Numerical investigation of bifurcation phenomena occurring in flows through planar sudden expansions

Summary A new convection scheme along with a multigrid algorithm were used for an in-depth study of bifurcation phenomena which occur in flows through planar sudden expansions.The predicted flow bifurcation parameters compare very well with experiments, especially for Reynolds numbers higher or lower than a critical Reynolds number Re_cr, the latter being the value beyond which flow asymmetry takes place. For Reynolds numbers below Re_cr the flow remains symmetric throughout all the period of its development, during which the recirculation lengths increase monotonically with time towards their final stationary values. For Reynolds numbers above Re_cr the flow changes from a symmetric structure at the beginning to an asymmetric one at the steady state where one of the main recirculation zones becomes longer than the other and a thid zone appears on the same side as the short primary recirculation zone.     

On the connection of isolated branches of a bifurcation diagram: the truss arch system

One of the important basic issues of bifurcation theory is the determination of the set of the fixed points of non-linear evolution equations as a function of its parameters. The branching of branches of solutions rarely occurs in real applications for which imperfections tend to distort these sharp transitions. Furthermore, bifurcation theory may a priori indicate that there are disjoint branches of solutions. In the present work, the truss arch system is considered and described. The bifurcation diagram is carried out numerically. It is shown that the truss arch system is a simple example of coexistence of disjoint branches. Moreover, it is shown that the emergence of the subcritical bifurcations of the non-shallow configuration is the result of the connection of these disjoint branches. The analytic solutions are derived and the connection of the branches is studied.     

Generic multiparameter bifurcation from a manifold

The geometry of generic k -parameter bifurcation from an n -manifold is discussed for all values of k , n with particular emphasis on the case n = 2 (the case n = 1 being dealt with in earlier work). Such bifurcations typically arise in the study of equilibrium states of dynamical systems with continuous (for example, spherical or toroidal) symmetry which undergo small symmetry-breaking perturbations, and in the use of Melnikov maps for detecting bifurcations of periodic orbits from resonance. Detailed analysis is given in the interesting case n = 2, k = 3 where the local geometry partly resembles unfolding of a degenerate wavefront or Legendrian collapse.     

An algorithm and a method to search bifurcation points in non‐linear problems

This paper is devoted to a numerical method able to help the determination of the bifurcation threshold in non‐linear time‐independent continuum mechanic problems. First, some theoretical results about uniqueness are recalled. In the framework of the large‐strain assumption, the differences between the classical finite‐step problem and the rate problem are presented. An iterative algorithm able to solve the rate problem is given. Using different initializations, it is seen in some numerical experiments that it is possible with this algorithm to get different solutions when the underlying mathematical problem solved does not enjoy a uniqueness property. The constitutive equations used have been chosen to be simple enough to deduce some theoretical knowledge about the corresponding uniqueness problems. Finally, a method is given which is able in some case to give an upper bound of the bifurcation threshold. Copyright © 2001 John Wiley & Sons, Ltd.     

Passage of a liquid drop through a bifurcation

Numerical simulations of the motion of a viscous liquid drop through a two-dimensional bifurcating channel are conducted using a boundary-element method for Stokes flow. The drop viscosity is assumed to be equal to the ambient fluid viscosity and the drop interface is assumed to exhibit uniform surface tension. The mean fluid pressures are prescribed at the channel inlet and two outlets, and the corresponding flow rates are computed as part of the solution. Preliminary simulations show that the shape of a two-dimensional drop moving through a channel with parallel walls is similar to that of an axisymmetric drop moving along the centerline of a circular tube. The ability of a drop to remain intact as it passes through the bifurcation is determined by the drop size and capillary number expressing the significance of surface tension. For a given drop size and channel inlet and outlet pressures, there is a critical capillary number above which a drop splits into two pieces connected by a thinning bridge. The presence of the drop has a weak effect on the inlet and outlet flow rates throughout its passage. Simulations based on a boundary-element method for a rigid particle with circular or elliptical shape reveal a significantly stronger effect due to the absence of interfacial mobility.     

The Freedericksz transition as a bifurcation problem

We consider the effects of the application of electric and magnetic fields to a homogeneously aligned nematic liquid crystal which lies between parallel plates. The average alignment of the fluid molecules is known as the director and it is set parallel to the top and bottom bounding surfaces. At a critical field strength a Freedericksz transition occurs which results in a distortion of the director field. We discuss this phenomenon from the point of view of bifurcation theory and use numerical bifurcation techniques to explore it. Imperfections in the cell, either in the applied field or the director orientation at the boundaries, are also considered. Finally, we investigate the effect of finite aspect in a two-dimensional version of the problem.     

Hopf bifurcation on a simple cubic lattice

We study Hopf bifurcation for diff erential equations defined on the space of functions on R3 which are triply periodic with respect to a simple (primitive) cubic lattice. The centre manifold theorem reduces the problem to a system of ordinary diff erential equations (ODEs) on the space (C+C)3 and symmetric under the group (O=Zc2) + T3. We abstract this group as the wreath product group O(2) /S3, and we use a general theory of symmetry - breaking bifurcations for wreath product groups to find (up to conjugacy) all branches of periodic solutions with maximal isotropy. The stability of these solutions is calculated . Branches of periodic solutions with sub-maximal isotropy can also exist. Some possibilities for bifurcations to heteroclinic cycles are explored.     

Adsorption-stimulated bifurcation transition in a silicon-porous-silicon system

It is shown that after a porous layer has been formed electrochemically on a silicon surface at elevated anode currents, the silicon-porous-silicon system is in a quasisteady nonequilibrium state from which it may be transferred under the influence of fairly weak adsorption effects. Pis’ma Zh. Tekh. Fiz. 24, 30–34 (November 12, 1998)     

Synthesis and characterization of the isolated straight polymer chain inside of single-walled carbon nanotubes

In this paper, a novel molecular wire comprising an isolated straight polystyrene (PS) chain encapsulated by a single-walled carbon nantube (SWNT) was prepared. The interaction of the PS chain with the SWNTs was systematically characterized using transmission electron microscopy (TEM), Fourier transform infrared (FTIR) spectroscopy, nuclear magnetic resonance (NMR) spectroscopy, and Raman spectroscopy.     

Vertical crack inside a semi-infinite plane

In this paper we consider the problem of determining the stress-intensity factors and the crack energy in a semi-infinite plane containing an inside crack perpendicular to the straight boundary of the plane. By the use of Mellin transform, we reduce the problem to solving a single singular integral equation. Approximate solution of the integral equation is obtained as a series of Chebyshev polynomials of the first kind. The coefficients $\text{B}{}_{\mathrm{n}}$ of the series are determined from a system of linear algebraic equations. Expressions for the stress-intensity factors at the edges of the crack, the shape of the crack and the crack energy are derived in terms of the coefficients $\text{B}{}_{\mathrm{n}}$. The numerical values of these quantities have been displayed graphically for three particular cases.     

Duffing-Van der pol系统的Hopf分岔

将保守Duffing系统作为未扰系统,并对它分四种情形进行了严格求解。用Melnikov函数方法研究了Duffing-Vanderpol系统的次谐分岔,获得了Duffing-Vanderpol系统的Hopf分岔条件。根据这些条件,在参数空间中确定了Hopf分岔曲线。在分岔曲线上取参数进行了数值模拟,所获得的奇、偶阶Hopf分岔与理论分析的结果完全一致。     

An experiment on a taylor vortex flow in a gap with a small aspect ratio bifurcation of flows in an asymmetric system

Summary Both Taylor vortex flows in a symmetric or asymmetric system exhibit various patterns (cell modes). They can be classified by the process of cell formation, the number of cells and the direction of flow for the cell, into primary modes or secondary modes, and normal modes or anomalous modes. Following the previous report in which flows in a symmetric end condition were classified, in the present work, for flows in an asymmetric end condition, the Reynolds number at which a secondary mode bifurcates into another mode is experimentally investigated, and the bifurcation of the Taylor vortex flows in an asymmetric system when the Reynolds number is gradually decreased is presented in a bifurcation diagram.List of symbols R1 Radius of inner cylinder (2R1=40.19±0.006 mm) - R2 Radius of outer cylinder (2R2=60.11±0.024 mm) - D Clearance between cylinders (R2-R1=9.96±0.025 mm) - L Height of working fluid - Aspect ratio=L/D - Rotational angular speed - Kinematic viscosity - Re Reynolds number=R1D/     

The non-smooth pitchfork bifurcation: a renormalization analysis

We give a renormalization group analysis of a system exhibiting a non-smooth pitchfork bifurcation to a strange non-chaotic attractor. For parameter choices satisfying two specified conditions, self-similar behaviour of the attractor on and near the bifurcation curve can be observed, which corresponds to a periodic orbit of an underlying renormalization operator. We examine the scaling properties for various parameter choices including the so-called pitchfork critical point. Finally, we study the autocorrelation function for the system and show that it is equivalent to that present in symmetric barrier billiards.     

厂区内拆除控制爆破

武钢厂区拆除爆破对象复杂,爆破环境恶劣,合理选取爆破参数,采取必要的防护措施,保证了周围环境的绝对安全.     

Heat transfer inside a rotating rectangular cavity

The qualitative liquid-flow pattern is considered for a rotating rectangular cavity in the case of heat transfer in a direction parallel to the axis of rotation; calculational relations describing the heat transfer are obtained.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 34, No. 1, pp. 89–93, January, 1978.     

The pressure inside a suspended gas bubble

Summary. An integral representation is derived for the pressure inside an inviscid, compressible or incompressible bubble suspended in a Newtonian fluid, in terms of the instantaneous bubble shape and structure of the ambient velocity field. The starting point in deriving this representation is the reciprocal theorem for Navier-Stokes flow, applied for the flow around the bubble and the flow due to a point source situated inside the bubble. The integral representation is used to evaluate the pressure inside a spherical bubble in potential flow to leading order with respect to the Weber number, and the pressure inside a circular bubble in Stokes flow to leading order with respect to the capillary number. The results are consistent with and generalize numerical and analytical solutions derived by previous authors.     

On a compressed spatial elastica constrained inside a tube

In this paper, we study the deformation of a clamped-clamped elastica under edge thrust and constrained inside a straight tube. The governing equations are formulated within the framework of director theory and solved by the shooting method. The deformations calculated from the elastica model are compared with those computed from small-deformation theory. For a relatively slender tube, the early stage of the deformation sequence is similar to the one obtained from small-deformation theory. They are one-point, two-point, three-point, and point-line-point contact deformations. However, some fundamental differences exist between these two theories even in this early stage of deformation. According to small-deformation theory, the constrained rod jumps from planar one-point contact to helical two-point contact suddenly. In elastica model, the planar one-point contact evolves to spatial one-point contact first and transforms to two-point contact smoothly. In small-deformation theory, the point-line-point contact is the final stage of deformation. In elastica model, on the other hand, the deformation may transform to other shapes as the edge thrust increases further. Generally speaking, the difference between small-deformation theory and elastica model grows as the radius of the constraining tube becomes larger. In the case when the ratio of the tube radius and the beam length is larger than 0.384, the constraining tube has no effect on the elastica deformation.     

主轴-滚动轴承系统三种分岔形式

摘 要：为了深入探讨主轴-滚动轴承这种具有接触非光滑因素的系统的失稳机理和分岔的产生机制,本文建立了一个基于Hertz接触力模型的6自由度系统动力学微分方程,对主轴系统从稳定到失稳的机制和途径进行了研究,探讨在非平衡力作用下,具有负游隙的机床主轴-滚动轴承系统的非线性动态特性。研究结果表明,如果轴承的内圈以很低的速度和滚子相接触,随控制参数频数比的变化,会产生擦边分岔。擦边分岔将导致系统响应从倍周期运动转迁为周期运动,从周期运动转迁为拟周期运动,从拟周期运动转迁到混沌。此外,倍周期分岔及环面倍化分岔也是使得主轴系统运动演化为混沌的重要形式。以上研究结果加深了我们对主轴-滚动轴承系统中混沌演化形式的理解,并丰富了机床主轴非线性动态理论的研究和应用。     

Motion of a spherical particle inside a liquid film

The motion of a spherical particle inside a liquid film coated on a plane wall is considered under conditions of Stokes flow in the limit of vanishing capillary number where the interfacial deformation is infinitesimal. The problem is formulated in terms of a system of one-dimensional integral equations for the velocity and traction Fourier coefficients along the trace of the interface, wall, and particle contour in a meridional plane, and the solution is found using a boundary-element method. Comprehensive results for the force and torque resistance coefficients are presented in the case of particle rotation and translation in quiescent fluids. The velocity of translation and angular velocity or rotation of a freely suspended particle in simple shear flow are computed and discussed over a broad range of conditions.