Different Types Of Bifurcation Diagrams Research Articles

Influence of modal coupling and geometrical imperfections on the nonlinear buckling of cylindrical panels under static axial load

The aim of this work is to analyse the influence of the nonlinear modal coupling and initial geometrical imperfection on the post-buckling path (perfect case) or nonlinear equilibrium path (imperfect case) of a simply supported, axially loaded cylindrical panel. The cylindrical panel is described by the Donnell nonlinear shallow shell theory and the lateral displacement field is based on a perturbation procedure, generating a precise low-dimensional model that satisfies out-of-plane boundary conditions and considers the forthcoming nonlinear modal coupling due to quadratic and cubic terms in a nonlinear equilibrium equation. The discretized equations of motion are determined by applying the standard Galerkin method. Various numerical techniques are employed to obtain the cylindrical panel’s nonlinear static equilibrium path with its structural stability analysis. The results show the influence of geometry on the nonlinear response of the cylindrical panel, unveiling an intricate competition between different types of bifurcation diagrams (stable symmetric, unsymmetrical, and unstable symmetric). Completing the presented results, the initial geometrical imperfection could change the stability of the initial nonlinear equilibrium path (from unstable to stable) depending on the applied amplitude of the imperfection and its shape. It is possible to observe that the influence of the geometrical imperfection on the nonlinear equilibrium path of the imperfect cylindrical panel is determinant.

Aspect Ratio Effect on Multiple Flow Solutions in a Two-Sided Parallel Motion Lid-Driven Cavity

AbstractA continuation method, accompanied with a linear stability analysis, is employed to investigate the bifurcation diagram of the flow solutions, as well as the multiple flow states in a cavity with different aspect ratios for parallel motion of two facing lids. The Reynolds number proportional to the wall velocity is used as the continuation parameter, and the evolution of the bifurcation diagrams in cases with different aspect ratios is illustrated. The induced flow patterns are highly dependent upon both the aspect ratios and the moving velocity of the walls. Three different types of bifurcation diagrams and their corresponding flow states are classified according to the aspect ratios. One stable symmetric flow state and one stable asymmetric flow state are identified. The stable asymmetric flow state is obtained at a high aspect ratio and a low Reynolds number. Meanwhile, the regions of stable and unstable flows are distinguished according to the different aspect ratios.

Semi-analytical solutions for cubic autocatalytic reaction-diffusion equations; the effect of a precursor chemical

Semi-analytical solutions for a cubic autocatalytic reaction, with linear decay and a precursor chemical, are considered. The model is coupled with diffusion and considered in a one-dimensional reactor. In this model the reactant is supplied by two mechanisms, diffusion via the cell boundaries and decay of an abundant precursor chemical present in the reactor. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for the reaction-diffusion cell. Singularity theory and a local stability analysis are used to determine the regions of parameter space in which the different types of bifurcation diagrams and Hopf bifurcations occur. The effect of the precursor chemical concentration is examined in detail and some novel behaviours are identified. References P. Gray and S. K. Scott. Chemical oscillations and instabilities: nonlinear chemical kinetics. Oxford University Press, London, 1990. P. Gray and S. K. Scott. Autocatalytic reactions in the isothermal continuous, stirred tank reactor: isolas and other forms of multistability. Chem. Engng. Sci., 38:29--43, 1983. doi:10.1016/0009-2509(83)80132-8 P. Gray and S. K. Scott. Autocatalytic reactions in the isothermal continuous, stirred tank reactor: oscillations and instabilities in the system $a + 2b \to 3b$; $b \to c$. Chem. Engng. Sci., 39:1087--1097, 1984. doi:10.1016/0009-2509(84)87017-7 S. R. Kay, S. K. Scott, and P. G. Lignola. The application of singularity theory to isothermal autocatalytic reactions: the influence of uncatalysed reactions. Proc. Roy. Soc. Lond A, 409:433--448, 1987. doi:10.1098/rspa.1986.0015 S. K. Scott. Isolas, mushrooms and oscillations in isothermal, autocatalytic reaction-diffusion equations. Chem. Engng. Sci., 42:307--315, 1987. doi:10.1016/0009-2509(87)85060-1 S. R. Kay and S. K. Scott. Multiple stationary states, sustained oscillations and transient behavior in autocatalytic reaction-diffusion equations. Proc. Roy. Soc. Lond A, 418:345--364, 1988. doi:10.1098/rspa.1988.0088 J. H. Merkin, D. J. Needham, and S. K. Scott. On the creation, growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme. SIAM Journal on Applied Mathematics, 47:1040--1060, 1987. http://www.jstor.org/stable/2101706 T. R. Marchant. Cubic autocatalytic reaction-diffusion equations: semi-analytical solutions. Proc. Roy. Soc. Lond A, 458:873--888, 2002. doi:10.1098/rspa.2001.0899 V. Balakotaiah and D. Luss. Multiplicity features of reacting systems. Chem. Engng. Sci., 38:1709--1721, 1983. doi:10.1016/0009-2509(83)85028-3 J. Guckenheimer and P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, New York, 1983. M. Golubitsky and D. G. Schaeffer. Singularites and groups in bifurcation theory. Springer-Verlag, New York, 1985.

Enhanced boiling heat transfer using radial fins

A numerical bifurcation analysis is carried out in order to determine the solution structure of radial fins subjected to multi-boiling heat transfer mode. One-dimensional conduction is employed throughout the thermal analysis. The fluid heat transfer coefficient is temperature dependent on the three regimes of phase-change of the fluid. Six fin profiles, defined in the text, are considered. Multiplicity structure is obtained to determine different types of bifurcation diagrams, which describe the dependence of a state variable of the system like the temperature or the heat dissipation on the fin design parameters, conduction–convection parameter (CCP) or base temperature difference (ΔT). Specifically, the effects of ΔT, CCP and Biot number are analyzed. The results are presented graphically, showing the significant behavioral features of the heat rejection mechanism.

Analysis of Multiplicity Phenomena in Longitudinal Fins Under Multi-Boiling Conditions

A numerical bifurcation analysis is carried out in order to determine the solution structure of longitudinal fins subject to multi-boiling heat transfer mode. The thermal analysis can no longer be performed independently of the working fluid since the heat transfer coefficient is temperature dependent and includes the nucleate, the transition and the film boiling regimes where the boiling curve is obtained experimentally for a specific fluid. The heat transfer process is modeled using one-dimensional heat conduction with or without heat transfer from the fin tip. Furthermore, five fin profiles are considered: the constant thickness, the trapezoidal, the triangular, the convex parabolic and the parabolic. The multiplicity structure is obtained in order to determine the different types of bifurcation diagrams, which describe the dependence of a state variable of the system (for instance the fin temperature or the heat dissipation) on a design (Conduction-Convection Parameter) or operation parameter (base Temperature Difference). Specifically the effects of the base Temperature Difference, of the Conduction-Convection Parameter and of the Biot number are analyzed and presented in several diagrams since it is important to know the behavioral features of the heat rejection mechanism such as the number of the possible steady states and the influence of a change in one or more operating variables to these states.

The bifurcation diagrams for the Ginzburg–Landau system of superconductivity

In this paper, we provide the different types of bifurcation diagrams for a superconducting cylinder placed in a magnetic field along the direction of the axis of the cylinder. The computation is based on the numerical solutions of the Ginzburg–Landau model by the finite element method. The response of the material depends on the values of the exterior field, the Ginzburg–Landau parameter and the size of the domain. The solution branches in the different regions of the bifurcation diagrams are analyzed and open mathematical problems are mentioned.

A simplified model for analyzing catalytic reactions in short monoliths

We present a new simplified model for analyzing catalytic reactions in short monoliths. The model is described by a two-point boundary value problem in the radial co-ordinate with the reaction terms appearing in the boundary conditions. For the case of linear kinetics, we compare the predictions of the new short monolith (SM) model with the more general two-dimensional model as well as other literature models such as the widely used one-dimensional two-phase model and the two-dimensional convection model (plug flow or parabolic velocity profile but without axial diffusion or conduction). For the case of monotone kinetics, we show that the steady-state behavior of the general model is bounded by the two limiting models, namely the SM model and the convection model (this is analogous to the homogeneous CSTR and PFR models bounding the behavior of the more general axial dispersion model). More importantly, for the case of an exothermic reaction, the SM model retains all the qualitative bifurcation features of the general two-dimensional model. We use the SM model to analyze and classify the steady-state bifurcation behavior of the catalytic monolith for the case of a single exothermic surface reaction and derive explicit analytical expressions for the ignition, extinction and hysteresis loci in terms of the system parameters. We show that there exist four qualitatively different types of bifurcation diagrams of exit temperature (or conversion) versus residence time when the fluid Lewis number is less than unity ( Le f <1). Some of the diagrams contain isolated high-temperature branches and solution profiles on these branches show a local maximum in the surface temperature. We also show that in the practically important mass transfer controlled regime, the predictions of the SM model are close to the more general two-dimensional model. Finally, we discuss the practical implications of the results presented in this work.

Singularity theory approach to ideal binary distillation

This work analyzes the multiplicity of states in binary distillation by rigorous application of the singularity theory (Balakotaiah and Luss, 1984; Golubitsky and Schaeffer, 1985). We consider the mass-reflux flow rate (L w ) as bifurcation parameter and compute codimension-2 varieties that divide the feed composition (z F )-boilup (V) parameter space into regions where different types of bifurcation diagrams exist. Finally, we investigate the effect of physical and design parameters on the location and extent of the multiplicity regions.

On the steady-state behavior of the nonadiabatic autothermal tubular reactor

Singularity theory, local bifurcation analysis and numerical simulations are combined to investigate the steady-state behavior of the nonadiabatic autothermal tubular reactor in which a first-order exothermic reaction occurs. The steady-state model of the reactor consists of three first-order nonlinear ordinary differential equations with unmixed boundary conditions and contains six dimensionless groups. It is found that this simple convection-reaction model can have an arbitrarily large number of steady states. In addition, when the residence time is varied, the different possible bifurcation diagrams that exist vary from all those that occur in the classical cooled CSTR model to those found in the classical nonadiabatic tubular reactor model. We use local bifurcation techniques to make these predictions and confirm them with numerical solution of the full model. Singularity theory is used to classify the different types of bifurcation diagrams for a case where there is a maximum of five steady states.

Multiplicity features of nonadiabatic, autothermal tubular reactors

AbstractSingularity theory is combined with asymptotic analysis to determine the exact uniqueness‐multiplicity boundary and ignition and extinction locus for the non‐adiabatic, autothermal tubular reactor model. It is found that the steady‐state behavior of the nonadiabatic reactor is described by the two limiting cases of adiabatic and strongly cooled models. The adiabatic case has been examined in a previous study. Here, we develop limiting models to describe the strongly cooled asymptotes. We also classify the different types of bifurcation diagrams of conversion vs. residence time using the results of singularity theory with a distinguished parameter. Analytical criteria are developed for predicting the conditions under which autothermal operation is feasible when heat losses are significant.

Multiplicity features of adiabatic autothermal reactors

AbstractSingularity theory, large activation energy asymptotics, and numerical methods are used to present a comprehensive study of the steady‐state multiplicity features of three classical adiabatic autothermal reactor models: tubular reactor with internal heat exchange, tubular reactor with external heat exchange, and the CSTR with external heat exchange. Specifically, we derive the exact uniqueness‐multiplicity boundary, determine typical cross‐sections of the bifurcation set, and classify the different types of bifurcation diagrams of conversion vs. residence time. Asymptotic (limiting) models are used to determine analytical expressions for the uniqueness boundary and the ignition and extinction points. The analytical results are used to present simple, explicit and accurate expressions defining the boundary of the region of autothermal operation in the physical parameter space.

Natural convection effects on thermal ignition in a porous medium. I. Semenov model

A one-dimensional diffusion-convection-reaction model is formulated to account for natural convection effects on thermal ignition in an open system consisting of a porous medium. Various limiting cases of the model are considered. A detailed analysis of the Semenov (lumped) model is presented. Explicit relations are derived for the dependence of the critical Semenov number ( ψ c ) on the Rayleigh number ( R *). It is shown that for R * → 0, ψ c approaches the classical (conduction) limit e -1 , while for R * ≫ 1, the ignition locus is given by the convection asymptote ψ c / R * = 4 e -2 . Inclusion of reactant consumption shows that the conduction asymptote disappears at B = 4 while the convection asymptote ceases to exist for B Ls &lt; 3 + 2√2, where Ls is a modified Lewis number and B is the heat of reaction parameter. It is shown that the Semenov model has five different types of bifurcation diagrams of temperature against Rayleigh number (particle size), (single-valued, inverse S , isola, inverse S + isola and mushroom). This behaviour is found to be qualitatively identical to that of the forced convection problem investigated by Zeldovich &amp; Zysin.

Natural convection effects on thermal ignition in a porous medium. II. Lumped thermal model-I

In this paper, we examine the disappearance of criticality, ignition locus and bifurcation diagrams of temperature against Rayleigh number of a one-dimensional diffusion-convection-reaction model with the assumption of infinite thermal conductivity and zero species diffusivity. The predictions of this model are compared with those of the Semenov model to determine the impact of the species diffusion term. It is shown that for large values of the Rayleigh number (R* ≫ 1), the ignition locus may be expressed in a parametric form B Ls = t /ln t + t /( t - 1) (1 &lt; t ≼ 3.4955), ψ / R * = ( B Ls ) 2 (( t - 1)/ t ) exp{ - B Ls + B Ls / t } ln t , where B is the heat of reaction parameter, ψ is the Semenov number and Ls is a (modified) Lewis number. Criticality is found to disappear at B Ls = 4.194. When these results are compared with those of the Semenov model, it is found that neglecting the species diffusion term gives conservative approximations to the ignition locus, and criticality boundary. It is found that the lumped thermal model-I has five different types of bifurcation diagrams of temperature against Rayleigh number (single­-valued, isola, inverse S , mushroom, inverse S + isola). These diagrams are qualitatively identical to the bifurcation diagrams of temperature against flow rate for the forced convection problem under the assumption of infinite thermal conductivity and zero species diffusivity.

A Steady‐State Multiplicity Analysis of a Kinetic Model for Zinc Electrodeposition: Mapping of Parameter Regions with Different Types of Bifurcation Diagrams

Singularity theory of a distinguished parameter has been applied to a kinetic model of zinc electrodeposition to map out the regions on the [Zn2+]‐pH plane having different types of bifurcation diagrams. The bifurcation diagrams show the dependency of the state variable, the fractional surface coverage of a zinc intermediate , on the electrode potential under steady‐state conditions. The analysis shows that the system can exhibit as many as 23 different types of steady‐state responses depending upon the pH and [Zn2+], many of which occur over small parameter ranges and differ subtly from one another. The power of singularity theory is well illustrated by the fact that only by its application can all of these regions be identified in a straightforward and systematic manner.

Stationary concentration patterns on an isothermal catalytic wire

Steady-state concentration patterns may exist on a catalytic wire on which an isothermal bimolecular Langmuir—Hinshelwood reaction occurs. The structure of these solutions and the intrinsic size of the pattern is determined as a function of the kinetic and transport parameters. Singularity theory with symmetry is used to predict the intricate structure and organiztion of the qualitatively different types of bifurcation diagrams of inhomogeneous solutions next to the singular points. A Liapunov—Schmidt reduction procedure enables a systematic finding of the various singular points and division of the parameter space into regions with qualitatively different bifurcation diagrams, such as one with isolated branches of stable inhomogeneous solutions. Secondary bifurcation can cause formation of branches of mixed-mode solutions.

INPUT-MULTIPLICITY IN LUMPED-PARAMETER SYSTEMS

Input multiplicity is a situation in which different inputs can give the same output. In this work we identify the conditions leading to input multiplicity and present an efficient technique for dividing the parameter space into regions with different number of input states as well as with different types of bifurcation diagrams. It is shown that bounds on the values of the manipulated inputs have an important impact on the number of feasible solutions. The technique is illustrated with examples.

Multiplicity features of porous catalytic pellets—I. Single zeroth-order reaction case

The steady state multiplicity features of porous catalytic pellets, the temperature of which is uniform but different from that of the surrounding, are determined for a zeroth-order reaction. Simple algebraic expressions are derived which divide the global parameter space into regions with different number of solutions as well as with different types of bifurcation diagrams of surface temperature vs ambient Thiele modulus. The model predicts that up to five steady state solutions exist. The region of parameters with five solutions shrinks with increasing external mass transfer resistance and as the geometry of the pellet changes from a slab to a sphere.

On bifurcation and stability of permanent rotations of a heavy solid body whose center of mass is close to the principal plane of its ellipsoid of inertia

All possible qualitatively different types of bifurcation diagrams for bodies whose center of mass is close to the principal planes of their triaxial ellipsoids of inertia are classified, and the most interesting of such diagrams are presented.