Velocity Vanishes Research Articles

Propagation of microwave beams through the stagnation zone in an inhomogeneous plasma

A study is made of the microwave beam evolution due to passing through the stagnation zone, where the group velocity vanishes, thus making the paraxial approximation for the wavefield inappropriate. An extension to the standard beam tracing technique is suggested that allows one to calculate the microwave beam parameters on either branch of its path apart from the stagnation zone, omitting the calculation of the wavefield inside it. Application examples of the extended technique are presented for the case of microwave reflection from the upper hybrid resonance layer in a tokamak plasma.

Theoretical Modelling of Film-Forming Mechanisms Under Transient Conditions: Application to Deceleration and Experimental Validation

This paper proposes an analytical model for line (1D) and point contacts (2D), based upon the Ertel’s hypothesis to predict the evolution of film thickness in steady-state and transient conditions in elastohydrodynamic lubrication. This theoretical approach, applied to a velocity ramp at constant deceleration, is perfectly supported by experimental results in terms of film thickness distribution during the deceleration process and in terms of central film thickness at the vanishing of the entrainment velocity. This work emphasizes the role of the transport effects of the lubricant at the instantaneous entrainment velocity on the time and length scales at which the film thickness disturbances induced by the deceleration process occur until the complete halting of the surfaces.

A multiphase Eulerian approach for modelling the polymer injection into a textured mould

Micro-injection moulding is frequently used for the fabrication of devices in many different fields such as micro-medical technologies, micro-optics and micro-mechanics thanks to its effectiveness for mass production. This work focuses mainly on offering numerical methodology to model the injection into textured moulds. Such approach can predict the different filling scenarios of the micro-details and consequently can provide optimal operating conditions (mould and melt temperatures, flow rate) according to the desired final part quality. In fact, numerical simulations made with industrial software can only describe the injection process at the macroscopic scale where the micro details are not detected. Although the melt temperature and front evolution are tracked throughout time, neither the micro details nor the local heat transfer are properly represented. Since the latter impacts the local viscosity and solidification, simulation of both mould and cavity temperature evolutions is primordial to insure a complete and accurate representation of textured mould filling. The present computations are made at both macro- and micro- scales by using a full Eulerian approach in which the three phases (melt, mould and air) are described by level-set functions. Our numerical approach is checked to the replication of a textured mould for which two dimensional computations are relevant. This replication is properly modelled by taking into account viscosity dependence with temperature in the thermal boundary layer at the melt/mould interface. In particular the expected solidification below a specific temperature is taken into account by either increasing drastically the viscosity or by imposing a vanishing velocity by penalty method. The influence of flow rate and mould temperature are also analysed whereas it is shown that the surface tension can be neglected during injection stage.

The asymptotic behavior of globally smooth solutions to the compressible magnetohydrodynamic equations with Coulomb force

The asymptotic stability of the steady state with the strictly positive constant density and the vanishing velocity and magnetic field to the Cauchy problem of the three-dimensional compressible viscous, heat-conducting magnetohydrodynamic equations with Coulomb force is established under small initial perturbations. Using a general energy method, we obtain the optimal time decay rates of the solution and its higher-order spatial derivatives by introducing the negative Sobolev and Besov spaces. As a corollary, the [Formula: see text]–[Formula: see text] [Formula: see text] type of the decay rates follows without requiring that the [Formula: see text] norm of initial data is small.

The dynamic generalization of the Eshelby inclusion problem and its static limit

The dynamic generalization of the celebrated Eshelby inclusion with transformation strain is the (subsonically) self-similarly expanding ellipsoidal inclusion starting from the zero dimension. The solution of the governing system of partial differential equations was obtained recently by Ni& Markenscoff (In press. J. Mech. Phys. Solids (doi:10.1016/j.jmps.2016.02.025)) on the basis of the Radon transformation, while here an alternative method is presented. In the self-similarly expanding motion, the Eshelby property of constant constrained strain is valid in the interior domain of the expanding ellipsoid where the particle velocity vanishes (lacuna). The dynamic Eshelby tensor is obtained in integral form. From it, the static Eshelby tensor is obtained by a limiting procedure, as the axes' expansion velocities tend to zero and time to infinity, while their product is equal to the length of the static axis. This makes the Eshelby problem the limit of its dynamic generalization.

Global existence of smooth solutions to 2D Chaplygin gases on curved space

This paper investigates the smooth solution of 2D Chaplygin gas equations on an asymptotically flat Riemannian manifold. Under the assumption that the initial data are close to a constant state and the vorticity of the initial velocity vanishes, we prove the global existence of smooth solutions to the Cauchy problem for two‐dimensional flow of Chaplygin gases on curved space. Copyright © 2016 John Wiley & Sons, Ltd.

Travelling and standing envelope solitons in discrete non-linear cyclic structures

Envelope solitons are demonstrated to exist in non-linear discrete structures with cyclic symmetry. The analysis is based on the Non-Linear Schrodinger Equation for the weakly non-linear limit, and on numerical simulation of the fully non-linear equations for larger amplitudes. Envelope solitons exist for parameters in which the wave equation is focussing and they have the form of shape-conserving wave packages propagating roughly with group velocity. For the limit of maximum wave number, where the group velocity vanishes, standing wave packages result and can be linked via a bifurcation to the non-localised non-linear normal modes. Numerical applications are carried out on a simple discrete system with cyclic symmetry which can be seen as a reduced model of a bladed disk as found in turbo-machinery.

Big brake singularity is accommodated as an exotic quintessence field

We describe a big brake singularity in terms of a modified Chaplygin gas equation of state $p=({\ensuremath{\gamma}}_{m}\ensuremath{-}1)\ensuremath{\rho}+\ensuremath{\alpha}{\ensuremath{\gamma}}_{m}{\ensuremath{\rho}}^{\ensuremath{-}n}$, accommodate this late-time event as an exotic quintessence model obtained from an energy-momentum tensor, and focus on the cosmological behavior of the exotic field, its kinetic energy, and the potential energy. At the background level the exotic field does not blow up, whereas its kinetic energy and potential both grow without limit near the future singularity. We evaluate the classical stability of this background solution by examining the scalar perturbations of the metric along with the inclusion of entropy perturbation in the perturbed pressure. Within the Newtonian gauge, the gravitational field approaches a constant near the singularity plus additional regular terms. When the perturbed exotic field is associated with $\ensuremath{\alpha}g0$ the perturbed pressure and contrast density both diverge, whereas the perturbed exotic field and the divergence of the exotic field's velocity go to zero exponentially. When the perturbed exotic field is associated with $\ensuremath{\alpha}l0$ the contrast density always blows up, but the perturbed pressure can remain bounded. In addition, the perturbed exotic field and the divergence of the exotic field's velocity vanish near the big brake singularity. We also briefly look at the behavior of the intrinsic entropy perturbation near the singular event.

Optimal time decay of the compressible micropolar fluids

This paper primarily studies the large-time behavior of solutions to the Cauchy problem on the compressible micropolar fluid system which is a generalization of the classical Navier–Stokes system. The asymptotic stability of the steady state with the strictly positive constant density, the vanishing velocity, and micro-rotational velocity is established under small perturbation in regular Sobolev space. Moreover, it turns out that both the density and the velocity tend time-asymptotically to the corresponding equilibrium state with rate (1+t)−3/4 in L2 and the micro-rotational velocity also tends to the equilibrium state with the faster rate (1+t)−5/4 in L2 norm. The proof is based on the spectrum analysis and time-weighted energy estimate.

Contact constraints and dynamical equations in Lagrangian systems

Constraints work in the simplest way among a variety of methods of modeling the mechanical behaviors on the interface between bodies. The constraints within a persistent point contact usually fall into the following three categories: geometry-dependent constraints due to non-penetration limitation between the two rigid bodies; velocity- or force-dependent constraints due to the vanishing of tangential velocity or sliding friction engaged in tangential interaction. Though those constraints may be intuitively obtained for some simple problems, they are essentially associated with the evolution of location parameters denoting the temporal position of the contact point. Focusing on a multibody system subject to a persistent point contact, we propose a uniform and programmable procedure to formulate the constraint equations. Kinematic analysis along the procedure can clearly expose the dependence of the constraint equations on the location parameters, unveil the reason why the velocity-dependent constraints may become nonholonomic, and exhibit the fulfillment of the Appell–Chetaev’s rule naturally. Furthermore, we employ d’Alembert–Lagrangian principle to yield the dynamical equations of the system via the method of Lagrange’s multipliers. The dynamical equations so obtained are then compared with those derived from a quite different method that characterizes the contact interplay as a pair of contact force vectors. Accordingly, the correlations between the Lagrange multipliers and the components of the real contact force can be clarified. The clarification enables us to correctly embed the force-dependent constraints into the dynamical equations. A classical example of a thin disk contacting a horizontal rough surface is provided to demonstrate the validation of the proposed theory and method.

WHAT CAN THE ALIGNMENTS OF THE VELOCITY MOMENTS TELL US ABOUT THE NATURE OF THE POTENTIAL?

ABSTRACT We prove that, if the time-independent distribution function  ( v ; r ) ?> of a steady-state stellar system is symmetric under velocity inversion such that  ( − v 1 , v 2 , v 3 ; r ) =  ( v 1 , v 2 , v 3 ; r ) ?> and the same is true for v 2 and v 3, where ( v 1 , v 2 , v 3 ) ?> is the velocity component projected onto an orthogonal frame, then the potential within which the system is in equilibrium must be separable (i.e., the Stäckel potential). Furthermore, we find that the Jeans equations imply that, if all mixed second moments of the velocity vanish; that is, ⟨ v i v j ⟩ = 0 ?> for any i ≠ j ?> , in some Stäckel coordinate system and the only non-vanishing fourth moments in the same coordinate are those in the form of ⟨ v i 4 ⟩ ?> or ⟨ v i 2 v j 2 ⟩ ?> , then the potential must be separable in the same coordinates. Finally we also show that all second and fourth velocity moments of tracers with an odd power to the radial component v r being zero is a sufficient condition to guarantee the potential to be of the form Φ = f ( r ) + r − 2 g ( θ , φ ) ?> .

J0410501 マイクロ・クラック発展方程式の新展開とそれによる延性破壊進展過程の評価・検証

The author proposed a new criterion of micro-crack nucleation, which was deduced theoretically by shock wave theory, based on a new concept whereby micro-crack nucleation is caused by a jump in the wave velocity along the intersection line between two different stationary discontinuity bands characterized by the vanishing velocity of an acceleration wave. Successively, in order to consider the crystal orientation dependence of the ductile fracture progress, algorithm of the proposed criterion was incorporated into the finite element crystal plasticity model. In the present paper, the new micro-crack evolution equations are derived based on the previously proposed evolution equation. The new evolution equation deduces the strain jump being infinite within finite time which corresponds occurrence of the displacement jump: macro-crack/void. Simulations of ductile fracture progress of FCC single crystal notched plates with different crystal orientations to the loading direction and also qualitative comparisons of simulated results with experimental observations are performed. Based on the simulations and the new evolution equations it is concluded that the strain gradient dominates the ductile fracture progress from micro-crack nucleation to macro-crack growth and fracture/collapse.

Stopping electrons with radio-frequency pulses in the quantum Hall regime

Most functionalities of modern electronic circuits rely on the possibility to modify the path fol- lowed by the electrons using, e.g. field effect transistors. Here we discuss the interplay between the modification of this path and the quantum dynamics of the electronic flow. Specifically, we study the propagation of charge pulses through the edge states of a two-dimensional electron gas in the quantum Hall regime. By sending radio-frequency (RF) excitations on a top gate capacitively coupled to the electron gas, we manipulate these edge state dynamically. We find that a fast RF change of the gate voltage can stop the propagation of the charge pulse inside the sample. This effect is intimately linked to the vanishing velocity of bulk states in the quantum Hall regime and the peculiar connection between momentum and transverse confinement of Landau levels. Our findings suggest new possibilities for stopping, releasing and switching the trajectory of charge pulses in quantum Hall systems.

Remark on Luo-Hou’s Ansatz for a Self-similar Solution to the 3D Euler Equations

In this note we show that Luo-Hou's ansatz for the self-similar solution to the axisymmetric solution to the 3D Euler equations leads to triviality of the solution under suitable decay condition of the blow-up profile. The equations for the blow-up profile reduces to an over-determined system of partial differential equations, whose only solution with decay is the trivial solution. We also propose a generalization of Luo-Hou's ansatz. Using the vanishing of the normal velocity at the boundary, we show that this generalized self-similar ansatz also leads to a trivial solution. These results show that the self-similar ansatz may be valid either only in a time-dependent region which shrinks to the boundary circle at the self-similar rate, or under different boundary conditions at spatial infinity of the self-similar profile.

Free-vibration acoustic resonance of nonlinear hyperelastic materials

The free-vibration acoustic resonance (FVAR) of nonlinear hyperelastic materials was investigated within the framework of nonlinear elasticity and the calculus of variations. Variational analysis revealed that displacement due to FVAR satisfies the following conditions: time reversal symmetry, time periodicity, periodic vanishing of displacement velocity, and surface traction-free conditions. Numerical analysis of two-dimensional St. Venant–Kirchhoff hyperelastic materials revealed the existence and structure of the color symmetry embedded therein. The nonlinear FVAR modes showed non-monotonic amplitude dependence of resonance frequency. The symmetries of three-dimensional nonlinear FVAR modes for triclinic, monoclinic, and orthorhombic elastic media are shown on the basis of the magnetic point groups. This classification shows excellent agreement with that obtained by the overtone analysis of molecular vibration.

An Origami Approximation to the Cosmic Web

AbstractThe powerful Lagrangian view of structure formation was essentially introduced to cosmology by Zel'dovich. In the current cosmological paradigm, a dark-matter-sheet 3D manifold, inhabiting 6D position-velocity phase space, was flat (with vanishing velocity) at the big bang. Afterward, gravity stretched and bunched the sheet together in different places, forming a cosmic web when projected to the position coordinates.Here, I explain some properties of an origami approximation, in which the sheet does not stretch or contract (an assumption that is false in general), but is allowed to fold. Even without stretching, the sheet can form an idealized cosmic web, with convex polyhedral voids separated by straight walls and filaments, joined by convex polyhedral nodes. The nodes form in ‘polygonal’ or ‘polyhedral’ collapse, somewhat like spherical/ellipsoidal collapse, except incorporating simultaneous filament and wall formation. The origami approximation allows phase-space geometries of nodes, filaments, and walls to be more easily understood, and may aid in understanding spin correlations between nearby galaxies. This contribution explores kinematic origami-approximation models giving velocity fields for the first time.

Rocking instability under ground motion of large statues freely standing atop elastically supported cantilevers

The rocking (overturning) instability of freely standing large statues atop the capital of elastically supported cantilevers subjected to ground motion is thoroughly discussed. The present analytical solution deals with a monolithic slender cantilever carrying a freely standing rigid block (simulating the statue) instead of the actual multi-drum cantilever of ancient columns that exhibit the loss of energy due to impact and sliding between drums. Hence, owing to the omission of the dissipation of energy, the results of this analysis based on a monolithic cantilever are quite conservative. Attention is focused on establishing the minimum-amplitude ground acceleration causing overturning of the rigid block with or without impact. Rocking instability criteria for determining such a minimum amplitude, which leads through the vanishing of the angular velocity to an escaped motion in the phase-plane portrait, are properly established by including the effect of time-dependent initial conditions. Closed-form solutions confirmed via numerical simulation bring to light important findings regarding the individual and combined effect of various parameters on the rocking instability of the rigid block.

Rocking instability of free-standing statues atop slender viscoelastic columns under ground motion

The highly complex rocking response of free-standing statues atop multi-drum columns underground excitation resulting in insuperable difficulties for obtaining reliable solution is reexamined analytically. This is achieved after simulating the columns by monolithic viscoelastic cantilevers having structural damping, based on experiments, equivalent to the energy dissipation due to impact and sliding of multi-drum columns. Subsequently, the conditions of rocking (overturning) instability of free-standing rigid blocks (representing the statues) after their uplift from the top surface of the laterally vibrating cantilevers, are established, including overturning with or without impact. Attention focuses on the minimum amplitude ground acceleration which leads to an escaped motion through the vanishing of the angular velocity and acceleration. Maximization of such a minimum amplitude (implying stabilization) of the rigid block is obtained by seeking the optimum combination of values of the slenderness ratio of the column and its height. Analytically derived results based on linearised analyses are in excellent agreement with those obtained via nonlinear numerical analyses.

INTERACTION OF ELECTROMAGNETIC WAVES WITH A MOVING SLAB: FUNDAMENTAL DYADIC METHOD

This paper concerns with the interaction of electromagnetic waves with a moving slab. Consider a homogeneous isotropic slab moving uniformly in an arbitrary direction surrounded by an isotropic medium (free space). In this paper a new simple and systematic method is proposed for analyzing re∞ection and transmission of obliquely incident electromagnetic waves by a moving slab based on the concept of propagators. In the previous works complex relations were arrived but using this novel method those complexities will not appear thus the method may be extended to more complex structures. In this method, flrst, electric and magnetic flelds are decomposed into their tangential and normal components then each constitutive dyadic is decomposed into a two-dimensional dyadic in transverse plane and two two-dimensional vectors in this plane. Substituting these dyadics into Maxwell's equations gives a flrst order difierential equation which contains fundamental dyadic of the medium. From the solution of this equation, flelds inside the slab may be expressed in terms of flelds at the front surface of the slab and the propagator matrix which is an exponential function of fundamental dyadic. Using this method the up-going and down-going tangential electromagnetic flelds may be obtained at the same time. As a limiting case a slab with vanishing velocity is discussed using this method, and re∞ection and transmission coe-cients of this slab are derived, which ends in Fresnel's equations. At last, several typical examples are provided to exemplify the applicability of the proposed method. Moreover, the results are compared with the method of Lorentz transformation. A good agreement is observed between the results which verifles the validity of the proposed method.

Ion Bernstein waves in a plasma with a kappa velocity distribution

Using a Vlasov-Poisson model, a numerical investigation of the dispersion relation for ion Bernstein waves in a kappa-distributed plasma has been carried out. The dispersion relation is found to depend significantly on the spectral index of the ions, κi, the parameter whose smallness is a measure of the departure from thermal equilibrium of the distribution function. Over all cyclotron harmonics, the typical Bernstein wave curves are shifted to higher wavenumbers (k) if κi is reduced. For waves whose frequency lies above the lower hybrid frequency, ωLH, an increasing excess of superthermal particles (decreasing κi) reduces the frequency, ωpeak, of the characteristic peak at which the group velocity vanishes, while the associated kpeak is increased. As the ratio of ion plasma to cyclotron frequency (ωpi/ωci) is increased, the fall-off of ω at large k is smaller for lower κi and curves are shifted towards larger wavenumbers. In the lower hybrid frequency band and harmonic bands above it, the frequency in a low-κi plasma spans only a part of the intraharmonic space, unlike the Maxwellian case, thus exhibiting considerably less coupling between adjacent bands for low κi. It is suggested that the presence of the ensuing stopbands may be a useful diagnostic for the velocity distribution characteristics. The model is applied to the Earth's plasma sheet boundary layer in which waves propagating perpendicularly to the ambient magnetic field at frequencies between harmonics of the ion cyclotron frequency are frequently observed.