their forced response when nonlinearities are taken into account. The method of averaging is used to study the nonlinear interactions between the pairs of modes with identical natural frequencies. The external harmonic excitation is spatially distributed like one of the two modes and is orthogonal to the other mode. A careful bifurcation analysis of the amplitude equations is undertaken in the case of resonant forcing. The response of the structure is dependent on the amplitude of forcing, the excitation frequency, and the damping present. For sufficiently large forcing, the response does not remain restricted to the directly excited mode, as both the directly excited and the orthogonal modes participate in it. These coupled-mode responses arise due to pitchfork bifurcations from the single-mode responses and represent traveling wave solutions for the structure. Depending on the amount of damping, the coupled-mode responses can undergo Hopf bifurcations leading to complicated amplitude-modulated motions of the structure. The amplitude-modulated motions exhibit period-doubling bifurcations to chaotic amplitude-modulations, multiple chaotic attractors as well as crisis. The existence of chaotic amplitude dynamics is related to the presence of Sil'nikov-type conditions for the averaged equations.     

A few-mode optical fiber retaining optical vortices

The ability of an optical fiber with axial losses to selectively suppress the fundamental HE ₁₁ mode, as well as the TE and TM waveguide modes, and, simultaneously, to transmit optical vortices with almost zero energy losses is considered. The attenuation coefficients for the corresponding eigenmodes and vortices are determined. It is shown that such a fiber operates as a mode filter for the feeding beam.     

Essential instabilities of fronts: bifurcation, and bifurcation failure

Various instability mechanisms of fronts in reaction-diffusion systems are analysed; the emphasis is on instabilities that have the potential to create modulated (i.e. time-periodic) waves near the primary front. Hopf bifurcations caused by point spectrum with associated localized eigenfunctions provide an example of such an instability. A different kind of instability occurs if one of the asymptotic rest states destabilizes: these instabilities are caused by essential spectrum. It is demonstrated that, if the rest state ahead of the front destabilizes, then modulated fronts are created that connect the rest state behind the front with small spatially periodic patterns ahead of the front. These modulated fronts are stable provided the spatially periodic patterns are stable. If, on the other hand, the rest state behind the front destabilizes, then modulated fronts that leave a spatially periodic pattern behind do not exist.     

Mechanism of modulated microwave absorption and flux distribution in granular YBa2Cu3Ox under high field sweep

The mechanism of modulated microwave absorption (MMA) is suggested from the modulation (H _m ) dependence in granular YBa₂Cu₃O_x superconductor under high-field sweep. It is proposed that the MMA signalS be decomposed into three shape factors to interpret the experimental result. The VN factor arises from the difference in the number of vortices between the add (A) mode and the subtract (S) mode corresponding to the alternating field. The CT factor arises from the appearance of the transition region of the shielding current in the Smode where the absorption is less. The SS factor arises from the difference in the surface slope of the flux distribution between the two modes. With increasingH _m , the positive VN increases linearly. The positive CT increases initially and then decreases. The slope for the Amode is very gentle, so that SS is negative, and it initially increases rapidly and then saturates. These factors can reproduce the complex behavior of theH _m dependence ofS fairly well.     

Pattern formation through reaction and diffusion in a simple pooled-chemical system

The formation of spatial patterns is considered for a reaction-diffusion system based upon the cubic autocatalator, A+2B→3B, B→C, with the reaction taking place inside a closed vessel, the reactant A being replenished by the slow decay of a precursor P via the simple step P→A.

Patterns are shown to form only when the dimensionless diffusion coefficient λ is sufficiently small, with the number of available patterns increasing as λ diminishes to zero. Two types of patterns occur, standing-wave patterns arising out of Hopf bifurcations, together with steady-wave patterns arising out of pitchfork bifurcations. The local behaviour on the bifurcating branches is obtained via weakly nonlinear theory. Close to its point of bifurcation, each pattern is shown to be partially stable; that is, it remains stable to small disturbances composed of its own, or any higher spacial wave numbers. However, the pattern is unstable to disturbances with smaller spatial wave number than its own. This partial stability is in line with observations of pattern formation in chemical systems for which the cubic autocatalator provides a rational model. Such patterns are not generally observed to have absolute local stability, but appear and disappear in a transient manner, influenced by the external disturbances to the system. Moreover, patterns do not appear after the system has been well stirred. This is also in line with the present theory, which demonstrates that a spatially uniform state is always the most stable. For a typical case, the bifurcation branches are extended by numerical integration, which reveals a connecting of two branches through a further symmetry-breaking bifurcation.     

Existence of extra vortex and twin vortex of anomalous mode in Taylor vortex flow with a small aspect ratio

Summary This experimental work on Taylor vortex flow in a gap with a small aspect ratio is concerned with two extra vortices and a twin vortex system, each of which depends on an anomalous cell of the anomalous mode. Extra vortices are smaller than other vortices such as defined cells. At any Reynolds number and aspect ratio extra vortices can be found at the corner of the end plate and inner rotating cylinder and at the corner of the end plate and outer stationary cylinder. For a one-cell flow (anomalous one-cell mode) in a symmetric system, an outer extra vortex develops and grows to the same size as the main cell, only in an aspect ratio of less than one. A twin vortex is observed to form when two vortices are aligned in the direction of the radius. There are three flow fields on the end plate; two are extra vortex flows and the other is the main cell flow. The flow direction of the anomalous cell is from the inner cylinder to the outer one, at the end plate opposite of the flow direction of the normal cell.Nomenclature R ₁ Radius of inner cylinder (2R ₁=40.19±0.006 mm) - R ₂ Radius of outer cylinder (2R ₂=60.11±0.024 mm) - R _r Radius ratio (R ₁/R ₂=0.669) - d Clearance between cylinders (R ₂–R ₁=9.96±0.025 mm) - L Height of working fluid - Aspect ratio=L/d - Rotational angular speed - Kinematic viscosity - Re Reynolds number=R ₁ d/ Other nomenclature is defined as it appears