Fundamental Dynamic Equations Research Articles

Finite‐element analysis of Taylor flow

AbstractA finite‐element analysis of Taylor flow in a cylindrical capillary was performed using a commercial FEM program (FIDAP) to solve the fundamental fluid dynamics equations together with the capillary forces at the gas–liquid interface. A moving‐surface formulation was used to calculate the bubble shape. The thickness of the liquid film surrounding the gas bubble, the degree of mixing in the liquid phase, and the slip velocity between the two phases were calculated. These parameters influence the performance of monolith reactors operating in the Taylor flow regime. On comparison with experimental results it was found that the FEM calculation generally predicts a thinner liquid film, which can possibly be explained in terms of a peripheral variation in surface tension. Moreover, the wavelength of the wiggles predicted in the liquid film near the tail end of the bubble was compared to those arising from a simplified mathematical analysis available in the literature. Good agreement was found for Ca < 0.005, while for higher Ca the FEM predicts significantly shorter wavelengths, indicating that the lubrication theory is not valid here.

FREE VIBRATION AND STABILITY OF THIN ELASTIC BEAMS SUBJECTED TO AXIAL FORCES

Natural frequencies and buckling loads of a simply supported beam with small length-to-depth ratio and sufficiently thin rectangular cross-sections subjected to initial axial tensile and/or compressive forces are analyzed. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a one-dimentional higher order beam theory for thin rectangular beams is derived through Hamilton's principle. Several sets of truncated approximate theories which can take into account the effects of both shear deformations with depth changes and rotary inertia are applied to solve the eigenvalue problem of a thin elastic beam. The Navier solution procedure is used to satisfy the boundary conditions of a simply supported thin rectangular beam. In order to assure the accuracy of the present theory, convergence properties of the minimum natural frequency and the buckling load for the axial and bending problems of thin beams are examined in detail. It is noticed that the present approximate theories can predict the natural frequencies and buckling loads of thin beams with small length-to-depth ratio more accurately than other refined higher order theories and the classical beam theory.

In-plane vibration and stability of shallow circular arches subjected to axial forces

Natural frequencies and buckling loads of a simply supported shallow circular arch with sufficiently small depth-to-radius of curvature ratio (H/R ⪡ 1) subjected to initial axial tensile and/or compressive forces are analysed. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a one-dimensional higher-order arch theory for in-plane vibration problems of shallow circular arches is derived through Hamilton's principle. Several sets of truncated approximate theories which can take into account the effects of both shear deformations with depth changes and rotary inertia are applied to solve the eigenvalue problems of an elastic arch. Convergence properties of the natural frequency and the buckling load of simply supported shallow circular arches are examined in detail. The present approximate theories can predict the natural frequencies and buckling loads of shallow circular arches with small length-to-depth ratio L/H more accurately compared with previously published results.

A107 PIVとCFDを組み合わせた流れ場の推定法

CFD and field measurement have been used widely to understand flow field. Although CFD can estimate full view of flow field easily, it contains inevitably numerical errors. Meanwhile, field measurement usually requires much labor and contains experimental errors, too. PIV can measure flow velocities at many points simultaneously, but it is still difficult to measure whole points for a large field.In this work, we propose a new technique to understand the whole flow field by combining PIV and CFD. In this technique, velocities at data absence points are calculated by CFD using locally obtained PIV data as a boundary condtion. And the PIV data are corrected simultaneously to reduce measurement errors using CFD results and fundamental equations of fluid dynamics. To examine this technique we performed some simulations using artificial data. And the two-dimensional flow in the staggered screens was retrieved using our combined technique. The results agreed well with visualized flow patterns. It can be possible to reproduce whole flow field adequately by using our new technique. But it is still necessary to examine the perfomance of our method in full detail.

The general equations of analytical dynamics

It is shown that the generalized Poincaré and Chetayev equations, which represent the equations of motion of mechanical systems using a certain closed system of infinitesimal linear operators, are related to the fundamental equations of analytical dynamics. Equations are derived in quasi-coordinates for the case of redundant variables; it is shown that when an energy integral exists the operator X 0 = ∂/ ∂t satisfies the Chetayev cyclic-displacement conditions. Using the energy integral the order of the system of equations of motion is reduced, and generalized Jacobi-Whittaker equations are derived from the Chetayev equations. It is shown that the Poincaré-Chetayev equations are equivalent to a number of equations of motion of non-holonomic systems, in particular, the Maggi, Volterra, Kane, and so on, equations. On the basis of these, and also of other previously obtained results, the Poincaré and Chetayev equations in redundant variables, applicable both to holonomic and non-holonomic systems, can be regarded as general equations of classical dynamics, equivalent to the well-known fundamental forms of the equations of motion, a number of which follow as special cases from the Poincaré and Chetayev equations.

Nonlinear inertia-gravity internal wave in stratified fluid

Based on the fundamental equations of geophysical fluid dynamics and on the consequence of vertical density stratification, travelling wave coordinates are used in this work to study the geometric topological structures of nonlinear permanent wave in phase plane. Rigorous mathematical mechanics demonstrate that the solution of permanent solitary wave does not exist. Hamilton functions and “action-angle” variables are used to express the travelling wave system in the simplest form and the analytic solution of the nonlinear inertia-gravity internal wave is obtained.

Review: Activated Carbon Filters in Respiratory Protective Equipment.

The review considers first the nature of airflow through granular carbon filters and the relationship between pressure drop and airflow rate. Equilibrium adsorption is then summarized, with particular emphasis on the volume filling of micropores, on which most of the theory of activated carbon performance is based. The dynamics of adsorption is studied, and the analytical solution of the fundamental dynamic adsorption equation is given, leading to the development of various equations used in practice for the description of breakthrough. The most fundamental dynamic parameter, the rate constant for adsorption, is discussed, and the effects of concentration, granule diameter, and air stream velocity are summarized. Extracts of a number of experimental observations with reference to specific adsorbates are given, followed by an account of the adsorption of mixtures of compounds. The effect of relative humidity is discussed. This is followed by an account of non-destructive tests, and the review ends with a summary and an indication of possible fruitful areas for research.

Free vibration and stability of thick elastic plates subjected to in-plane forces

Natural frequencies and buckling loads of a simply supported thick plate subjected to in-plane initial tensile and/or compressive forces are analysed. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order plate theory is derived through Hamilton's principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of a thick plate. In order to assure the accuracy of the present theory, convergence properties of the minimum natural frequency and the buckling load for the out-of-plane problem of a simply supported square plate are examined in detail. It is noticed that the present approximate theories can predict the frequencies and buckling loads of an extremely thick plate more accurately compared to other refined theories and classical plate theory.

Resonance frequency and damping constant of a spherical gas-vapor bubble in an infinite medium

The resonance frequency f0 and damping constant δ for free volume oscillations of a spherical bubble in an infinite medium are derived from the linearized, fundamental equations of fluid dynamics. There are three possible modes (acoustic, thermal, and mass diffusion) of wave propagation that satisfy the governing equations for the gas–vapor mixture inside the bubble. Outside it, only the acoustic and thermal diffusion modes need consideration because gas diffusion across the bubble surface into the liquid appears negligible for free oscillations. The parameters f0 and δ are values such that the boundary conditions can be satisfied. The dependence of f0 and δ on bubble size, ambient pressure, and temperature is examined. When a bubble contains a high concentration of gas, the values of f0 and δ agree with the results of Devin [J. Acoust. Soc. Am. 31, 1654–1667 (1959)]. When the concentration of vapor is high, it is found that there exists only one resonance frequency for free oscillations [compare the results of R. D. Finch and E. A. Neppiras, J. Acoust. Soc. Am. 53, 1402–1410 (1973)]. [Work supported by ONR.]

New formulations of the primitive equations of atmosphere and applications

The primitive equations are the fundamental equations of atmospheric dynamics. With the purpose of understanding the mechanism of long-term weather prediction and climate changes, the authors study as a first step towards this long-range project what is widely considered as the basic equations of atmospheric dynamics in meteorology, namely the primitive equations of the atmosphere. The mathematical formulation and attractors of the primitive equations, with or without vertical viscosity, are studied. First of all, by integrating the diagnostic equations they present a mathematical setting, and obtain the existence and time analyticity of solutions to the equations. They then establish some physically relevant estimates for the Hausdorff and fractal dimensions of the attractors of the problems.

Some Numerical Results in 3-D Transient Linear Naval Hydrodynamics

Some numerical results obtained by the code MELINA are given. This code has been designed for the solution of three-dimensional steady and transient flows past a floating or submerged body; the fundamental equations of dynamics are taken into account in the case where the body is freely floating. Only the case of an ocean of infinite depth is addressed here. It is shown that convergence to a steady state is achieved in the case of a periodic forced motion if the history of the motion is correctly taken into account. Also, the case of a freely floating sphere with initial vertical displacement from equilibrium is treated and compared with experiments.

Dynamic Equations of Polymers with Deformations in Semidilute Regions

We present fundamental dynamic equations describing collective diffusive motions of deformed polymers. If we incorporate Doi's recent idea on the polymer diffusion in flow, the equations turn out to be different from those previously proposed by the present author (Phys. Rev. Lett. 62 (1989) 2472). In the absence of flow, on the other hand, they are consistent with a theory of Brochard and de Gennes. A seriously ambiguous assumption in the derivation remains.

A New Total Heart Design Via Implantable Impeller Pumps

Hemolysis and thrombosis have been considered as the main limitations of the impeller pump and other centrifugal pumps to total heart applications. One of the solutions is to choose the impeller shroud and vane according to the stream surfaces of blood flow, so as to vanish the turbulence and stagnation which cause the hemolysis and thrombosis, respectively, in the pump. This paper describes the method of deducing the stream surfaces together with the velocity distributions in the impeller from the fundamental dynamical equations, and presents a prototype design of the impeller total heart which promises to be an ideal alternative to the problematic diaphragm total heart.

Relativistic radiative hydrodynamics

In conventional rediative hydrodynamics, the dissipative fluxes are proportional to the gradients of the temperature, velocity, etc., with proportionality constants inferred from the linearized radiative transfer equation. The dynamical evolution is determined by the conservation of matter, energy, and momentum. Conventional theory breaks down if the geometry under consideration is photospheric, i.e., if emission/absorption processes are weak compared to scattering and photons can “random walk” over large distances without being absorbed. The fundamental dynamical equations of motion include the relativistic transfer equation and form the starting point for establishing extended versions of radiative hydrodynamics. The transfer equation is replaced by a hierarchy of relativistic moment equations and the dissipative fluxes are treated as dynamical variables in their own right. The general moment equations involve source terms which are a priori undetermined. If the radiation is almost thermal the source terms can be expressed with Rosseland means. In this case, the moment equations reproduce the structure equations postulated by Israel for general dissipative continuous media. Quantitative aspects are discussed. In the limit of slow motion, the moment equations are studied separately and interpreted as hyperbolic diffusion equations. These are solved with a relativistic random walk model and the results are applied to time-dependent radiative transfer. Time profiles in every scattering order are given and compared with Monte Carlo simulations. The viscous damping of acoustic modes is reconsidered. Equations of motion for spherical accretion in a Schwarzschild geometry are established and compared with conventional equations used in the literature. Ill-known limitations of conventional radiative hydrodynamics are highlighted.

NATURAL FREQUENCY OF A FILL DAM BY MEANS OF TWO DIMENSIONAL TRUNCATED WEDGE TAKING SHEAR AND BENDING MOMENT EFFECTS INTO ACCOUNT

The discussion, in this paper, is confined to the natural vibration of the fill dam which can be described by the truncated wedge of inhomogeneous rigidity: G=Gm·(z/h)n where G: modulus of rigidity, z: distance from the top of dam, h: dam height, n: rigidity index, and with rectangular canyon shape. Equilibrium of the bending moment and the shearing force in the vertical direction, together with the shearing force and twisting moment in the transverse direction, leads to the fundamental dynamic equations of the prescribed model of dam.Finite difference method is used to determine the natural frequencies and natural modes of the dam. The frequencies and the modes are computed for the various rigidity index n, the canyon width and the bottom width, the results are put in order in graphical or tabular forms.

Towards a Self-consistent Model of a Galaxy

Using a theoretical model for the functional distribution of stars in phase space in a spherically symmetric galactic system, it is found upon solving the fundamental equations of stellar dynamics that the rotation curves produced by the model are flat for large distances within the system. the properties of the stellar orbits within such systems are investigated and an N-ring axially and equatorially symmetric model for simulating its dynamics is presented. Poisson's equation is solved by expanding density and potential in Legendre polynomials (c.f. van Albada and van Gorkom, 1977). It is helped to follow the time development of such a system under various forces.

Fluid Transmission Lines—Distributed Parameter Models Part 1: A Review of the State of the Art

This paper presents a hierarchical classification of linear, distributed parameter models which have been used in the study of rigid, uniform fluid transmission lines. The development of the models from the fundamental equations of fluid dynamics is discussed, and the assumptions necessary in obtaining each model are shown. The solutions to the models are presented in general Laplace forms, and the terms that appear in the solutions are discussed with reference to the initial and subsequent assumptions. The existence of seven distinct forms of models was identified and verified by comparison with a vast number of apparently different models quoted in over three hundred research papers covering various engineering, medical, and biological applications. Significant contributions by various researchers over the last eighty years to the development of distributed parameter models of fluid transmission lines are reviewed. The application of the various models based upon comparisons made in the literature is dealt with, and the effect of the initial assumptions on each model's accuracy is discussed. The aim of the paper is to assist in the identification of areas of research where future efforts should be directed and hopefully to prevent needless duplication of research effort.

Multiple-Scale Approach to Nonlinear Phase Dynamics

A fundamental equation of the phase dynamics is derived for a class of periodic structures by a systematic application of the multiple-scale expansion method, in which the instantaneous rate of change of the phase is shown to be determined only by the local values of spatial derivatives of the phase itself. The analys is shows that the phase equation is inapplicable to conservative systems. The expansion method is also applied to a class of forced periodic structures, and the same type of equation is derived although the dependent variable then does not correspond to the phase of the periodic structure.

粒子間に付着力が働く場合の粉体力学の基礎方程式

粒子間に付着力が働く場合の粉体力学の基礎方程式

Multicriteria dynamic programming with an application to the integer case

Fundamental dynamic programming recursive equations are extended to the multicriteria framework. In particular, a more detailed procedure for a general recursive solution scheme for the multicriteria discrete mathematical programming problem is developed. Definitions of lower and upper bounds are offered for the multicriteria case and are incorporated into the recursive equations to aid problem solution by eliminating inefficient subpolicies. Computational results are reported for a set of 0–1 integer linear programming problems.