Symmetric Bifurcation Research Articles

Planar sudden symmetric expansion flows and bifurcation phenomena of purely viscous shear-thinning fluids

The goal of a present study is to investigate the effects of the generalized Newtonian fluids on the threshold of the transition from flow symmetry to its asymmetry for the flow through a 1:3 planar sudden expansion. We consider purely viscous shear-thinning fluids covering a range of the index n of Power law model n = 0.6 , 0.8 and compare them with the Newtonian fluid ( n = 1.0 ). Sudden expansion fluid flow is studied numerically by solving the two-dimensional momentum equations along with the continuity equation and the Power law rheological model. We report a systematic results in a range of generalized Reynolds number 10 ≤ R e g e n ≤ 150 with a focus on its critical value. The subsequent effects are investigated by means of a different flow parameters, i.e. generalized Reynolds number ( R e g e n ), modified Reynolds number ( R e m o d ) and the Reynolds number defined in accordance with the inlet wall viscosity ( R e w a l l ). Results indicate that the shear-thinning viscous behaviour increases the onset of bifurcation phenomena and the critical value of Reynolds number.

SU-GG-J-25: Aerosols/Nanoparticle Therapy for Malignancy Involving the Lung - a Transport Phenomenon Study

Purpose: Study the nanoparticles/aerosols localized deposition patterns in morphometrically realistic lung bifurcations geometry using computational fluid dynamics code FLUENT®. Method and Materials: Chest CT scan data from an unidentified patient, with 3 mm slices in the axial direction are used to reconstruct the lung airway bifurcation geometry using Amira 4.0 (Mercury Inc.) program. The 3D rendered airway bifurcations are exported to FLUENT® preprocessor GAMBIT where further meshing was performed and exported for FLUENT® computation. Species Transport and Reaction model in FLUENT® was used to explore the local deposition patterns of nanoparticles/aerosols. At this time, only the inspiratory phase of the breathing cycle has been studied in realistic bifurcation geometry for 3rd and 4th generations. Modeling using the CT scan data is currently in progress. N2 has been assumed to be the bulk carrier gas in the flow fields carrying arbitrary particles uniformly sized (smoke, radon daughters, radiopharmaceuticals, etc.) with defined physical and diffusional properties. In this initial study a mass fraction of 10−8 was used for particle mass concentration. The particles interacting with the the airway wall surface are assumed to be deposited. Results: Results from preliminary studies of localized particle deposition velocities and deposition fractions using unrealistic symmetric airway bifurcations geometries for 3rd and 4th generations using FLUENT® model for Po-218 nanoparticles are in close agreement with the experimental data of Kinsara et al. (Health Phys. 68(3), 1995). Further, results obtained with the realistic bifurcation geometry for 3rd and 4th generations, suggest that the particle deposition is maximum at the carinal region of the bifurcations. Conclusion: Maximum particle deposition concentration was observed to occur at the carinal regions in the bifurcations. Further, the deposition patterns provide very useful information for the design of the inhalation drugs, with detail on where the particles end up after one breath in.

Invariants, equivariants and characters in symmetric bifurcation theory

In the analysis of stability in bifurcation problems it is often assumed that the (appropriate reduced) equations are in normal form. In the presence of symmetry, the truncated normal form is an equivariant polynomial map. Therefore, the determination of invariants and equivariants of the group of symmetries of the problem is an important step. In general, these are hard problems of invariant theory and, in most cases, they are tractable only through symbolic computer programs. Nevertheless, it is desirable to obtain some of the information about invariants and equivariants without actually computing them, for example, the number of linearly independent homogeneous invariants or equivariants of a certain degree. Generating functions for these dimensions are generally known as ‘Molien functions’. We obtain formulae for the number of linearly independent homogeneous invariants or equivariants for Hopf bifurcation in terms of characters. We also show how to construct Molien functions for invariants and equivariants for Hopf bifurcation. Our results are then applied to the computation of the number of invariants and equivariants for Hopf bifurcation for several finite groups and the continuous group O(3).

Conditions for symmetric, antisymmetric, and zero-stiffness bifurcation in view of imperfection sensitivity and insensitivity

In general, it is not clear how to design structures such that they are ab initio imperfection insensitive, i.e., without modifications of the original design after the diagnosis of imperfection sensitivity. Symmetry and antisymmetry of bifurcation paths, representing qualitative properties of a structure, not only simplify the postbuckling analysis, but also have an influence on the behavior of real, i.e., imperfect structures. The special case of a zero-stiffness postbuckling path incorporates both, symmetry and antisymmetry. Mathematical definitions of the three categories symmetric, antisymmetric, and zero-stiffness equilibrium paths are given. It is shown that symmetry as well as antisymmetry causes the vanishing of specific coefficients in an asymptotic series expansion, following from Koiter’s initial postbuckling analysis. Thereupon, methods of checking for the three categories of bifurcation behavior are discussed. Finally, the three categories are investigated in terms of necessary and/or sufficient conditions for imperfection insensitivity. For instance, a horizontal tangent of the postbuckling path at the bifurcation point is required for imperfection insensitivity. Four examples illustrate non-symmetric, symmetric, and antisymmetric bifurcation as well as zero-stiffness postbuckling behavior. For the first two examples, the approach of increasing the stiffness results in the conversion of the system from imperfection sensitivity into insensitivity. Remarkably, this happens without changing the prebuckling behavior and the buckling load of the structure. The third example demonstrates that imperfection sensitivity is an inevitable implication of antisymmetric bifurcation.

Phase bifurcations of electron gas at the conditions of diamagnetic instability

In a framework of catastrophe theory we investigate the equilibrium set for an electron gas at the conditions of strong dHvA effect and show that discontinuities accompanied the diamagnetic phase transition (DPT) is handled by Riemann–Hugoniot catastrophe. We show that the applicability of the standard condition for observation of DPT a ≥ 1 where a is the differential magnetic susceptibility is valid only in the nearest vicinity of the centre of dHvA period corresponding to triple degenerate point, but for arbitrary value of magnetic field in every period of dHvA oscillations this condition is modified in accordance with the bifurcation set of cusp catastrophe. While at the center of dHvA period the symmetrical supercritical pitchfork bifurcation gives rise to the second-order phase transition on temperature, the deviation of magnetic field from the value corresponding to the centre of dHvA period results in the change of the phase transition order from the second to the first one both on temperature and magnetic field. In the framework of developed theory we obtain good agreement with available experimental data.

Comparison of Axisymmetric and Three-Dimensional Models for Gas Uptake in a Single Bifurcation During Steady Expiration

Reactive gas uptake is predicted and compared in a single bifurcation at steady expiratory flow in terms of Sherwood number using an axisymmetric single-path model (ASPM) and a three-dimensional computational fluid dynamics model (CFDM). ASPM is validated in a two-generation geometry by comparing the average gas-phase mass transfer coefficients with the experimental values. ASPM predicted mass transfer coefficients within 20% of the experimental values. The flow and concentration variables in the ASPM were solved using Galerkin finite element method and in the CFDM using commercial finite element software FIDAP. The simulations were performed for reactive gas flowing at Reynolds numbers ranging from 60 to 350 in both symmetric bifurcation for three bifurcation angles, 30 deg, 70 deg, and 90 deg, and in an asymmetric bifurcation. The numerical models compared with each other qualitatively but quantitatively they were within 0.4-8% due to nonfully developed flow in the parent branch predicted by the CFDM. The radially averaged concentration variation along the axial location matched qualitatively between the CFDM and ASPM but quantitatively they were within 32% due to differences in the flow field. ASPM predictions compared well with the CFDM predictions for an asymmetric bifurcation. These results validate the simplified ASPM and the complex CFDM. ASPM predicts higher Sherwood number with a flat velocity inlet profile compared to a parabolic inlet velocity profile. Sherwood number increases with the inlet average velocity, wall mass transfer coefficient, and bifurcation angle since the boundary layer grows slower in the parent and daughter branches.

Non-linear buckling and postbuckling behavior of thin-walled beams considering shear deformation

The static stability of thin-walled composite beams, considering shear deformation and geometrical non-linear coupling, subjected to transverse external force has been investigated in this paper. The theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field (accounting for bending and warping shear) considering moderate bending rotations and large twist. This non-linear formulation is used for analyzing the prebuckling and postbuckling behavior of simply supported, cantilever and fixed-end beams subjected to different load condition. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton–Rapshon method. The numerical results show that the beam loses its stability through a stable symmetric bifurcation point and the postbuckling strength is in relation with the buckling load value. Classical predictions of lateral buckling are conservative when the prebuckling displacements are not negligible and the non-linear buckling analysis is required for reliable solutions. The analysis is supplemented by investigating the effects of the variation of load height parameter. In addition, the critical load values and postbuckling response obtained with the present beam model are compared with the results obtained with a shell finite element model (Abaqus).

COMPLEX INTERMITTENCY IN SWITCHING CONVERTERS

Intermittent instability is commonly observed in switching power supplies during the design and development phase. It manifests as symmetrical period-doubling bifurcation in the time domain with long intermittent periods. Such intermittent operation is considered undesirable in practice and is usually avoided by appropriate adjustments of circuit parameters. This paper explores the mechanism and conditions for the emergence of intermittency in a common voltage-mode controlled buck converter. It is found that interference at frequencies near the switching frequency or its rational multiples will induce intermittent operation. The strengths and frequencies of the interfering signals determine the type and period of intermittency. The problem is analyzed by transforming the conventional parameter-bifurcation analysis to a time-bifurcation analysis. Analytical results are verified by simulations and experimental measurements.

Dynamic analysis and suppressing chaotic impacts of a two-degree-of-freedom oscillator with a clearance

A two-degree-of-freedom impact oscillator is considered. The maximum displacement of one of the masses is limited to a threshold value by the symmetrical rigid stops. Impacts between the mass and the stops are described by an instantaneous coefficient of restitution. Dynamics of the system is studied with special attention to periodic-impact motions and bifurcations. Period-one double-impact symmetrical motion and transcendental impact Poincaré map of the system is derived analytically. Stability and local bifurcations of the period-one double-impact symmetrical motions are analyzed by using the impact Poincaré map. The Lyapunov exponents in the vibratory system with impacts are calculated by using the transcendental impact map. The influence of the clearance and excitation frequency on symmetrical double-impact periodic motion and bifurcations is analyzed. A series of other periodic-impact motions are found and the corresponding bifurcations are analyzed. For smaller values of clearance, period-one double-impact symmetrical motion usually undergoes pitchfork bifurcation with decrease in the forcing frequency. For large values of the clearance, period-one double-impact symmetrical motion undergoes Neimark–Sacker bifurcation with decrease in the forcing frequency. The chattering-impact vibration and the sticking phenomena are found to occur in the region of low forcing frequency, which enlarges the adverse effects such as high noise levels, wear and tear and so on. These imply that the dynamic behavior of this system is very rich and complex, varying from different types of periodic motions to chaos, even chattering-impacting vibration and sticking. Chaotic-impact motions are suppressed to minimize the adverse effects by using external driving force, delay feedback and feedback-based method of period pulse.

Carotid geometry effects on blood flow and on risk for vascular disease

It has been widely observed that atherosclerotic diseases occur at sites with complex hemodynamics, such as artery bifurcations, junctions, and regions of high curvature. These regions usually have very low or highly oscillatory wall shear stress (WSS). In the present work, 3D pulsatile blood flow through a model of the carotid artery bifurcation was simulated using a finite volume numerical method. The goal was to quantify the risk of atherogenesis associated with different carotid artery geometries. A risk scale based on the average WSS on the sinus wall of the internal carotid artery was proposed—a scale that can be used to quantify the effect of the carotid geometry on the relative risk for developing vascular disease. It was found that the bifurcation angle and the out-of-plane angle of the internal carotid artery affect the formation of low stress regions on the carotid walls. The main conclusions are: (a) larger internal carotid artery angles ( θ IC) generally increase the frequency and the area of blood recirculation and lower the WSS on the sinus wall, hence increasing the risk of plaque build-up; (b) off-plane angles were found to lower the WSS on the sinus for geometries with θ IC⩾25°. Larger off-plane angles generally increase the danger of plague build-up; (c) for θ IC<25°, the off-plane angle does not have an obvious effect on the hemodynamic WSS; (d) symmetric bifurcations were found to increase the WSS on the sinus wall and ease the risk of vascular disease.

River bifurcations: Experimental observations on equilibrium configurations

In this work we have investigated the equilibrium configurations of a Y‐shaped fluvial bifurcation through a laboratory analysis. Three series of experimental runs have been performed in a wide flume, where a symmetrical bifurcation has been constructed joining three branches with fixed banks and movable bed made of a well sorted quartz sand; the angle between the two downstream distributaries was equal to 30 degrees. The experiments have been carried out with different values of longitudinal bed slope and water discharge, in order to investigate a range of the relevant morphodynamic parameters typical of gravel bed braided rivers. The equilibrium configuration of the bifurcation has been characterized through the measure of the discharge partition in downstream branches and of the local bed structure at the node. The existence of unbalanced equilibrium configurations has been observed and the role of migrating alternate bars has been pointed out. The experimental results confirm the theoretical predictions which have been recently obtained through the simple model of Bolla Pittaluga et al. (2003). Moreover, interpreting the measured data in the light of the concept of morphodynamic influence provides a new perspective in the analysis of the equilibrium configurations of a bifurcation.

Bifurcation dynamics in lineage-commitment in bipotent progenitor cells

Lineage specification of multipotent progenitor cells is governed by a balance of lineage-affiliated transcription factors, such as GATA1 and PU.1, which regulate the choice between erythroid and myelomonocytic fates. But how ratios of lineage-determining transcription factors stabilize progenitor cells and resolve their indeterminacy to commit them to discrete, mutually exclusive fates remains unexplained. We used a simple model and experimental measurements to analyze the dynamics of a binary fate decision governed by a gene-circuit containing auto-stimulation and cross-inhibition, as embodied by the GATA1–PU.1 paradigm. This circuit generates stable attractors corresponding to erythroid and myelomonocytic fates, as well as an uncommitted metastable state characterized by coexpression of both regulators, explaining the phenomenon of “multilineage priming”. GATA1 and PU.1 mRNA and transcriptome dynamics of differentiating progenitor cells confirm that commitment occurs in two stages, as suggested by the model: first, the progenitor state is destabilized in an almost symmetrical bifurcation event, resulting in a poised state at the boundary between the two lineage-specific attractors; second, the cell is driven to the respective, now accessible attractors. This minimal model captures fundamental features of binary cell fate decisions, uniting the concepts of stochastic (selective) and deterministic (instructive) regulation, and hence, may apply to a wider range of binary fate decision points.

Analyses of impact motions of harmonically excited systems having rigid amplitude constraints

Two harmonically excited systems having symmetrical rigid constraints are considered. Repeated impacts usually occur in these systems due to the rigid amplitude constraints. The impact forms occurring in two systems are different, i.e., the components of one system collide with each other, and one of components of the other system collides with rigid obstacles. Dynamics of these systems are studied with special attention to Neimark–Sacker bifurcations associated with several periodic-impact motions. Period-one double-impact symmetrical motions and associated Poincaré maps of two systems are derived analytically. Stability and local bifurcations of the period-one double-impact symmetrical motions are analyzed by using the Poincaré maps. Neimark–Sacker bifurcations associated with several periodic-impact motions are found by numerical simulation, and the corresponding routes from quasi-periodic impact motions to chaos are also stated. The influence of the clearance and excitation frequency on symmetrical double-impact periodic motion and bifurcations is analyzed. Studies show that the vibratory systems having symmetrical rigid amplitude constraints may exhibit complex and rich quasi-periodic impact behavior under different system parameter conditions.

Applications of equivariant degree for gradient maps to symmetric Newtonian systems

We consider G = Γ × S 1 with Γ being a finite group, for which the complete Euler ring structure in U ( G ) is described. The multiplication tables for Γ = D 6 , S 4 and A 5 are provided in the Appendix. The equivariant degree for G -orthogonal maps is constructed using the primary equivariant degree with one free parameter. We show that the G -orthogonal degree extends the degree for G -gradient maps (in the case of G = Γ × S 1 ) introduced by Gȩba in [K. Gȩba, W. Krawcewicz, J. Wu, An equivariant degree with applications to symmetric bifurcation problems I: Construction of the degree, Bull. London. Math. Soc. 69 (1994) 377–398]. The computational results obtained are applied to a Γ -symmetric autonomous Newtonian system for which we study the existence of 2 π -periodic solutions. For some concrete cases, we present the symmetric classification of the solution set for the systems considered.

Fiber Deposition in the Tracheobronchial Region: Experimental Measurements

Fiber deposition efficiency in the tracheobronchial region was studied by using two human airway replicas. Carbon fibers with a monodisperse diameter of 3.66 μ were delivered into lung casts at the flow rates of 15, 45, and 60 L/min. Deposition efficiencies in each airway bifurcation were obtained individually. Even though the data shows large variability among the bifurcations within the same generation, there is a general trend for the data obtained in these two replicas. Our results show that fiber deposition for carbon fibers increases with the Stokes number, indicating that inertial impaction is the dominant mechanism. Also, fiber deposition in the tracheobronchial region is lower than that of spherical particles at a given Stokes number, and is similar to the deposition found in the nasal and oral airways as reported previously. This finding implies that it is easier for fibers to penetrate the upper respiratory tract and to reach the lower airways as compared to spherical particles. Our measurements are in general agreement with data obtained with asbestos fiber as reported in another study using a hollow airway replica. Our study appears to give a higher deposition efficiency than the results obtained using idealized single symmetrical bifurcation models. This demonstrates, again, the importance of using a realistic airway model, including larynx, to simulate the airway geometry and flow pattern of the human respiratory tract.

複数の部材座屈と極限点が重複するアーチ状トラスの最悪不整

Imperfection sensitivity properties are investigated for an arch that has multiple member buckling at a limit point, which is classified as hilltop branching with multiple symmetric bifurcation points. The critical loads of imperfect structures are shown to be governed by a piecewise linear law of the imperfection parameter. Antioptimization problems are formulated to obtain the worst imperfection mode that most drastically reduces the critical load. A formula for the worst nodal imperfection as a linear combination of the critical modes is also presented. The validity of the formula is ensured by path-following analysis of arches with randomly generated imperfections.

Constructal dendritic geometry and the existence of asymmetric bifurcation

This paper considers the fundamental question of whether the optimal geometry of dendritic tree networks is symmetric or asymmetric. Asymmetry of dendritic networks is a result of the relation between flow resistance and flow fraction at each bifurcation node. Asymmetry of bifurcation (asymmetry of Y-shaped assemblies) appears when the flow fraction at each bifurcation node is not equal to one-half. Asymmetric bifurcation provides lower flow resistance than symmetric bifurcation. Murray’s law of bifurcation (Di+1∕Di=2−1∕3) is valid only when the flow fraction at every bifurcation node in dendritic networks is one-half and the networks are symmetric. General rules to construct asymmetric trees are developed and reported in this paper. It is shown that even through pressure drops across round and square cross-sectional shape channels are different; their flow resistances can be expressed by similar relations.

Eddy genesis and transformation of Stokes flow in a double-lid-driven cavity. Part 2: Deep cavities

This paper extends an earlier work [1] on the development of eddies in rectangular cavities driven by two moving lids. The streamfunction describing Stokes flow in such cavities is expressed as a series of Papkovich-Faddle eigenfunctions. The focus here is deep cavities, i.e. those with large height-to-width aspect ratios, where multiple eddies arise. The aspect ratio of the fully developed eddies is found computationally to be 1.38 &gt; 0.05, which is in close agreement with that obtained from Moffatt's [2] analysis of the decay of a disturbance between infinite stationary parallel plates. Extended control space diagrams for both negative and positive lid speed ratios are presented, and show that the pattern of bifurcation curves seen previously in the single-eddy cavity is repeated at higher aspect ratios, but with a shift in the speed ratio. Several special speed ratios are also identified for which the flow in one or more eddies becomes locally symmetric, resulting in locally symmetric bifurcation curves. By superposing two semi-infinite cavities and using the constant velocity damping factor found by Moffatt, a simple model of a finite multiple-eddy cavity is constructed and used to explain both the repetition of bifurcation patterns and the local symmetries. The speed ratios producing partial symmetry in the cavity are shown to be integer powers of Moffatt's velocity damping factor.

Postbuckling analysis of columns resting on an elastic foundation

The postbuckling response of perfect and geometrically imperfect elastic columns resting on an elastic Winkler type foundation is thoroughly discussed. This is established by employing an approximate analytic technique leading to very reliable results in the vicinity of the critical state. It was found that the critical state of perfect columns is a stable symmetric bifurcation point and consequently there is no sensitivity to initial geometrical imperfections. Moreover, a simple but readily analyzed mechanical model is proposed to simulate the salient features of buckling mechanism of the column on elastic foundation with those of the model. The simplicity, reliability and efficiency of the proposed analysis as well as the successful modeling of the buckling mechanism of the column by that of a single mode mechanical model are illustrated with the aid of numerical examples.

MEMS sensors to resolve spatial variations in shear stress in a 3-D blood vessel bifurcation model

Coronary artery disease (CAD) preferentially develops at the arterial branching points or bifurcations. Hemodynamics, particularly wall shear stress, plays an important role in regulating the development of CAD. The advent of the microelectromechanical systems (MEMS) sensor provides a potential entry point to overcome the difficulty in measuring temporal and spatial variations in shear stress. We, hereby, demonstrate the application of a MEMS sensor to resolve circumferential variations in shear stress using a three-dimensional symmetric bifurcation model. Reynolds numbers ranging from 1.34 to 6.7 were chosen to simulate flow at the microcirculation level. At these low Reynolds numbers, the wall shear stress was highest at the divider of bifurcation, and relaxed to a lower value downstream from the bifurcation. Skin friction coefficient values (C/sub f/), defined as local wall shear stress normalized by the upstream dynamic pressure, varied circumferentially by a factor of 2 or more from the medial wall at the divider to the lateral wall of bifurcation. These experimental skin friction coefficients at various positions were in close agreement with values derived from the Navier-Stokes solution.