Bifurcations of heterodimensional cycles with one orbit flip and one inclination flip

In this paper, the bifurcations of heterodimensional cycles are investigated by setting up a suitable local coordinate system in a four-dimensional system. Under some ungeneric conditions—one orbit flip and one inclination flip—the persistence of heterodimensional cycles, the existence of homoclinic orbit and a family of periodic orbits, the coexistence of heterodimensional loop and periodic orbit are obtained. Also, the relevant bifurcation surfaces and their existing regions are given.     

Hyperbolic phenomena in a strongly degenerate parabolic equation

We consider the equation u _t =((u) (u _x )) _x , where >0 and where is a strictly increasing function with lim _s = <. We solve the associated Cauchy problem for an increasing initial function, and discuss to what extent the solution behaves qualitatively like solutions of the first-order conservation law u _t = ((u)) _x . Equations of this type arise, for example, in the theory of phase transitions where the corresponding free-energy functional has a linear growth rate with respect to the gradient.     

Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation

Communicated by M. Gurtin     

Global Weak Solutions to a Generic Two-Fluid Model

This paper deals with mathematical properties of a generic two-fluid flow model commonly used in industrial applications. More precisely, we address the question of whether available mathematical results in the case of a single-fluid governed by the compressible barotropic Navier–Stokes equations may be extended to such a two-phase model. We focus on existence of global weak solutions, linear theory and determination of eigenvalues and invariant regions.     

Stabilization of parabolic systems via a degenerate nonnegative feedback

A degenerate nonnegative linear operatorB is added to an unbounded self-adjoint linear operatorL in a Hilbert space. The perturbed operator is of the formL + kB, k being a positive parameter. A sufficient condition is sought so that we can choose the minimum eigenvalue ofL + kB arbitrarily large. It is also studied from the numerical analytic viewpoint how the minimum eigenvalue behaves ask increases in relation to an approximation ofB. Modifications of the classical theory of spectrum perturvations are required, since a new type of information is relevant concerning bounds on the multiplicative constant in the familiar formulations of exponential decay. The result is then applied to a new stabilization problem for a class of linear and/or semilinear parabolic differential equations so that their evolutions can be stabilized and at the same time not deviate very much from the initial states for everyt>0, wherekB is considered as a feedback operator.     

Impact on a degenerate torus floating in a fluid of infinite depth

The vertical impact problem is considered for a degenerate torus, a solid obtained by rotation of a circle about its tangent, half-immersed in an ideal fluid of infinite depth. The impact is assumed to be such that the flow past the torus is attached (separationless impact). After odd continuation of the velocity potential across the free surface, the problem is divided into two subproblems. The first corresponds to the translational motion of the torus in an unbounded fluid along the axis of symmetry, and the second to the rotation of the torus. For a torus of general form the problem of translational motion along the axis of symmetry was considered in [1], in which a solution was constructed in toroidal coordinates in the form of a series in terms of Legendre functions, using the method of separation of variables, and an expression for the kinetic energy was found. In the present paper, the velocity potential is found in quadratures for the translational motion of the degenerate torus. For the rotational flow the problem is reduced to the solution of a Fredholm integral equation of the second kind. Degenerate bipolar coordinates are used. The numerical values of the apparent mass and the apparent moment of inertia are found. The separationless impact condition is derived.Rostov-on-Don. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 161–165, September–October, 1995.     

Bifurcation and universal unfolding for a rotating shaft with unsymmetrical stiffness

The 1/2 subharmonic resonance bifurcation and universal unfolding are studied for a rotating shaft with unsymmetrical stiffness. The bifurcation behavior of the response amplitude with respect to the detuning parameter was studied for this class of problems by Xiao et al. Obviously, it is highly important to research the bifurcation behavior of the response amplitude with respect to the unsymmetry of stiffness for this problem. Here, by means of the singularity theory, the bifurcation and universal unfolding of amplitude with respect to the unsymmetrical stiffness parameter are discussed. The results indicate that it is a high codimensional bifurcation problem with codimension 5, and the universal unfolding is given. From the mechanical background, we study four forms of two parameter unfoldings contained in the universal unfolding. The transition sets in the parameter plane and the bifurcation diagrams are plotted. The results obtained in this paper show rich bifurcation phenomena and provide some guidance for the analysis and design of dynamic buckling experiments of this class of system, especially, for the choice of system parameters. The project supported by the National Natural Science Foundation of China (19990510), the National Key Basic Research Special Foundation (G1998020316) and Liuhui Center for Applied Mathematics, Nankai University and Tianjin University     

Numerical Investigation of Active Flow Control Around a Generic Truck A-Pillar

Large Eddy Simulations (LES) are conducted to study the actuated flow field around a bluff body. The model is a simplified section of a truck. The aim of the work is to model the separation of the flow acting at the front rounded corners, the so called A-pillars, and to minimize the separation of the flow by means of Zero Net Mass Flux synthetic jets. LES data show the interaction of the flow main structures, the separation mechanism and the effects of the actuation on the flow field. The flow is post processed using modal and frequency decompositions. Relevant results in terms of drag reduction were observed for the actuated flow. The principle flow mechanisms are discussed and an optimal actuation frequency, in terms of induced fluctuations and drag reduction, is identified.     

Generic properties of rheological flow curves

The interpretation and comparison of viscosity data for fluids in planar Couette flow are discussed using a generic rheological function, familiar from studies of polymeric liquids. The particular data are generated by non-equilibrium molecular dynamics (NEMD) simulations of model fluids, however the analysis has wider validity. The mathematical properties of the established flow curves are investigated and, inter alia, a consistent definition of the critical shear rate is established. It is also shown that the use of inappropriate, but commonly used, independent variables leads to curves whose qualitative behaviour is extremely sensitive to the physical parameters. Such a range of qualitative behaviour can lead to erroneous quantitative conclusions when interpreting physical (or model) data. A particular example is the assumption of a linear dependence of the viscosity on the square-root of the shear rate, as often used in the NEMD literature. The developed framework also enables the systematic comparison of simulations of the same molecular fluid performed at different state points. This is illustrated for the case of three recently published and apparently conflicting NEMD results for n-butane.     

Boundary-value problem for a degenerate system of parabolic equations in boundary-layer theory

A problem is considered for the system describing gas flows with plate boundary layer separation in Mises variables in boundary-layer theory. The existence of generalized solutions of the problem is proved. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 36–41, July–August, 2008.     

Study of Flow and Mixing in a Generic GT Combustor Using LES

While the basic capability of large eddy simulation (LES) has been amply demonstrated on a number of relatively simple academic configurations, there is still a lack of works applying LES to practical systems also performing detailed quantitative comparisons based on experimental data. In this paper, we tend to approach the simulation of real gas turbine combustor step by step as we first present results of the isothermal LES of a generic gas turbine combustor rig. The available detailed measurements are used to thoroughly validate the LES results. Beyond a pure validation, the LES is used to analyze the influence of the precessing vortex core present in the studied configuration on the mixing of fuel and oxidizer, and possible implications for reacting conditions are discussed.     

Generic non-linear modelling of a bi-material composite beam with partial shear interaction

Composite members composed of two materials joined by shear connection find widespread use in engineering infrastructure, in both traditional practice and innovative applications. Studies in the literature dating back nearly 60 years have elucidated the mechanics of the behaviour of these composite structural members in which the solution for the slip at the interface between the materials was determined by solving a linear differential equation. However, these solutions are based on a linear formulation of the strain-displacement relationship, and in some applications this relationship must be represented in non-linear form, so that the second order effects in the member can be quantified correctly. This paper presents such a study for a composite member with two materials, being typical of a steel-concrete composite beam in structural engineering. It quantifies the restraint of the member ends by longitudinal and rotational elastic springs, so that the axial tension developed is a function of the transverse loading, material properties, cross-sectional properties and the restraint stiffness. The problem is treated using minimisation of the total potential stored in the two members, the elastic shear connection at their interface, the restraints at the ends and the work done by the transverse forces, for which the differential equations for the deformations can be determined from routine variational calculus. The non-linear equation of equilibrium relating the external loading to the internal actions is stated in closed form by invoking the static and kinematic boundary conditions for the member. The solution is compared with closed form treatments derived elsewhere, and a representative member is analysed so that the influences of the non-linearity, end restraint stiffness and degree of partial shear interaction on its behaviour can be examined.     

On the decomposition of a degenerate singularly perturbed linear system of differential equations

We study the problem of decomposition of degenerate singularly perturbed systems of differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 3, pp. 401–415, July–September, 2006.     

Blowup properties for a class of nonlinear degenerate diffusion equation with nonlocal source

A nonlinear degenerate parabolic equation with nonlocal source was considered. It was shown that under certain assumptions the solution of the equation blows up in finite time and the set of blowup points is the whole region. The integral method is used to investigate the blowup properties of the solution.     

A degenerate Hopf–saddle-node bifurcation analysis in a family of electronic circuits

This paper presents a study of a three-parameter unfolding of a degenerate case in the Hopf--saddle-node singularity. This analysis shows that this nonlinear degeneracy is a source of interesting bifurcations of periodic orbits as well as global bifurcations of equilibria. The results achieved are applied to the study of a simple autonomous electronic circuit, which has just only one nonlinearity. The numerical results include the analysis of interesting resonance behaviors.     

Multiscale mechanics of macromolecular materials with unfolding domains

We propose a general multiscale approach for the mechanical behavior of three-dimensional networks of macromolecules undergoing strain-induced unfolding. Starting from a (statistically based) energetic analysis of the macromolecule unfolding strategy, we obtain a three-dimensional continuum model with variable natural configuration and an energy function analytically deduced from the microscale material parameters. The comparison with the experiments shows the ability of the model to describe the complex behavior, with residual stretches and unfolding effects, observed in different biological materials.     

Imperfection sensitivity of degenerate hilltop branching points

Imperfection sensitivity is investigated for a degenerate hilltop branching point, where a degenerate bifurcation point exists at a limit point. A degenerate hilltop branching point is important as it is a byproduct of optimization of shallow shell structures under non-linear buckling constraints. A systematic procedure is presented for asymptotic sensitivity analysis based on enumeration of vertices of a convex region defined by linear inequality constraints on the orders of the variables. The effectiveness of the proposed method is demonstrated by sensitivity analysis of degenerate hilltop branching points, considering minor and major imperfections, corresponding to an unstable-symmetric or asymmetric bifurcation point at the limit point. It is found that a hilltop branching point can be imperfection sensitive.