A parametric method of constructing Poincaré mappings in hydrodynamic systems

The plane-parallel motion of the particles of an incompressible medium reduces to an investigation of a Hamilton system. The stream function is a Hamilton function. A Hamilton function, which depends periodically on time and corresponds to the agitation of an incompressible medium in a domain which varies periodically with time, is considered. This agitation of the medium is due to dynamic chaos. The transition to dynamic chaos is described by investigating the location of the Lagrangian particles over time intervals which are multiples of the period (Poincaré points (PP)). The set of PP can be obtained using a Poincaré mapping in the phase flow. The method which has been developed is used to investigate the plane-parallel motion of the particles in an incompressible fluid in a thin layer, bounded by a flat bottom, rectilinear side walls and an upper boundary which is deformed according to a specified periodic law. The motion of the particles is determined from Hamilton's system of equations. The Hamiltonian (the stream function) is found in the thin-layer approximation and depends on two dimensionless parameters: the amplitude of deformation and the tangential velocity in the deforming boundary. The characteristic boundary, which separates the domain of the chaotic motion of the PP from the domain of ordered motion, is determined numerically in the domain of the two parameters. The topological structure of the phase trajectories up to the transition to chaotic conditions is analysed using the Poincaré mapping, found with an accuracy up to the third order with respect to the amplitude. The phase trajectories of the PP, found analytically, turn out to be close to the trajectories of the initial Hamilton system, determined numerically. The mapping found in the domain of the two dimensionless parameters enables one to determine, qualitatively, the boundary of the transition to chaos.     

On metastable deflagration in porous medium combustion

A regime of low velocity deflagration in hydraulically resisted flows such as those occurring in porous beds is discussed. An asymptotic expression for the deflagration velocity is derived. The obtained dependency elucidates the mechanism controlling the gradual enhancement of the propagation velocity prior to the abrupt transition from slow to fast combustion. This enhancement is caused by the drag-induced diffusion of pressure ahead of the advancing front. The time of transition from the slow to fast propagation is estimated.     

Phase transition for dielectric breakdown model

In this paper, the phase transition for DBM when ν varies is investigated by using real-space renormalization-group method. The result demonstrates that there are phase transitions for almost all the value of ν, and we find a new result that the larger ν is, the larger the value of phase transition point q_c is.     

Tempered stable Lévy motion and transient super-diffusion

The space-fractional diffusion equation models anomalous super-diffusion. Its solutions are transition densities of a stable Lévy motion, representing the accumulation of power-law jumps. The tempered stable Lévy motion uses exponential tempering to cool these jumps. A tempered fractional diffusion equation governs the transition densities, which progress from super-diffusive early-time to diffusive late-time behavior. This article provides finite difference and particle tracking methods for solving the tempered fractional diffusion equation with drift. A temporal and spatial second-order Crank-Nicolson method is developed, based on a finite difference formula for tempered fractional derivatives. A new exponential rejection method for simulating tempered Lévy stables is presented to facilitate particle tracking codes.     

Instability of the Phase Transition Front during Water Injection into High-Temperature Rock

Water injection into a high-temperature geothermal reservoir saturated with superheated vapor is investigated. A solution to the one-dimensional problem in the form of a traveling wave is found. It is shown that there exist two types of solutions which correspond to the boiling of water and the condensation of vapor. In the condensation regime with high initial pressure, vapor ahead of the phase transition front is shown to be in a supercooled state. For moderate or law initial pressure, solutions with condensation and boiling are thermodynamically consistent. Linear stability of the phase transition surface between the water and vapor regions is analyzed. It is shown that the phase transition front moving at constant velocity is always unstable.     

Numerical exploration of new features on three-dimensional transition of a cylinder wake

The three-dimensional transition of the wake flow behind a circular cylinder is studied in detail by direct numerical simulations using 3D incompressible N-S equations for Reynolds number ranging from 200 to 300. New features and vortex dynamics of the 3D transition of the wake are found and investigated. At Re = 200, the flow pattern is characterized by mode A instability. However, the spanwise characteristic length of the cylinder determines the transition features. Particularly for the specific spanwise charac-     

One type of waves in media with loosely bonded interface

A new type of waves that propagate in media having boundaries with slip is described. The group velocity of these waves is greater than the P — wave velocity and the less than the S-wave velocity, and the phase velocity equals the velocity of the longitudinal wave. The frequency of -waves is constant and is determined by cosntructive interference conditions. The waves under consideration are similar to Crary waves recorded in floating ice.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN USSR, Vol. 173, pp. 113–122, 1988.     

Multiscale Modeling of Nanocrystalline Materials: A Variational Approach

This paper presents a variational multi-scale constitutive model in the finite deformation regime capable of capturing the mechanical behavior of nanocrystalline (nc) fcc metals. The nc-material is modeled as a two-phase material consisting of a grain interior (GI) phase and a grain boundary (GB) phase. A rate-independent isotropic porous plasticity model is employed to describe the GB phase, whereas a crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the GI phase. Assuming the rule of mixtures, the overall behavior of a given grain is obtained via volume averaging. The scale transition from a single grain to a polycrystal is achieved by Taylor-type homogenization. It is shown that the proposed model is able to capture the inverse Hall-Petch effect. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical exploration of new features on three-dimensional transition of a cylinder wake

The three-dimensional transition of the wake flow behind a circular cylinder is studied in detail by direct numerical simulations using 3D incompressible N-S equations for Reynolds number ranging from 200 to 300. New features and vortex dynamics of the 3D transition of the wake are found and investigated. At Re = 200, the flow pattern is characterized by mode A instability. However, the spanwise characteristic length of the cylinder determines the transition features. Particularly for the specific spanwise characteristic length linear stable mode may dominate the wake in place of mode A and determine the spanwise phase difference of the primary vortices shedding. At Re = 250 and 300 it is found that the streamwise vortices evolve into a new type of mode’“dual vortex pair mode” downstream. The streamwise vortex structures switch among mode A, mode B and dual vortex pair mode from near wake to downstream wake. At Re = 250, an independent low frequency f _m in addition to the vortex shedding frequency f _s is identified. Frequency coupling between f _m and f _s occurs. These result in the irregularity of the temporal signals and become a key feature in the transition of the wake. Based on the formation analysis of the streamwise vorticity in the vicinity of cylinder, it is suggested that mode A is caused by the emergence of the spanwise velocity due to three dimensionality of the incoming flow past the cylinder. Energy distribution on various wave numbers and the frequency variation in the wake are also described.     

Full velocity difference and acceleration model for a car-following theory

In order to describe the car-following behavior more actually in real traffic, a full velocity difference and acceleration model (for short, FVDAM) is proposed by synthetically taking into account headway, velocity difference and acceleration of the leading car on the basis of full velocity difference model. The analytical method and numerical simulation results show that the proposed model can describe the phase transition of traffic flow and estimate the evolution of traffic congestion, that incorporating the acceleration of the leading car into car-following model can stabilize traffic flow, suppress the traffic jam and increase capacity, and that the following car in FVDAM can accelerate more quickly than in FVDM.     

A thermodynamically consistent Ginzburg–Landau model for superfluid transition in liquid helium

In this paper, we propose a thermodynamically consistent model for superfluid-normal phase transition in liquid helium, accounting for variations of temperature and density. The phase transition is described by means of an order parameter, according to the Ginzburg–Landau theory, emphasizing the analogies between superfluidity and superconductivity. The normal component of the velocity is assumed to be compressible, and the usual phase diagram of liquid helium is recovered. Moreover, the continuity equation leads to a dependence between density and temperature in agreement with the experimental data.     

A mixed finite element method for the Stokes equations

A new mixed finite element for the Stokes equations is considered. This new finite element is based on a mixed formulation of the Stokes problem in which the gradient of the velocity is introduced and the velocity is approximated by the Raviart-Thomas element [1]. Optimal error estimates are derived. The number of degrees of freedom, for this element, is the lowest possible, and the local conservation of the mass is assured. A hybrid version of the mixed method is also considered. Finally, some numerical results for the incompressible Navier-Stokes equations are presented. © 1994 John Wiley & Sons, Inc.     

Phase transition thresholds for some Friedman-style independence results

We classify the phase transition thresholds from provability to unprovability for certain Friedman-style miniaturizations of Kruskal's Theorem and Higman's Lemma. In addition we prove a new and unexpected phase transition result for ε₀. Motivated by renormalization and universality issues from statistical physics we finally state a universality hypothesis. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Perturbation theory with convergent series for calculating physical quantities specified by finitely many terms of a divergent series in traditional perturbation theory

A new high-accuracy method is suggested for calculating physical quantities for which only finitely many terms of the divergent series in a traditional perturbation theory are known. The method is based on approximating the desired quantity with the sum of finitely many terms of an absolutely convergent series. As an example, the β-function in the ϕ ₄ ⁴ model and the critical exponent α characterizing the behavior of the He⁴ heat capacity near the phase transition point are calculated. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 452–461, June, 2000.     

Quantum Phase Transition in an Interacting Fermionic Chain

We rigorously analyze the quantum phase transition between a metallic and an insulating phase in (non-solvable) interacting spin chains or one-dimensional fermionic systems. In particular, we prove the persistence of Luttinger liquid behavior in the presence of an interaction even arbitrarily close to the critical point, where the Fermi velocity vanishes and the two Fermi points coalesce. The analysis is based on two different multiscale analysis; the analysis of the first regime provides gain factors which compensate the small divisors due to the vanishing Fermi velocity.     

Zeroth-order phase transitions and Zipf law quantization

From the standpoint of thermodynamic averaging of fission microprocesses, we investigate the origin of radioactive release in an NPP after an accident or after resource depletion. The genesis of the NPP release is interpreted as a new thermodynamic phenomenon, a zeroth-order phase transition. This problem setting results in a problem in probabilistic number theory. We prove the corresponding theorem leading to quantization of the Zipf law for the frequency of a zeroth-order phase transition with different values of the jump of the Gibbs thermodynamic potential. We introduce the notion of hole dimension. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 1, pp. 118–142, January, 2007.     

Singlet-triplet fermion coupling theory

At zero temperature, the superfluid Fermi liquid self-consistency equations that describe superposition of states with singlet and triplet fermion coupling are obtained and analyzed. The conditions for creating such states are found. Equations for the critical temperature of the phase transition from a one-gap (singlet or triplet) superfluid state to a singlet-triplet superfluid state and the temperature dependence of the order parameters near the transition temperature are derived. A mechanism for creating new states is proposed, and the thermodynamic stability of these states is studied. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 115, No. 3, pp. 459–476, June, 1998.     

Optimal Trajectories for Earth-to-Mars Flight

This paper deals with the optimal transfer of a spacecraft from a low Earth orbit (LEO) to a low Mars orbit (LMO). The transfer problem is formulated via a restricted four-body model in that the spacecraft is considered subject to the gravitational fields of Earth, Mars, and Sun along the entire trajectory. This is done to achieve increased accuracy with respect to the method of patched conics.The optimal transfer problem is solved via the sequential gradient-restoration algorithm employed in conjunction with a variable-stepsize integration technique to overcome numerical difficulties due to large changes in the gravitational field near Earth and near Mars. The optimization criterion is the total characteristic velocity, namely, the sum of the velocity impulses at LEO and LMO. The major parameters are four: velocity impulse at launch, spacecraft vs. Earth phase angle at launch, planetary Mars/Earth phase angle difference at launch, and transfer time. These parameters must be determined so that V is minimized subject to tangential departure from circular velocity at LEO and tangential arrival to circular velocity at LMO.For given LEO and LMO radii, a departure window can be generated by changing the planetary Mars/Earth phase angle difference at launch, hence changing the departure date, and then reoptimizing the transfer. This results in a one-parameter family of suboptimal transfers, characterized by large variations of the spacecraft vs. Earth phase angle at launch, but relatively small variations in transfer time and total characteristic velocity.For given LEO radius, an arrival window can be generated by changing the LMO radius and then recomputing the optimal transfer. This leads to a one-parameter family of optimal transfers, characterized by small variations of launch conditions, transfer time, and total characteristic velocity, a result which has important guidance implications. Among the members of the above one-parameter family, there is an optimum–optimorum trajectory with the smallest characteristic velocity. This occurs when the radius of the Mars orbit is such that the associated period is slightly less than one-half Mars day.     

A Ginzburg–Landau model for the phase transition in Helium II

The paper deals with the second-order phase transition in Helium II by a Ginzburg–Landau model, in which any particle has simultaneously a normal and superfluid velocity. This pattern is able to describe the classical effects of Helium II as the phase diagram, the vortices, the second sound and the thermomechanical effect. Finally, the vorticities and turbulence are described by an extension of the model in which the material time derivative is used.