Fractional Derivatives Applied to Phase-Space Reconstructions

The concept and application of phase-space reconstructions are reviewed. Fractional derivatives are then proposed for the purpose of reconstructing dynamics from a single observed time history. A procedure is presented in which the fractional derivatives of time series data are obtained in the frequency domain. The method is applied to the Lorenz system. The ability of the method to unfold the data is assessed by the method of global false nearest neighbors. The reconstructed data is used to compute recurrences and correlation dimensions. The reconstruction is compared to the commonly used method of delays in order to assess the choice of reconstruction parameters, and also the quality of results.     

The effect of initial soil water content and vegetative cover on surface soil disturbance by tracked vehicles

A field trial was conducted to examine the effect of initial soil water content and vegetative cover on soil disturbance caused by tracked military vehicles. Disturbance was measured as depth of rut produced. Soil disturbance increased as the number of passes and turns increased, but especially after turning manoeuvres. Greater disturbance occurred on reduced vegetation plots and with increasing initial soil water content. Simple regression analysis was undertaken to determine which factor had the greatest influence on disturbance.     

Theory of a calorimeter at a permeable surface

The calorimetric method for determining the heat flux at a permeable (or sublimating) surface requires the solution of several specific heat-transfer problems, since the calorimeter, bringing about discontinuities in the boundary conditions at the wall (the cessation of blowing, a jump in the temperature of the wall and of its catalytic properties, etc.), introduces perturbations into the boundary layer and measures a heat flux differing from the flux in the absence of a calorimeter. Within the framework of the boundary layer, schematization of such problems is usually based on the isolation of an internal boundary layer (sublayer), which is the region of the effect of the new phenomena at the wall, and develops in the main boundary layer [1–5]. To take account of the effect of the inhomogeneity of the flow in the main boundary layer on heat transfer through the sublayer, here the method of mean-mass values is used, which, as has been demonstrated using various examples in [4] and in the present work, has a good degree of accuracy (even in the neighborhood of the breakaway point) and is suitable for the profiles of an inhomogeneous flow of rather general form. Based on this, for a laminar boundary layer finite formulas are obtained below for the heat flux to a calorimeter of relatively small size at a permeable wall, which can be used for the analysis of experiments.     

Interaction between the effects of local and overall imperfections on the buckling of elastic columns

A. van derNeut has studied the buckling of elastic columns made from thin-walled members, with particular reference to the effects of imperfections. In this paper, a graphical method is used to extend his work and to make an exhaustive study of the combined effects of both ‘local’ and ‘overall’ imperfections. The resulting picture is remarkably simple, and the effects of imperfections are well described by the celebrated Perry formula in conjunction with a single imperfection parameter compounding simply the local and overall imperfections. Experiments on small-scale rubber model columns substantiate the main results of the theoretical analysis.     

Gas flow in cylindrical capillaries under intermediate conditions

At present, there are sufficient solutions of the problem of free-molecular gas flow through a short cylindrical channel, for example, [1–3]. In intermediate flow conditions, for Knudsen number Kn 1, solutions have been obtained for the limiting cases: an infinitely long channel [4] and a channel of zero length (an aperture) [5]. However, no solution is known for short channels for Kn 1. The present work reports a calculation by the Monte Carlo method of the macroscopic characteristics of the gas flow through a short cylindrical channel (for various length—radius ratios), taking into account intermolecular collisions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 187–190, January–February, 1977.     

MHD flow in the vicinity of the stagnation point in a purely azimuthal magnetic field

Axisymmetric MHD flow in the vicinity of the stagnation point in the presence of a purely azimuthal nonhomogeneous magnetic field B {0, B, 0} is studied. This problem belongs to the class of MHD problems whose solutions are known as solutions of the layer type [1]. This class also includes, in particular, the classical exact solutions of the Navier-Stokes equations.The approximate solutions of the analogous MHD problems for the limiting cases of large and small values of the diffusion number ==/ have been considered in [2–5]. In this case it is possible to divide the flow into the so-called viscous and current layers, for each of which the approximate equations, simpler than the exact equations, are solved numerically or in quadratures. Using this technique it is possible to avoid the basic mathematical difficulty, which is that the sought solution of the boundary-value problem must be selected from a family of two-parameter solutions. The approximate method permits dividing the problem into two stages (corresponding to the two boundary layers) in each of which one unknown parameter is determined (in place of their simultaneous determination by direct integration of the basic equations).The drawback of the approximate methods [2–5] is their nonapplicability in the most interesting case, when the thicknesses of the current and viscous layers are of comparable magnitude, i. e., when the kinematic and magnetic viscosities ( and ) are quantities of the same order. We should also note the poor accuracy of the methods in the framework of the considered approximations for a comparatively large volume of the calculations required, which, in turn, prevents obtaining more exact solutions.The present paper presents a numerical integration of the equations describing MHD flow in the vicinity of the stagnation point over a wide range of S and numbers (Alfvén and diffusion numbers), without the assumption of their smallness, with preliminary determination of the unknowns at the zero of the derivatives of the sought functions with the aid of the method of asymptotic integration.A critical value of the Alfvén number is found, for which the retardation of the fluid by the magnetic field (for the first considered configuration of the magnetic field) at the wall is so intense that the friction vanishes everywhere on the surface of the solid body. It is also found that with further increase of the number S a region of reverse flow appears near the wall, which is separated from the remaining flow by a plane on which the z-component of the velocity is equal to zero.     

Comparison of stabilization modes of a symmetrical third-order non-linear oscillatory unit

The possibility of constructing symmetrical stabilized oscillator models that operate in several phases instead of operating in two phases, as is more usual, is considered in the present paper. In particular, symmetrical oscillator models operating in three phases are discussed comprehensively. An interesting feature of their dynamic behaviour, e.g. the appearance of limit cycles which are regionally stable and not orbitally stable, as is the case in the two phase counterpart, is suggested and dealt with.     

On the moment of forces acting on a body in a ferrofluid

Attention is drawn to the appreciable torque which acts on a nonmagnetic body in a ferrofluid when a field is applied. The contribution to the torque made by an external inhomogeneity of the field and the perturbation introduced by the body is considered. It is shown that in fields obtained in practice they may be comparable. A method of calculating the resultant force and the resultant torque is considered. The moment of the forces that act on a cylinder of elliptic section in homogeneous and inhomogeneous fields is calculated. An experimental investigation confirmed the theoretical calculations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 132–136, July–August, 1980.     

Heat mass transfer in a turbulent layer above permeable membranes

The temperature and concentration fields in a boundary layer above perforated membranes are presented, and their relationship with the velocity fields given in [1] is established. Measurements of the thermal state of membranes are made with various geometric and thermophysical properties and various coolant drafts. Empirical formulas are also presented for thermal flux and temperature of the permeable walls.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 22–31, July–August, 1973.     

Translation of Walter Noll’s “Derivation of the Fundamental Equations of Continuum Thermodynamics from Statistical Mechanics”

This article represents a translation of the original paper, “Die Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der Statistischen Mechanik”, which was written by Walter Noll and appeared in the Journal of Rational Mechanics and Analysis 4 (1955), 627–646. In the original paper, Noll addressed and analyzed the seminal paper of Irving & Kirkwood, published five years earlier, on “The statistical mechanical theory of transport processes. IV, The Equations of Hydrodynamics.” Noll gave new interpretations and provided a firm setting for ideas advanced by Irving & Kirkwood that clearly and directly related to the basic principles of continuum mechanics. This translation aims to expose the important contribution of Noll to a wider community of researchers at a time when the atomistic modeling of material behavior is being advanced. Noll’s use of elementary mathematics to discover physical effects, to explain physical concepts, and to draw conclusions of a physical nature is exhibited. Noll’s paper emerged from a report that he presented in a seminar at Indiana University in the summer of 1954. The seminar was organized by Clifford Truesdell, whose inspiration Noll gratefully acknowledged.