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Bifurcation Diagrams (bifurcation + diagram)
Selected AbstractsBifurcation and stability analysis of laminar flow in curved ductsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2010Werner Machane Abstract The development of viscous flow in a curved duct under variation of the axial pressure gradient q is studied. We confine ourselves to two-dimensional solutions of the Dean problem. Bifurcation diagrams are calculated for rectangular and elliptic cross sections of the duct. We detect a new branch of asymmetric solutions for the case of a rectangular cross section. Furthermore we compute paths of quadratic turning points and symmetry breaking bifurcation points under variation of the aspect ratio , (,=0.8,1.5). The computed diagrams extend the results presented by other authors. We succeed in finding two origins of the Hopf bifurcation. Making use of the Cayley transformation, we determine the stability of stationary laminar solutions in the case of a quadratic cross section. All the calculations were performed on a parallel computer with 32×32 processors. Copyright © 2009 John Wiley & Sons, Ltd. [source] Cover Picture: Anisotropy and Dynamic Ranges in Effective Properties of Sheared Nematic Polymer Nanocomposites (Adv. Funct.ADVANCED FUNCTIONAL MATERIALS, Issue 12 2005Mater. Abstract Forest and co-workers report on p.,2029 that nematic polymer nanocomposite (NPNC) films can be processed in steady shear flows, which generate complex orientational distributions of the nanorod inclusions. Distribution functions for a benchmark NPNC (11,vol.-% of 1,nm,×,200,nm rods) are computed for a range of shear rates, yielding a bifurcation diagram with steady states at very low (logrolling) and high (flow-aligning) shear rates, and limit cycles (tumbling, wagging, kayaking) at intermediate shear rates. The orientational distributions dictate the effective conductivity tensor of the NPNC film, which is computed for all distribution functions, and extract the maximum principal conductivity enhancement (Emax, averaged in time for periodic distributions) relative to the matrix. The result is a "property bifurcation diagram" for NPNC films, which predicts an optimal shear rate that maximizes Emax. Nematic, or liquid-crystalline, polymer nanocomposites (NPNCs) are composed of large aspect ratio, rod-like or platelet, rigid macromolecules in a matrix or solvent, which itself may be aqueous or polymeric. NPNCs are engineered for high-performance material applications, ranging across mechanical, electrical, piezoelectric, thermal, and barrier properties. The rods or platelets possess enormous property contrasts relative to the solvent, yet the composite properties are strongly affected by the orientational distribution of the nanophase. Nematic polymer film processing flows are shear-dominated, for which orientational distributions are well known to be highly sensitive to shear rate and volume fraction of the nematogens, with unsteady response being the most expected outcome at typical low shear rates and volume fractions. The focus of this article is a determination of the ranges of anisotropy and dynamic fluctuations in effective properties arising from orientational probability distribution functions generated by steady shear of NPNC monodomains. We combine numerical databases for sheared monodomain distributions[1,2] of thin rod or platelet dispersions together with homogenization theory for low-volume-fraction spheroidal inclusions[3] to calculate effective conductivity tensors of steady and oscillatory sheared mesophases. We then extract maximum scalar conductivity enhancement and anisotropy for each type of sheared monodomain (flow-aligned, tumbling, kayaking, and chaotic). [source] Numerical analysis of optical feedback phenomenon and intensity noise of fibre-grating semiconductor lasersINTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 3 2007Moustafa Ahmed Abstract This paper demonstrates numerical analysis of the dynamics and intensity noise of fibre-grating semiconductor lasers (FGSLs). The induced phenomenon of strong optical feedback (OFB) is analysed. The simulations are based on an improved time-delay rate equations model of a single-mode laser that takes into account the multiple round-trips of the lasing field in the fibre cavity. The analyses are performed in terms of the temporal trajectory of the laser intensity, bifurcation diagram and relative intensity noise (RIN). We explore influence of the fibre-cavity length on the dynamics and RIN. The results show that when the fibre cavity is short, the regime of strong OFB is characterized by either continuous-wave (CW) operation or periodic pulsation. The pulsation frequency is locked at the frequency separation of either the compound-cavity modes or the external fibre-cavity modes. The corresponding RIN level is close to or higher than the level of the solitary laser depending on pulse symmetry. When the fibre cavity is long, the laser exhibits unstable dynamics over wider range of OFB. Moreover, the strong-OFB pulsation becomes beating quasi-periodic at the relaxation oscillation frequency and the fibre-cavity mode-separation frequency. Copyright © 2007 John Wiley & Sons, Ltd. [source] Discrete thermodynamics of chemical equilibria and classical limit in thermodynamic simulationISRAEL JOURNAL OF CHEMISTRY, Issue 3-4 2007Boris Zilbergleyt This article sets forth comprehensive basic concepts of the discrete thermodynamics of chemical equilibrium as balance between internal and external thermodynamic forces. Conditions of chemical equilibrium in the open chemical system are obtained in the form of a logistic map, containing only one new parameter that defines the chemical system's resistance to external impact and its deviation from thermodynamic equilibrium. Solutions to the basic map are bifurcation diagrams that have quite traditional shape but the diagram areas feature specific meanings for chemical systems and constitute the system's domain of states. The article is focused on two such areas: the area of "true" thermodynamic equilibrium and the area of open chemical equilibrium. The border between them represents the classical limit, a transition point between the classical and newly formulated equilibrium conditions. This limit also separates regions of the system ideality, typical for isolated classical systems, and non-ideality due to the limitations imposed on the open system from outside. Numerical examples illustrating the difference between results of classical and discrete thermodynamic simulation methods are presented. The article offers an analytical formula to find the classical limit, compares analytical results with these obtained by simulation, and shows the classical limit dependence upon the chemical reaction stoichiometry and robustness. [source] Game theoretic approach to multiobjective designs: Focus on inherent safetyAICHE JOURNAL, Issue 1 2006Anjana Meel Abstract A method for designing processes that are inherently safer,with the primary focus on disturbances having the potential for unbounded hazardous responses,is introduced. In cases where safety is not threatened (as in isothermal fermentation reactors), but product quality can rapidly degrade, this method provides designs that ensure high product quality (as in pharmaceutical processes). Using game theory, the method accounts for the trade-offs in profitability, controllability, safety and/or product quality, and flexibility. For nonlinear processes that are hard to control; that is, have an unstable and/or nonminimum-phase steady state, over a wide range of operating conditions, extended bifurcation diagrams are introduced. When a steady state is nonminimum phase, the process may exhibit inverse response. The steady states of processes are classified on the basis of instability and nonminimum-phase behavior to segregate the operating regimes into distinct zones. Locally optimal designs, one corresponding to each zone, are obtained first. These are compared with other locally optimal designs at alternate operating conditions, and/or process reconfigurations, to obtain the globally optimal design using game theory. Four indices,profitability, controllability, safety and/or product quality, and flexibility,characterize the optimality of a design. A novel index for safe operation and/or product quality at a steady state is formulated as a function of the eigenvalues of the Jacobian of the process model and the Jacobian of the process zero dynamics, providing a quantitative measure of instability and nonminimum-phase behavior. The application of the proposed method to an isothermal, continuous stirred-tank reactor (CSTR) with van der Vusse reactions, an exothermic CSTR, and an anaerobic fermentor with substrate and product inhibition is presented. © 2005 American Institute of Chemical Engineers AIChE J, 2006 [source] Nonlinear modal interactions and low order modeling of a kicked flexible rodPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2007Joseph P. Cusumano A magnetically kicked flexible-beam oscillator is studied experimentally. Spectral bifurcation diagrams of the attracting steady states are obtained, and the system is seen to exhibit periodic, quasiperiodic and chaotic responses. The proper orthogonal decomposition (POD) is applied to spatial strain gauge data, and a specific nonlinear two-mode interaction is shown to be present in all responses. Based on these experimental observations, a two degree of freedom piecewise linear autonomous model is developed, which captures much of the behavior observed in the experiments. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] | |||||||||||||