Lagrangean Structure and Propagation of Singularities in Multidimensional Compressible Flow

We study the propagation of singularities in solutions of the Navier–Stokes equations of compressible, barotropic fluid flow in two and three space dimensions. The solutions considered are in a fairly broad regularity class for which initial densities are nonnegative and essentially bounded, initial energies are small, and initial velocities are in certain fractional Sobolev spaces. We show that, if the initial density is bounded below away from zero in an open set V, then each point of V determines a unique integral curve of the velocity field and that this system of integral curves defines a locally bi-Hölder homeomorphism of V onto its image at each positive time. This “Lagrangean structure” is then applied to show that, if the initial density has a limit at a point of such a set V from a given side of a continuous hypersurface in V, then at each later time both the density and the divergence of the velocity have limits at the transported point from the corresponding side of the transported hypersurface, which is also a continuous manifold. If the limits from both sides exist, then the Rankine–Hugoniot conditions hold in a strict pointwise sense, showing that the jump in the divergence of the velocity is proportional to the jump in the pressure. This leads to a derivation of an explicit representation for the strength of the jump in the logarithm of the density, from which it follows that discontinuities persist for all time, convecting along fluid particle paths, and in the case that the pressure is strictly increasing in density, having strengths which decay exponentially in time.     

Two Theories of Complex Bodies, one Lagrangean, the other Kinetic, Have a Common Ground

I have found in previous works that most special models proposed to represent bodies with some type of microstructure can be classified easily under the general umbrella of a theory where each element of the continuum is thought of as a Lagrangian system. To study phenomena in ‘kinetic’ continua I proposed an apparently different approach; the outcome is again a set of evolution equations. They mimic equations familiar in continua with affine microstructure: a Cauchy’s equation and an equation of balance of tensor moment of momentum, with the addition, however, of an equation of balance for a ‘Reynolds’ tensor’, an equation which, in a sense, shifts the boundary between kinetic and thermal properties of matter. I will show that there is no contrast between the two approaches. The latter one is based on an adequate and appropriately justified expression of the kinetic energy of the continuum, comprising the trace of the quoted Reynolds’ tensor and thus importing into the mechanical energy a term usually accounted by additional heat.     

Uniqueness and Existence Results on the Hele-Shaw and the Stefan Problems

(Accepted November 4, 2002) Published online April 4, 2003 Communicated by L.C. Evans     

Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

Continuum mechanics with dislocations, with the Cattaneo-type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov-type system of the first-order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov-type formulation brings the mathematical rigor (the local well posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization).     

Dynamic and stability analysis of the vibratory feeder and parts considering interactions in the hop and the hop-sliding regimes

Nonlinear Dynamics - In this paper, the interactions of a translational vibratory feeder and the parts in the hop and the hop-sliding regimes are studied by means of an improved multi-term...     

Influences of the fish-mouth project and the groins on the flow and sediment ratio of the Yangtze River waterway

A depth-integrated two-dimensional numerical model of current, salinity andsediment transport was proposed and calibrated by the observation data in the Yangtze RiverEstuary. It was then applied to investigate the flow and sediment ratio of the navigationchannel, i. e. the North Channel of the Yangtze River Estuary, before and after the firstphase waterway project is implemented. Particularly, the influences of groin length and theorientation of the submerged dam on the flow ratio and sediment load discharging into theNorth Channel were discussed. The numerical results demonstrate that less sediment loaddischarges into the navigation channel, which unburdens the waterway dredging, but in themeantime the flow ratio is also decreased. The flow and sediment ratio can be adjusted bychanging layout and dimensions of the hydro-structures, such as the groin length, the topheight, etc. The effect of the orientation of the submerged dam is more obvious than thegroin lengh.     

Correlation between the crossover modulus and the molecular weight distribution using the Doi-Edwards theory of reptation and the Rouse theory

The crossover modulus in the dynamic mechanical spectrum was linked with the molecular weight distribution (MWD) over a wide range of polydispersity ( = 1 to 6). Reasonable agreement was obtained between theory and experiment for near monodisperse and narrow MWD polymers using the Doi-Edwards theory of reptation. For broad MWD polymers, the Doi-Edwards theory was found to be inapplicable. Using the Doi-Edwards theory for the shorter chains and a Rouse type mechanism for the longer chains, a good fit was obtained between the calculated and experimentally measured crossover modulus values.     

On the Lamb vector and the hydrodynamic charge

This work is an attempt to test the concept of the hydrodynamic charge (analogous to the electric charge in electromagnetism) in the simple case of a coherent structure such as the Burgers vortex. We provide experimental measurements of both the so-called Lamb vector and its divergence (the charge) by two-dimensional particles images velocimetry. In addition, we perform a Helmholtz–Hodge decomposition of the Lamb vector in order to explore its topological features. We compare the charge with the well-known Q-criterion in order to assess its interest in detecting and characterizing coherent structure. Usefulness of this concept in studies of vortex dynamics is demonstrated.

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On the singularity index and the unfolding of the diffraction catastrophes

The caustics of high-frequency wave propagation may be classified using catastrophe theory. The wavefield in the vicinity of any caustic is described by the corresponding diffraction catastrophe. The singularity index, β, is a measure of the rate at which such a wavefield diverges as ω→∞ at the point where all control parameters and moduli are set equal to zero. It is shown that away from this point β also describes a balance between two different measures of the unfolding of the wavefield in each control direction, β = σ_nϱ_n. The indices σ_n and ϱ_n describe, respectively, the rate at which individual ray arrivals separate in time and decay as a function of control parameter.     

Reevaluation of the adhesive relationship between the tire and the soil

The author proved in an earlier article that the shear diagram is not in accord with its mechanical definition. The shear stress cannot be zero at the beginning of the initial rising portion of the curve. Shearing is not an increasing loading process, rather it is a limiting case to which a finite shear stress belongs. On the other hand the sheared surface varies under the tire. There are kinematic reasons for this. Points on the tire surface describe a looped cycloid and they slip in a backward direction (opposite to the direction of travel) while contacting the soil. Thus the driving force, which points in the direction of travel, is the product of the shear stress of finite magnitude and the sheared area. The latter increases proportionally with slip. The author describes his equation which is based on the principles discussed above. He supports his theory with a numerical example.     

Diffusion and Electrical Processes in the Turbulent Boundary Layer and in the Neighborhood of the Stagnation Point

The problem of electric current (engine current) formation in aircraft jet engine ducts as a result of the development of electrical diffusion boundary layers on the surfaces of the duct and internal engine components is investigated. It is assumed that the outer flow containing electrons and positive ions is quasi-neutral and that the electrical quasi-neutrality is violated (and the electric engine current develops) in the wall flow zone as a result of the difference between the electron and ion diffusion coefficients. The problem of the development of an electrical diffusion boundary layer inside the turbulent gasdynamic boundary layer on a plane surface is formulated and solved. The engine current distribution along the duct is found for various values of a turbulent viscosity on the boundary of the gasdynamic boundary layer which affect the laminar-turbulent transition point.The electrical diffusion processes that occurs when an electrically quasi-neutral hydrodynamic stream impinges on a plane surface (simulation of the flow in the neighborhood of a stagnation point) is studied. In this case the Navier-Stokes equations have a self-similar solution. It is shown that the system of electrohydrodynamic equations also has a self-similar solution. The electrical parameter fields are determined and the engine current is found on the basis of this solution.     

地球与月球关系的研究

基于鹦鹉螺外壳生长线条数的变化是月亮绕地球运动周期随年代变化反映的假设, 采用外推 法得出两个结论: (1) 月球最初被地球俘获绕地球运动时距地球的距离为目前月地距离的0.2 倍左右; (2) 月球最初被地球俘获绕地球运动时的时间距今为7亿年左右. 这 为进一步研究地球与月球的关系提供了有益的启示.     

Rossby inertia gravity solitary wave and the remote correlation between the East- and the West-Pacific subtropical high

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Flow and coupled heat transfer in the cavity between the rotor and stator

The turbulent flow and coupled heat transfer in the cavity between the rotor and stator is numerically simulated. Reynolds-averaged Navier-Stokes equations closed with equations of the k-ɛ turbulence model are used to calculate the viscous compressible gas flow characteristics and heat transfer; the unsteady heat conduction equation is used to calculate the temperature field in the metal. The influence of the mass flow rate of the coolant on the flow structure and efficiency of cooling of the rotor and stator walls is studied. The calculated results are compared with experimental data.