Hopf Bifurcation in the presence of symmetry

Using group theoretic techniques, we obtain a generalization of the Hopf Bifurcation Theorem to differential equations with symmetry, analogous to a static bifurcation theorem of Cicogna. We discuss the stability of the bifurcating branches, and show how group theory can often simplify stability calculations. The general theory is illustrated by three detailed examples: O(2) acting on R ², O(n) on R ⁿ , and O(3) in any irreducible representation on spherical harmonics.The work of second author was also supported by a visiting position in the Department of Mathematics, University of Houston     

Mixed convection flow of second grade fluid along a vertical stretching flat surface with variable surface temperature

In the present study we have explored the effects of thermal buoyancy on flow of a viscoelastic second grade fluid past a vertical, continuous stretching sheet of which the velocity and temperature distributions are assumed to vary according to a power-law form. The governing differential equations are transformed into dimensionless form using appropriate transformations and then solved numerically. The methods here employed are (1) the perturbation method together with the Shanks transformation, (2) the local non-similarity method with second level of truncation and (3) the implicit finite difference method for values of ξ ( = Gr _x /Re _x ², defined as local mixed convection parameter) ranging in [0, 10]. The comparison between the solutions obtained by the aforementioned methods found in excellent agreement. Effects of the elasticity parameter λ on the skin-friction and heat transfer coefficients have been shown graphically for the fluids having the values of the Prandtl number equal to 0.72, 7.03 and 15.0. Effects of the viscoelastic parameter and the mixed convection parameter, ξ, on the temperature and velocity fields have also been studied. We notice that with the increase in visco-elastic parameter λ, velocity decreases whereas temperature increases and that velocity gradient is higher than that of temperature. On leave of absence from the Department of Mathematics, University of Dhaka, Bangladesh.     

Rotational instability in second order fluids

Summary Using Rivlin and Erickson constitutive equation, the rotational instability of a second order fluid contained between two concentric rotating cylinders has been examined. It is found that the marginal instability is governed by a sixth order eigen value problem. The critical Taylor's number which determines the onset of instability has been determined as a function of m (=₂/₁), a (wave length) and S (=2S ₁+S ₂), with the help of a modified Galerkin technique. Figures 1 to 4 show the effect of second order terms on Taylor's number, radial velocity and streamlines.On study leave at the Department of Applied Mathematics, Indian Institute of Technology, Kharagpur, India.     

Two-dimensional Hierarchical Models for Prismatic Shells with Thickness Vanishing at the Boundary

We construct variational hierarchical two-dimensional models for elastic, prismatic shells of variable thickness vanishing at boundary. With the help of variational methods, existence and uniqueness theorems for the corresponding two-dimensional boundary value problems are proved in appropriate weighted functional spaces. By means of the solutions of these two-dimensional boundary value problems, a sequence of approximate solutions in the corresponding three-dimensional region is constructed. We establish that this sequence converges in the Sobolev space H¹ to the solution of the original three-dimensional boundary value problem. Mathematics Subject Classifications (2000) 74K20, 74K25.     

Characterization of Elasticity-Tensor Symmetries Using SU(2)

A symmetry class of an elasticity tensor, c, is determined by the variance of this tensor with respect to a subgroup of the special orthogonal group, SO(3). Using the double covering of SO(3) by the special unitary group, SU(2), we determine the subgroups of SU(2) that correspond to each of the eight symmetry classes. A family of maps between C² and R³ that preserve the action of the two groups is constructed. Using one of these maps and three associated polynomials, we derive new methods for characterizing the symmetry classes of elasticity tensors. Mathematics Subject Classifications (2000) 74B05, 74E10.     

Wave propagation in inhomogeneous elastic media; normal loading of spherical and cylindrical surfaces

The propagation of waves in inhomogeneous elastic media with spherical or cylindrical symmetry, when the curved surface of the solid is given a uniform normal loading — the stress and displacement components within the solid then may be assumed to depend on one space coordinate and time alone —, is considered. The particular case in which the elastic parameters are proportional to (radius) ⁿ is considered as a special case.Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No.: DA-31-124-ARO-D-462.     

Inertial manifolds and inertial sets for the phase-field equations

The phase-field system is a mathematical model of phase transition, coupling temperature with a continuous order parameter which describes degree of solidification. The flow induced by this system is shown to be smoothing in H¹×L² and a global attractor is shown to exist. Furthermore, in low-dimensional space, the flow is essentially finite dimensional in the sense that a strongly attracting finite-dimensional manifold (or set) exists.This work was completed while the authors were visiting the Institute for Mathematics and its Applications at the University of Minnesota, Minneapolis, Minnesota 55455.     

Melt-particle mixing in gas-stirred ladles with throughflow

An experimental study is performed on a gas-particle stirred ladle system with throughflow, using a simplified water model. Narrow ladles are used to produce 2-D flows. Flow visualization by the direct photographic method is employed to investigate the effects of ladle geometry, throughflow rate, air flow rate and its injection location on the melt-particle mixing performance. Image processing is applied to aid in determining the mixing performance. It is disclosed that an efficient mixing may be achieved if the gas at a higher flow rate is injected with particles through a nozzle near the bottom corner of the ladle wall on the melt inlet side. The mixing performance is better in a rectangular ladle (aspect ratio of 2) than in a square ladle (aspect ratio of unity). The effect of throughflow rate on mixing is minor. The study has an important application in manufacturing processes, such as continuous casting process, and materials processing.List of symbols AR aspect ratio - B width of water vessel, m - Bn Nozzle location on bottom surface of water vessel, m - H height of water vessel or height between bottom surface and free surface of water vessel, m - Hn Nozzle location on vertical (inlet side) surface of water vessel, m - Q _g volumetric rate of gas, m³/s - Q _l volumetric rate of water, m³/s - Q _s volumetric rate of particle, m³/s - x transverse coordinate, m - y longitudinal coordinate, m Visiting scholar on leave from the Mechanical Engineering Department, Kagoshima University, Japan     

Stability analysis and transition prediction of hypersonic boundary layer over a blunt cone with small nose bluntness at zero angle of attack

Stability and transition prediction of hypersonic boundary layer on a blunt cone with small nose bluntness at zero angle of attack was investigated. The nose radius of the cone is 0.5 mm; the cone half-angle is 5°, and the Mach number of the oncoming flow is 6. The base flow of the blunt cone was obtained by direct numerical simulation. The linear stability theory was applied for the analysis of the first mode and the second mode unstable waves under both isothermal and adiabatic wall condition, and eN method was used for the prediction of transition location. The N factor was tentatively taken as 10, as no experimentally confirmed value was available. It is found that the wall temperature condition has a great effect on the transition location. For adiabatic wall, transition would take place more rearward than those for isothermal wall. And despite that for high Mach number flows, the maximum amplification rate of the second mode wave is far bigger than the maximum amplification rate of the first mode wave, the transition location of the boundary layer with adiabatic wall is controlled by the growth of first mode unstable waves. The methods employed in this paper are expected to be also applicable to the transition prediction for the three dimensional boundary layers on cones with angle of attack.     

Function theoretic solutions to problems of orthotropic elasticity

The plane strain problem for a two dimensional orthotropic elastic body is investigated. In particular analytic representations for the solution of the displacement boundary value problem and the stress boundary value problems are found. To this end, the Navier equations are reduced by means of composite transformations to normal form. These are the so-called equations for bianalytic function of the type (k). The generalized Cauchy integral formula for this function theory is used to obtain representation formulae. A simplified method to solve these problems by bianalytic function theory is given for certain situations of plane strain for an orthotropic elastic body. AMS (MOS): 35A20, 35CO5, 35G15, 35J55.Applied Mathematics Institute Technical Report No. 140A, July 1983.The work of this author was supported in part by grant no. DE-AC01-81ER-10967 from the Department of Energy.     

Beugungsprobleme der mathematischen Akustik

Ohne Zusammenfassung Vorgelegt von C.Müller Diese Arbeit entstand w?hrend eines Aufenthaltes am Mathematics Research Center, U.S. Army, University of Wisconsin, in Madison.     

Relations Between Energy and Enstrophy on the Global Attractor of the 2-D Navier-Stokes Equations

We examine how the global attractor of the 2-D periodic Navier–Stokes equations projects in the normalized, dimensionless energy–enstrophy plane (e, E). We treat time independent forces, with the view of understanding how the attractor depends on the nature of the force. First we show that for any force, is bounded by the parabola E = e^1/2 and the line E=e. We then show that for to have points near enough to the parabola, the force must be close to an eigenvector of the Stokes operator A; it can intersect the parabola only when the force is precisely such an eigenvector, and does so at a steady state parallel to this force. We construct a thin region along the parabola, pinched at such steady states, that the attractor can never enter. We show that 0 cannot be on the attractor unless the force is in H^m for all m. Different lower bound estimates on the energy and enstrophy on are derived for both smooth and nonsmooth forces, as are bounds on invariant sets away from 0 and near the line E = e. Motivation for the particular attention to the regions near the parabola and near 0 comes from turbulence theory, as explained in the introduction. Mathematics Subject Classification (2000): 35Q30, 76F02.     

Vortex shedding in cylinder flow of shear-thinning fluids: I. Identification and demarcation of flow regimes

An experimental study on the flow of non-Newtonian fluids around a cylinder was undertaken to identify and delimit the various shedding flow regimes as a function of adequate non-dimensional numbers. The measurements of vortex shedding frequency and formation length (l_f) were carried out by laser-Doppler anemometry in Newtonian fluids and in aqueous polymer solutions of CMC and tylose. These were shear thinning and elastic at weight concentrations ranging from 0.1 to 0.6%. The 10 and 20 mm diameter cylinders (D) used in the experiments had aspect ratios of 12 and 6 and blockage ratios of 5 and 10%, respectively. The Reynolds number (Re^*) was based on a characteristic shear rate of U_∞/(2D) and ranged from 50 to 9×10³ thus encompassing the laminar shedding, the transition and shear-layer transition regimes. Increasing fluid elasticity reduced the various critical Reynolds numbers (Re_etr^*, Re_lf^*, Re_bbp^*) and narrowed the extent of the transition regime. For the 0.6% tylose solution the transition regime was even suppressed. On the other end, pseudoplasticity was found to be indirectly responsible for the observed reduction in Re_otr^*: it increases the Strouhal number which in turn increases the vortex filaments, precursors of the transition regime. Elasticity was better quantified by the elasticity number Re′/We than by the Weissenberg number. This elasticity number involves the calculation of the viscosity at a high characteristic shear rate, typical of the boundary layer, rather than at the average value (U_∞/(2D)) used for the Reynolds number, Re^*.     

不同交通流状态下桥梁气动导数的风洞试验研究

为了考虑实际运营车辆对桥梁气动导数的影响,根据车辆密度模拟了三种交通流状态,基于强迫振动装置,分别对每个交通流和无车状态下的桥梁气动导数进行风洞试验研究,讨论了不同攻角下不同车流的车辆对桥梁气动导数的影响,探究了车辆对气动导数影响的百分比以及气动导数变化量的变化规律。研究结果表明:不同攻角下不同车流的车辆均对直接导数A*2、H*4和交叉导数A*4、H*2影响显著,A*2、A*3变化量随着折减风速有一定的变化规律。虽然不同攻角下不同车流的车辆对气动导数的影响程度及影响规律不同,并且车流的繁忙程度对大多数气动导数的影响规律不明显,但是车辆对桥梁气动导数的影响不容忽略。     

Rheo-dielectric behavior of low molecular weight liquid crystal 2. Behavior of 8CB in nematic and smectic states

Rheo-dielectric behavior was examined for 4−4^′−n-octyl-cyanobiphenyl (8CB) having large dipoles parallel to its principal axis (in the direction of the C≡N bond). In the quiescent state at all temperatures (T) examined, orientational fluctuation of the 8CB molecules was observed as dielectric dispersions at characteristic frequencies ω_c>10⁶ s⁻¹. In the isotropic state at high T, no detectable changes of the complex dielectric constant ɛ^*(ω) were found under slow flow at shear rates ˙γ≫ω_c. In the nematic state at intermediate T, the terminal relaxation intensity of ɛ^*(ω) was decreased under such slow flow. In the smectic state at lower T, the flow effect became much less significant. These results were related to the flow-induced changes of the liquid crystalline textures in the nematic and smectic states, and the differences of the rheo-dielectric behavior in these states are discussed in relation to a difference of the symmetry of molecular arrangements in the nematic and smectic textures. Received: 1 October 1998 Accepted: 13 January 1999     

不同展长比旗帜在均匀来流中摆动的实验研究

为研究展长比对旗帜摆动特性的影响,在低速风洞内开展了质量比M*为0.6,展长比H*为0.2～1.5的柔性旗帜在不同来流速度下的失稳实验.利用高速摄影机和图像处理技术分析了无量纲速度U*和展长比H*对旗帜运动学特征(频率和振幅)的影响;利用自行研制的天平测量了旗帜受到的阻力,首次给出了旗帜的动力学特征.结果发现,保持展长...     

Flow characteristics of gas-liquid-particle mixing in a gas-stirred ladle system with throughflow

An experimental study is performed on air-liquid-particle mixing, resulting from an air-particle mixture injected into a liquid flowing through a slender ladle. Flow visualization combined with image processing is employed to investigate the bubble and particle behavior at the nozzle outlet. Effort is directed to particle discrimination in both the liquid and the bubbles to determine particle distribution, which affects the mixing performance of gas bubbles, solid particles and liquid. A real-time movement of bubble and particle behavior can be visualized by means of image processing with the use of a slow-motion video recording. It is disclosed that the particles injected through the nozzle may stick on the inner surface of the gas bubble, break through the bubble surface, or mingle with the gas stream to form a two-phase jet, depending on the particle-to-gas mass flow rate ratio. It is observed that when a solid-gas two-phase jet penetrates deeper in the horizontal direction, the particles and bubbles rise along the vertical sidewall and simultaneously spread in the transverse direction, thus promoting a better liquid-particle mixing. The application of the slow-motion video recording results in quantitative evaluations of both the penetration depth of particles or of gas-particles from the injection nozzle and the velocity distribution along the sidewall.List of symbols B Width of water vessel, m - B _n Nozzle location on bottom surface of water vessel, m - d _o Diameter of a gas-particle injection nozzle, m - H Height of water vessel, m - H _n Nozzle location on vertical surface of water vessel, m - L Penetration length of particles or of particles and gas from the nozzle, m - Q _g Volumetric flow rate of gas, m³/s - Q _l Volumetric flow rate of water, m³/s - Q _s Volumetric flow rate of particle, m³/s - Re _g Gas Reynolds number based on inner diameter of the air-particle injection nozzle - t Time, sec. - W Thickness of water vessel, m - x Transverse coordinate, m - y Longitudinal coordinate, m - Mass flow rate ratio of particles to gas Visiting scholar on leave from the Mechanical Engineering Department, Kagoshima University, Kagoshima, JapanThe work reported was supported by the National Science Foundation under the Grant No. CTS-8921584     

An experimental—Numerical evaluation of the<Emphasis Type="Italic">T</Emphasis><Stack><Subscript>ε</Subscript><Superscript>*</Superscript></Stack> integral for a three-dimensional crack front

TheT _ε ^* integral was calculated on the surface of single edge notched, three-point bend (SE(B)) specimens using experimentally obtained displacements. Comparison was made withT _ε ^* calculated with the measured surface displacements andT _ε ^* calculated at several points through the thickness of a finite element (FE) model of the SE(B) specimen. Good comparison was found between the surfaceT _ε ^* calculated from displacements extracted from the FE model and the surfaceT _ε ^* calculated from experimentally obtained displacements. The computedT _ε ^* integral was also observed to decrease as the crack front was traversed from the surface to the mid-plane of the specimen. Mid-planeT _ε ^* values tend to be approximately 10% of the surface values.     

Gurtin-type properties associated with solutions of the transient plate bending problem

The general motion of a thin elastic plate executing small deflections normal to its surface is considered. The edge of the plate is part simply supported, part clamped, part free and part along which the tractions are prescribed. Variational and reciprocity principles are established for the problem.

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Zusammenfassung Es wird die allgemeine Bewegung einer dünnen elastischen Platte die kleine Ablenkungen senkrecht zu ihrer Oberfläche ausführt betrachtet. Der Rand der Platte ist teilweise einfach gelagert, teilweise frei, teilweise eingespannt und teilweise sind am Rand Kräfte vorgegeben. Für das Problem werden Variations- und Reziprozitätsprinzipien angegeben.

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Mathematics Department, University of Southampton     

Large-scale flow structures and heat transport of turbulent forced and mixed convection in a closed rectangular cavity

Results of an experimental study of flow structure formation and heat transport in turbulent forced and mixed convection are presented. The experiments were conducted in a rectangular cavity with a square cross section, which has an aspect ratio between length and height of Γ_xz = 5. Air at atmospheric pressure was used as working fluid. The air inflow was supplied through a slot below the ceiling, while exhausting was provided by another slot, which is located directly above the floor. Both vents extend over the whole length of the cell. In order to induce thermal convection the bottom of the cell is heated while the ceiling is maintained at a constant temperature. This configuration allows to generate and study mixed convection under well defined conditions. Results of forced convection at Re = 1.07 × 10⁴ as well as mixed convection at 1.01 × 10⁴ ? Re ? 3.4 × 10⁴ and Ra = 2.4 × 10⁸ (3.3 ? Ar ? 0.3), which were obtained by means of Particle Image Velocimetry and local temperature measurements, are presented. For purely forced convection a 2D mean wind, which can be approximated by a solid body rotation, is found. With increasing Archimedes number this structure becomes unstable, leading to a transition of the solid body rotation into additional smaller convection rolls. Proper orthogonal decomposition of the instantaneous velocity fields has been performed for further analysis of these coherent large-scale structures. Their fingerprint is found in the spatial temperature distribution of the out flowing air at the end of the outlet channel, which reveals a temporally stable profile with two maxima over the length of the outlet. Moreover a maximum in the global enthalpy transport by the fluid is found at Ar ≈ 0.6.