Showing papers on "Equations of motion published in 1992"

Drop Formation in a One-Dimensional Approximation of the Navier-Stokes Equation

Abstract: We consider the viscous motion of a thin axisymmetric column of fluid with a free surface. A one-dimensional equation of motion for the velocity and the radius is derived from the Navier–Strokes equation. We compare our results with recent experiments on the breakup of a liquid jet and on the bifurcation of a drop suspended from an orifice. The equations form singularities as the fluid neck is pinching off. The nature of the singularities is investigated in detail.

Direct simulation of particle dispersion in a decaying isotropic turbulence

Abstract: Results of a numerical investigation of the dispersion of solid particles in decaying isotropic turbulence are presented. The 3D time-dependent velocity field of a homogeneous nonstationary turbulence is computed using the method of direct numerical simulation (DNS). The dispersion characteristics of three different solid particles (corn, copper, and glass) injected in the flow are obtained by integrating the complete equation of particle motion along the instantaneous trajectories of 22-cubed particles for each particle type, and then by performing ensemble averaging. Good agreement was achieved between the present DNS results and the measured time development of the mean-square displacement of the particles. Questions of how and why the dispersion statistics of a solid particle differ from those of its corresponding fluid point and surrounding fluid and what influences inertia and gravity have on these statistics are also discussed.

The Euler Equations of Motion with Hydrostatic Pressure as an Independent Variable

Abstract: A novel form of the Euler equations is developed through the use of a different vertical coordinate system. It is shown that the use of hydrostatic pressure as an independent variable has the advantage that the Euler equations then take a form that parallels very closely the form of the hydrostatic equations cast in isobaric coordinates. This similarity holds even when topography is incorporated through a further transformation into terrain-following coordinates. This leads us to suggest that hydrostatic-pressure coordinates could be used advantageously in nonhydrostatic atmospheric models based on the fully compressible equations.

A New Perspective on Constrained Motion

Abstract: The explicit general equations of motion for constrained discrete dynamical systems are obtained. These new equations lead to a simple and new fundamental view of lagrangian mechanics.

Direct simulation of fluid particle motions

Abstract: Continuum models of two-phase flows of solids and liquids use constitutive assumptions to close the equations. A more fundamental approach is a “molecular dynamic” simulation of flowing “big” particles based on reliable macroscopic equations for both solid and liquid. We developed a package that simulates the unsteady two-dimensional solid-liquid two-phase flows using the Navier-Stokes equations for the liquid and Newton's equations of motion for the solid particles. The Navier-Stokes equations are solved using a finite-element formulation and Newton's equations of motion are solved using an explicit-implicit scheme. We show that the simplest fully explicit scheme to update the particle motion using Newton's equations is unstable. To correct this instability we propose and implement and Explicit-Implicit Scheme in which, at each time step, the positions of the particles are updated explicitly, the computational domain is remeshed, the solution at the previous time is mapped onto the new mesh, and finally the nonlinear Navier-Stokes equation and the implicitly discretized Newton's equations for particle velocities are solved on the new mesh iteratively. The numerical simulation reveals the effect of vortex shedding on the motion of the cylinders and reproduces the drafting, kissing, and tumbling scenario which is the dominant rearrangement mechanism in two-phase flow of solids and liquids in beds of spheres which are constrained to move in only two dimensions.

Two-Dimensional Rigid-Body Collisions With Friction

Abstract: This paper derives solutions for frictional planar rigid body collisions, using Routh's impact process diagrams, for both Newtonian and Poisson restitution.

Overview No. 98 I—Geometric models of crystal growth

Abstract: Recent theoretical advances in the mathematical treatment of geometric interface motion make more tractable the theory of a wide variety of materials science problems where the interface velocity is not controlled by long-range-diffusion. Among the interface motion problems that can be modelled as geometric are certain types of phase changes, crystal growth, domain growth, grain growth, ion beam and chemical etching, and coherency stress driven interface migration. We provide an introduction to nine mathematical methods for solving such problems, give the limits of applicability of the methods, and discuss the relations among them theoretically and their uses in computation. Comparisons of some of them are made by displaying how the same physical problems are treated in the various applicable methods.

Memory effects in the relaxation of quantum open systems

Abstract: A close examination of the validity of the Markovian approximation in the context of relaxation theory is presented. In particular, we examine the question of positivity of various approximations to the reduced dynamics of an open system in interaction with a heat reservoir. It is shown that the Markovian equations of motion obtained in the weak coupling limit (Redfield equations) are a consistent approximation to the actual reduced dynamics only if supplemented by a slippage in the initial conditions. This slippage captures the effects of the non‐Markovian evolution that takes place in a short transient time, of the order of the relaxation time of the isolated bath.

General-relativistic celestial mechanics. II. Translational equations of motion.

Abstract: The translational laws of motion for gravitationally interacting systems of {ital N} arbitrarily composed and shaped, weakly self-gravitating, rotating, deformable bodies are obtained at the first post-Newtonian approximation of general relativity. The derivation uses our recently introduced multi-reference-system method and obtains the translational laws of motion by writing that, in the local center-of-mass frame of each body, relativistic inertial effects combine with post-Newtonian self- and externally generated gravitational forces to produce a global equilibrium (relativistic generalization of d'Alembert's principle). Within the first post-Newtonian approximation (i.e., neglecting terms of order ({ital v}/{ital c}){sup 4} in the equations of motion), our work is the first to obtain complete and explicit results, in the form of infinite series, for the laws of motion of arbitrarily composed and shaped bodies. We first obtain the laws of motion of each body as an infinite series exhibiting the coupling of all the (Blanchet-Damour) post-Newtonian multipole moments of this body to the post-Newtonian tidal moments (recently defined by us) felt by this body. We then give the explicit expression of these tidal moments in terms of post-Newtonian multipole moments of the other bodies.

Weakly nonlinear gravitational instability for arbitrary Omega

Abstract: The weakly nonlinear evolution of particle trajectories in Friedmann-Lemaitre models with zero cosmological constant is investigated. The matter is assumed to be a nonrelativistic pressureless fluid. A perturbative expansion in Lagrangian coordinates is used and analytic expressions for the second-order solutions with arbitary density parameter Ω are derived. This perturbative expansion is valid provided the gradients of the displacement field are small, a much weaker condition than the usual Eulerian requirement of smallness of the density perturbations

Vortex motion and the Hall effect in type-II superconductors: A time-dependent Ginzburg-Landau theory approach

Abstract: Vortex motion in type-II superconductors is studied starting from a variant of the time-dependent Ginzburg-Landau equations, in which the order-parameter relaxation time is taken to be complex. Using a method due to Gor'kov and Kopnin, we derive an equation of motion for a single vortex (B\ensuremath{\ll}${\mathit{H}}_{\mathit{c}2}$) in the presence of an applied transport current. The imaginary part of the relaxation time and the normal-state Hall effect both break ``particle-hole symmetry,'' and produce a component of the vortex velocity parallel to the transport current, and consequently a Hall field due to the vortex motion. Various models for the relaxation time are considered, allowing for a comparison to some phenomenological models of vortex motion in superconductors, such as the Bardeen-Stephen and Nozi\`eres-Vinen models, as well as to models of vortex motion in neutral superfluids. In addition, the transport energy, Nernst effect, and thermopower are calculated for a single vortex. Vortex bending and fluctuations can also be included within this description, resulting in a Langevin-equation description of the vortex motion. The Langevin equation is used to discuss the propagation of helicon waves and the diffusional motion of a vortex line. The results are discussed in light of the rather puzzling sign change of the Hall effect which has been observed in the mixed state of the high-temperature superconductors.

Deriving the equations of motion for porous isotropic media

Abstract: The equations of motion and stress/strain relations for the linear dynamics of a two‐phase, fluid/solid, isotropic, porous material have been derived by a direct volume averaging of the equations of motion and stress/strain relations known to apply in each phase. The equations thus obtained are shown to be consistent with Biot’s equations of motion and stress/strain relations; however, the effective fluid density in the equation of relative flow has an unambiguous definition in terms of the tractions acting on the pore walls. The stress/strain relations of the theory correspond to ‘‘quasistatic’’ stressing (i.e., inertial effects are ignored). It is demonstrated that using such quasistatic stress/strain relations in the equations of motion is justified whenever the wavelengths are greater than a length characteristic of the averaging volume size.

Intense Dynamic Loading Of Condensed Matter

Abstract: MECHANICS OF CONTINUOUS MEDIA Stresses and Strains in a Solid Body Equations of One-Dimensional Motion of Compressible Media, Shock Waves Interpretation of Detection Data on Compression and Rarefaction Waves EXPERIMENTAL TECHNIQUES OF THE PHYSICS OF HIGH DYNAMIC PRESSURE Explosive Generation of Dynamic Pressure Ballistic Experiments with a Shock Wave Promising Sources of High Dynamic Pressure Discrete Measurement of Wave and Mass Velocities Pressure Profiles Recorded with Manganin Sensors Measurement of the Velocity of Matter ELASTOPLASTIC PROPERTIES OF SHOCK-LOADED SOLIDS Basic Relationships and Models Moduli of Elasticity and the Velocity of Sound in Shock Compressed Metals Dynamic Yield Point Structure of Plastic Compression Waves Compression and Rarefaction Waves in Shock-Compressed Metals Compaction of Porous Media in Shock Waves Catastrophic Thermoplastic Shear under Dynamic Deformation. Impact Compression of Brittle Materials Methods of Studying High Dynamic Deformations Microscopic Models of Strain Dynamics EVOLUTION OF LOAD PULSES IN MEDIA WITH POLYMORPHIC PHASE TRANSITIONS The Structure of Compression and Release Waves in Iron The Graphite to Diamond Transition under Shock Compression Properties of the Phase Transition Induced by Shock FRACTURE UNDER PULSED LOADING. SPALLING STRENGTH Dynamics of Wave Interactions During Spalling Spalling Strength of Metals Work of Spalling Fracture Determination if Tensile Stress Behind the Spall Plane Resistance of Polymers, Brittle Materials, and Liquids to Spalling Fracture Mechanism and Kinetics of Dynamic Fracture of Metals SHOCK AND DETONATION WAVES IN SOLID EXPLOSIVES Basic Concepts and Models Kinetics of Dissociation of Explosives Deduced from Analyses of Evolution of Shock Waves Semiempirical Macrokinetic Equations of Solid Explosives MODEL EQUATIONS OF STATE FOR A WIDE RANGE OF PRESSURE AND TEMPERATURE General Analysis of Phase Diagram Quasi-Harmonic Model of a Solid Equation of State for Condensed Phase at High Temperatures Evaporation Effects and Generalized Equations of State Tabulated and Approximate Equations of States GENERALIZED EQUATIONS OF STATE FOR METALS Cold Compression Curve Electron Component Thermal Excitation of the Crystal Lattice Liquid Phase Procedure for Constructing Semiempirical Equations of State for Metals THERMODYNAMIC PROPERTIES OF METALS Aluminum Copper Lead Lithium FURTHER BRIEF NOTES ON CONTINUUM MECHANICS Equation of Motion Shock Waves Characteristic Form of Gas-Dynamic Equations. Simple Waves The Structure of a Shock Wave Decay of a Random Discontinuity in Hydrodynamics Equation of Motion for Porous Condensed Media DYNAMICS OF CONDENSED MEDIA WITH ALLOWANCE FOR THEIR STRENGTH Equations of Motion. Divergence Form Characteristics Form on Equations of Motion Simple Waves Shock Waves in a Hyperelastic Medium Decay of an Arbitrary Discontinuity Equations of Motion for a Hyperelastic Medium in an Arbitrary Curvilinear Set of Coordinates Equations of Motion of a Maxwell Viscoelastic Medium Nonlinear Waves in a Viscoelastic Medium BRIEF REVIEW OF COMPUTATIONAL TECHNIQUES FOR THE DYNAMICS OF CONDENSED MEDIA Method of Particles in Cells Method of Large Particles Godunov's Method Lagrangian Methods NUMERICAL MODELING OF CONDENSED MEDIA UNDER INTENSIVE PULSED LOADING Tabulated Form of the Equation of State High-Speed Collisions Irregular Collisions of Strong Shock Waves in Metals Numerical Modeling of the Effect of Relativistic High-Current and High-Energy Ion Beams on Metal Targets Effect of an Explosion on an Iron Plate Impactor Penetration of an Obstacle of Finite Thickness Impact of a Micrometerorite on a Spacecraft Shield References Index

On the continuum equations for dispersed particles in nonuniform flows

Abstract: The continuum equations of a dispersed phase of solid, noncolliding particles in a nonuniform turbulent gas flow are derived from a kinetic equation for the transport of the average phase space density 〈W(v,x,t)〉 for particles with velocity v and position x at time t. The crucial feature of this equation is the form given for the phase space diffusion current j representing the net acceleration of a particle from interactions with turbulent eddies. This is based on Kraichnan’s Lagrangian history direct interaction approximation which gives j=−[(∂/∂v)⋅μ+(∂/∂x)⋅λ+γ]〈W(v,x,t)〉, where μ, λ, and γ are dispersion tensors dependent upon displacements in the velocity and position of a particle about v,x in times of order of the time scale of the fluctuating aerodynamic driving force. Most important these tensors are affected by spatial variations in the mean carrier flow velocity and external force as well as inhomogeneities in the carrier phase turbulence; γ is zero for homogeneous turbulence. This form for j is invariant under a random Galilean transformation. The dispersed phase momentum and energy equations deduced from the kinetic equation contain local gradient forms for the net fluctuating interphase force (per unit volume) and its rate of working. The former contains in general an asymmetric stress component (which adds to the particle Reynolds stress) as well as a body force dependent upon inhomogeneities in the turbulence. Conditions are examined under which the mass and momentum equations reduce to a convection–diffusion equation. The convection velocity in this case is the local carrier flow velocity plus a drift velocity proportional to local gradients of the turbulence (zero if the particles follow the flow). The diffusion coefficient is linearly related to the pressure/mean density of the dispersed phase via the particle response time. It is shown to be influenced by the local mean shearing of the carrier flow. Conditions are derived when this contribution can be ignored, the diffusion coefficient reducing to the form for homogeneous turbulence.

A new conservation law constructed without using either Lagrangians or Hamiltonians

Abstract: A new conservation theorem is derived. The conserved quantity is constructed in terms of a symmetry transformation vector of the equations of motion only, without using either Lagrangian or Hamiltonian structures (which may even fail to exist for the equations at hand). One example and implications of the theorem on the structure of point symmetry transformations are presented.

System for predicting behavior of automotive vehicle and for controlling vehicular behavior based thereon

Abstract: A system for detecting a physical amount of behavior of a vehicle includes, acceleration sensors arranged on at least two longitudinal axes of the vehicle, the vertical axis Of the vehicle and the lateral axis of the vehicle, a plurality of the acceleration sensors being disposed on each of the axes. A unit is provided for establishing a conversion equation for determining acceleration values of linear motion at an arbitrary point of the vehicle in the direction of each axis of an arbitrary coordinate system and acceleration values of rotational motion with respect to the each axis of the coordinate system while simultaneously using acceleration values detected by the acceleration sensors disposed on at least two of the vehicular longitudinal axes, the vertical axis and the lateral axis. There is also provided a unit for calculating the conversion equation to obtain the acceleration values of linear motion at an arbitrary point of the vehicle in the direction of each axis of the arbitrary coordinate system and acceleration values of rotational motion with respect to each axis of the coordinate system, a unit for establishing a motion equation expressing a plurality of freedom motions, and a unit for calculating the motion equation with the acceleration values of linear motion at an arbitrary point of the vehicle in the direction of each axis of the arbitrary axis of the arbitrary coordinate system and acceleration values of rotational motion with respect to the each axis of the coordinate system to obtain the physical amount associated with the behavior of the vehicle.

Isotropic-nematic transition in shear flow: State selection, coexistence, phase transitions, and critical behavior.

Abstract: Macroscopic fluid motion can have dramatic consequences near the isotropic-nematic transition in thermotropic liquid-crystalline fluids. We explore some of these consequences using both deterministic and stochastic descriptions involving coupled hydrodynamic equations of motion for the nematic order parameter and fluid velocity fields. By analysing the deterministic equations of motion we identify the locally stable states of homogeneous nematic order and strain rate, thus determining the homogeneous nonequilibrium steady states which the fluid may adopt

Motion of a magnetic domain wall traversed by fast‐rising current pulses

Abstract: A Bloch wall is predicted to undergo finite displacements when traversed by a current pulse with short rise time ≤20 ns and long fall time, in thin films of metallic ferromagnets. In Ni–Fe films of thickness 85–150 nm, pulses with peak current density ≂1×107 A/cm2 are expected to induce wall displacements of order 0.1–1 μm. This effect originates from the s‐d exchange interaction. It is phenomenologically similar to the well‐known ‘‘wall streaming’’ motion of Bloch walls subjected to fast‐rising pulses of hard‐axis magnetic field. The effect is related to the existence of a novel, current‐induced, term in the expression for the momentum of a magnetic domain wall.

Nonlinear oscillations of suspended cables containing a two-to-one internal resonance

Abstract: The near-resonant response of suspended, elastic cables driven by planar excitation is investigated using a two degree-of-fredom model. The model captures the interaction of a symmetric in-plane mode and an out-of-plane mode with near commensurable natural frequencies in a 2:1 ratio. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. The existence and stability of periodic solutions are investigated using a second order perturbation analysis. The first order analysis shows that suspended cables may exhibit saturation and jump phenomena. The second order analysis, however, reveals that the cubic nonlinearities and higher order corrections disrupt saturation. The stable, steady state solutions for the second order analysis compare favorably with results obtained by numerically integrating the equations of motion.

A theoretical approach for analyzing the restabilization of wakes

Abstract: Recently reported experimental results demonstrate that restabilization of the low-Reynolds-number flow past a circular cylinder can be achieved by the placement of a smaller cylinder in the wake of the first at particular locations. Traditional numerical procedures for modeling such phenomena are computationally expensive. An approach is presented here in which the properties of the adjoint solutions to the linearized equations of motion are exploited to map quickly the best positions for the small cylinder's placement. Comparisons with experiment and previous computations are favorable. The approach is shown to be applicable to general flows, illustrating how strongly control mechanisms that involve sources of momentum couple to unstable (or stable) modes of the system.

Hydromagnetic flow of a dusty fluid over a stretching sheet

Abstract: Analysis of hydromagnetic flow of a dusty fluid over a stretching sheet is carried out with a view to throw adequate light on the effects of fluid-particle interaction, particle loading, and suction on the flow characteristics. The equations of motion are reduced to coupled non-linear ordinary differential equations by similarity transformations. These coupled non-linear ordinary differential equations are solved numerically on an IBM 4381 with double precession, using a variable order, variable step-size finite-difference method. The numerical solutions are compared with their approximate solutions, obtained by a perturbation technique. For small values of β the exact (numerical) solution is in close agreement with that of the analytical (approximate) solution. It is observed that, even in the presence of a transverse magnetic field and suction, the transverse velocity of both the fluid and particle G phases decreases with an increase in the fluid-particle interaction parameter, β, or the particle-loading parameter, k . Moreover, the particle density is maximum at the surface of the stretching sheet, and the shearing stress increases with an increase in β or k .