Identification of regions of fastest mixing in a system of point vortices

This paper describes a numerical method for efficiently identifying the regions of fastest mixing of a passive dye in a flow due to a system of point vortices. Results obtained from computations are presented for systems of three and four point vortices, both in the unbounded domain and inside a circular cylinder. The flow is two‐dimensional and the fluid is incompressible. The regions where mixing is possible are found by studying the largest Lagrangian Lyapunov exponent distribution with respect to various initial positions of tracer particles. The regions of fastest mixing are then identified from the Lyapunov exponent distribution at small times. The results of the method are verified by quantifying the mixing by using a traditional box counting technique. The technique is then applied to several different initial configurations of vortices and some interesting results are obtained. Some numerical findings about the nature of the exponents computed are also discussed. Copyright © 2002 John Wiley Sons, Ltd.     

Diagonalization of a three-dimensional system of equations in terms of displacements of the linear theory of elasticity of transversely isotropic media

A dynamic three-dimensional system of linear equations in terms of displacements of the theory of elasticity of transversely isotropic media is given explicit expressions for phase velocities and polarization vectors of plane waves. All the longitudinal normals are found. For some values of the elasticity moduli, the system of equations is reduced to a diagonal shape. For static equations, all the conditions of the system ellipticity are determined. Two new representations of displacements through potential functions that satisfy three independent quasi-harmonic equations are given. Constraints on elasticity moludi, at which the corresponding coefficients in these representations are real, different, equal, or complex, are determined. It is shown that these representations are general and complete. Each representation corresponds to a recursion (symmetry) operator, i.e., a formula of production of new solutions.     

Numerical Analysis of the Branched Forms of Bending for a Rod

Nonlinear boundary–value problems of plane bending of elastic arches subjected to uniformly distributed loading are solved numerically by the shooting method. The problems are formulated for a system of sixth–order ordinary differential equations that are more general than the Euler equations. Four variants of rod loading by transverse and longitudinal forces are considered. Branching of the solutions of boundary–value problems and the existence of intersected and isolated branches are shown. In the case of a translational longitudinal force, the classical Euler elasticas are obtained. The existence of a unique (rectilinear) form of equilibrium upon compression of a rod by a following longitudinal force is shown.     

Cases of Integrability of Three-Dimensional Dynamic Equations for a Solid

A dynamic model of the interaction of a rigid body with a jet flow of a resistant medium is considered. This model allows us to obtain three-dimensional analogs of plane dynamic solutions for a solid interacting with the medium and to reveal new cases where the equations are Jacobi integrable. In such cases, the integrals are expressed in terms of elementary functions. The classical problems of a spherical pendulum in a flow and three-dimensional motion of a body with a servoconstraint are shown to be integrable. Mechanical and topological analogs of these problems are found     

Numerical study of self-induced wing rock of a low-aspect-ratio delta wing in the presence of external disturbances

Self-induced wing rock of a delta wing, in particular, in the presence of external disturbances are studied by means of numerical simulations of a separated flow of an ideal incompressible fluid around a delta wing. The results obtained are compared with experimental data. The vortex nature and the mechanism of self-induced oscillations are studied. Regions of synchronization of the aerodynamic self-oscillatory system in the presence of external disturbances are identified. Methods of suppression of self-induced wing rock are proposed.     

Theory of dry friction of rubbery materials

Friction of solids involves short-range forces between adjacent surface layers, which are largely determined by the shape and structure of those layers, which are themselves determined to a considerable extent by the relative velocity. A theory of friction thus involves the microstructure and the detailed physical phenomena near the surfaces.However, most existing theories are based on phenomenological (essentially macroscopic) concepts (see [1] for a survey), though the explicit use of microscopic concepts is presented in [2], where it is shown that one elastic body sliding over another gives rise to elastic waves that carry energy away from the contact surface. This loss may be treated formally as due to a tangential force resisting the motion. The force defined in this way has a falling velocity characteristic.There is much evidence that the friction differs greatly from that for ordinary elastic bodies if one body (or both) should be highly elastic (rubber, polymer, etc) [3]. A model describing these differences would be of considerable interest.Here we consider the somewhat idealized ease of a rubbery body sliding over a crystalline one; the frictional force is deduced as a function of the velocity and other parameters. The surfaces are taken as smooth and clean, while the bodies are homogeneous. Various simplifying assumptions are made, but these are unimportant from the qualitative standpoint.We are indebted to G. I. Barenblatt for a discussion.     

Steady motion of a rigid disk of finite thickness on a horizontal plane

The article discusses the steady motion of a rigid disk of finite thickness rolling on its edge on a horizontal plane under the influence of gravity. The governing equations are presented and two cases allowing for a steady-state solution are considered: rolling on consistently rough ground and rolling on perfectly smooth ground. The conditions of steady motion are derived for both kinds of ground and it is shown that the possible steady motion of a disk is either on a straight line or in a circle. Oscillations about steady state are discussed and conditions for stable motion established. The bifurcations of steady motions on a smooth surface are also considered.     

Approximate symmetries of creeping flow equations of a second grade fluid

Creeping flow equations of a second grade fluid are considered. Two current approximate symmetry methods and a modified new one are applied to the equations of motion. Approximate symmetries obtained by different methods and the exact symmetries are contrasted. Approximate solutions corresponding to the approximate symmetries are derived for each method. Symmetries and solutions are compared and advantages and disadvantages of each method are discussed in detail.     

Regimes of sedimentation of a consolidated system of solid spherical particles

The results of the experimental investigation of gravitational sedimentation of a consolidated system of solid monodisperse spherical particles in a viscous liquid are represented over wide ranges of the particle number density and the Reynolds and Stokes numbers. Empirical dependences of the velocity of sedimentation of a particle aggregate and the drag coefficient of a system of particles as functions of the initial volume number density are obtained. The boundary values of the particle number density separating the sedimentation regimes are determined.     

Stability of the in-plane bending mode of thin-walled framed structures

The paper outlines an approach to solving the stability problem for framed structures under arbitrary transverse loading. The available methods are limited by one law of variation in the bending moment responsible for loss of stability. The equilibrium equations for a thin-walled bar are integrated assuming that the bending moment is constant. The solution of the Cauchy problem is given in normal form. The arbitrary varying bending moment is approximated by a piecewise-constant function, which will be a little different from the original if the bar is partitioned into a great number of segments. The equations of the boundary-value problem for a discretized framed structure are derived using the boundary-element method. The critical forces and moments are determined from a transcendent equation. Numerical solutions are presented to demonstrate the high accuracy and efficiency of the approach. The solutions of test problems are in agreement with those obtained by Timoshenko     

Aeroelastic prediction of the limit cycle oscillations of a cropped delta wing

The flutter and limit cycle oscillation (LCO) behavior of a cropped delta wing are investigated using a newly developed computational aeroelastic solver. This computational model includes a well-validated Euler finite difference solver coupled to a high-fidelity finite element structural solver. The nonlinear structural model includes geometric nonlinearities which are modelled using a co-rotational formulation. The LCOs of the cropped delta wing are computed and the results are compared to previous computations and to experiment. Over the range of dynamic pressures for which experimental results are reported, the LCO magnitudes computed using the current model are comparable to those from a previous computation which used a lower-order von Karman structural model. However, for larger dynamic pressures, the current computational model and the model which used the von Karman theory start to differ significantly, with the current model predicting larger deflections for a given dynamic pressure. This results in a LCO curve which is in better qualitative agreement with experiment. Flow features which were present in the previous computational model such as a leading edge vortex and a shock wave are enhanced in the current model due to the prediction of larger deflections and rotations at the higher dynamic pressures.     

Stability for manifolds of equilibrium states of fractional generalized Hamiltonian systems

For a fractional generalized Hamiltonian system, in terms of Riesz derivatives, stability theory for the manifolds of equilibrium states is presented. The gradient representation and second order gradient representation of a fractional generalized Hamiltonian system are studied, and the conditions under which the system can be considered as a gradient system and a second order gradient system are given, respectively. Then, equilibrium equations, disturbance equations, and first approximate equations of a fractional generalized Hamiltonian system are obtained. A theorem for the stability of the manifolds of equilibrium states of the general autonomous system is used to a fractional generalized Hamiltonian system, and three propositions on the stability of the manifolds of equilibrium states of the system are investigated. As the special cases of this article, the conditions which a fractional generalized Hamiltonian system can be reduced to a generalized Hamiltonian system, a fractional Hamiltonian system and a Hamiltonian system are given, respectively, and the stability theory for the manifolds of equilibrium states of these systems are obtained. Further, a fractional dynamical system and a fractional Volterra model of the three species groups are given to illustrate the method and results of the application. Finally, by using the method in this paper, we construct a new kind of fractional dynamical model, i.e. the fractional Hénon–Heiles model, and we study its stability of the manifolds of equilibrium states.     

Influence of surface tension on the stability of heterogeneous equilibrium of barotropic liquid phases

On the basis of the results of earlier work of the author [1] a study is made of the equilibrium and stability of a two-phase single-component heterogeneous liquid system with respect to perturbations of arbitrary shape. Allowance is made for the influence of surface tension, which plays a critical part in the formation of nucleating centers of a new phase [2]. Conditions of equilibrium are derived, and also a criterion of radial stability of a nucleating center of a new phase bounded by a closed spherical boundary. It is shown that radial stability of spherical nucleating centers also guarantees stability with respect to perturbations of arbitrary shape. The part played by the finite size of the system and the boundary conditions is elucidated. For this, two different cases are studied: a) a system under a constant external pressure, b) a system with fixed volume. In the first case, all equilibrium states are unstable. In the second, there are both unstable and stable configurations (depending on the corresponding values of two dimensionless parameters). The equation of the hyperbola of neutral stability is derived. The limits of a very small coefficient of surface tension and a very large size of the container are considered. The first situation corresponds to stable configurations, the second to unstable. For simplicity, the considered systems are assumed to be isothermal, and the equilibrium and stability are analyzed on the basis of the mechanical analog of Gibbs's principle, namely, the principle of a minimum of the mechanical potential energy of the barotropic heterogeneous liquid system. The case of nonisothermal perturbations leads to similar results, but the expressions for the corresponding dimensionless parameters are more cumbersome and less physically perspicuous.     

Effects of the curvature radius of a cylindrical shell on the stress intensity factors of surface cracks

The effects of the curvature radius of a cylindrical shell on stress intensity factors are investigated in circumferential (inner and outer) semielliptical surface cracks in a cylindrical shell. What is new in this paper is to have given: (1) The stress intensity factors for surface cracks in a cylindircal shell are determined by photoelastic technique. (2) By a special method photoelastic slices are handled for obtaining a clear caustic curve, and the stress intensity factors for surface cracks in a cylindrical shell are determined by the caustic method. (3) An approximate equation of curvature correction factor F_c is proposed. (4) Effects of the curvature radius R of a cylindrical shell on the stress intensity factors of surface cracks are obtained. The results of this paper are in fair agreement with already existing analytical results. The approximate equation of curvature correction factor F_c can be widely used for engineering purposes.     

Integration of the quasilinear equations of motion of a mixture through a porous nondeformable medium

Numerical methods are analyzed of solving the quasilinear system of partial differential equations describing the motion of a sorbed gas (liquid) mixture through a porous, saturated, nondeformable medium consisting of porous grains. Conditions are obtained for convergence of the iteration process of a difference scheme. Conditions are found under which the system attains invariant solutions of the running-wave type. Estimates are obtained of times and coordinates, during which and through whose passage the solutions of the boundary-value problem become invariant.     

Control of chaotic oscillations of a satellite

Analytical conditions and practical methods of their realization are proposed to solve a problem of a command signal tracking for a nonlinear disturbed system.Non- linear disturbed plants consisting of linear dynamic block and nonlinear block in feedback are considered.Nonlinear part of the plant and disturbance are unknown and bounded. The paper illustrates a possibility of applications of proposed algorithms to control libra- tion angle of satellite.     

Direct modeling of flow of FENE fluids

Direct simulations of macromolecular fluids are carried out for flows between parallel plates and in expanding and contracting channels. The macromolecules are modeled as FENE dumbbells with soft disks or Lennard-Jones dumbbell-dumbbell interactions. The results are presented in terms of profiles and contour plots of velocity, pressure, temperature, density, and flow fields. In addition the data for potential energy, shear stress, and the normal components of the stress tensor are collected. In general, an excellent agreement is found between the simulated profiles and the well-known flow structures, such as flow separation and formation of viscous eddies, indicating that micro-hydrodynamics is a viable tool in linking macroscopic phenomena with the underlying physical mechanisms. The simulations are performed in the Newtonian regime, for medium-size systems comprising up to 3888 dumbbells. This number is sufficiently large to control boundary and particle number effects. The flow is induced by gravity. The traditional stochastic (thermal) and periodic boundary conditions are employed. Also, diffusive boundary conditions, which could include a stagnant fluid layer and repulsive potential walls, are developed. The scaling problems, which are related to the application of a large external force in a microscopic system (of the size of the order 100 Å), result in extreme pressure and temperature gradients. In addition, the viscosity and thermal conductivity coefficients obtained from velocity and temperature profiles of the channel flow are presented. These results are confirmed independently from modeling of Couette flow by the SLLOD equations of motion and from the Evans algorithm for thermal conductivity.     

Analytical computation of aerodynamic characteristics of bodies of revolution under locality law conditions

An analytical method of computing the aerodynamic forces acting on a convex body of revolution whose motion satisfies locality law conditions. i.e., the force pulse acting on a surface element depends only on the flow mode and the local angle between the velocity and the normal to the surface, is elucidated. The solution is represented in a form where the influence of the parameters characterizing the shape of the streamlined body, the angle of attack, and the flow mode are explicitly extracted. Universal expressions are obtained for a number of coefficients, which are valid for any body of revolution. The aerodynamic characteristics of a four-parameter family of bodies of revolution in the hypersonic-flow mode are computed in a gas with a different degree of rarefaction.     

Measurement of the Viscoelastic Properties of Bitumen Using Instrumented Spherical Indentation

Indentation testing as a tool for determination of the viscoelastic mechanical properties of bitumen is examined in some detail using theoretical, numerical as well as experimental methods. In particular Brinell indentation is analysed and simple but rigorous formulae for a complete characterization of linear viscoelastic materials are presented. Numerical methods (finite element methods) are used in order to verify and substantiate these relations for an experimental situation. Indentation experiments are then performed on bitumen and special efforts are made in order to avoid size effects, i. e. anomalous results due to the fact that the indented specimens are too small and as a result, far field boundary conditions will influence the interpretation of the experimental output. The mechanical properties determined experimentally by indentation are compared with corresponding results from standard mechanical tests, and the results are encouraging considering the fact that non-linear effects are also influencing the outcome of the experiments.     

Two-dimensional elastica analysis of equilibrium shapes of single-anchor inflatable dams

Several types of inflatable dams are considered. These are long, air-inflated, cylindrical structures on a rigid foundation. Sometimes one of the long edges of a sheet is folded back to the other edge, and then the two edges are clamped to the foundation along a single anchoring line. A second configuration can be modeled as two sheets attached along two long edges, with one edge anchored and the other free to lift as air pressure is applied between the sheets. Another device treated here is a hinged spillway gate lifted by an inflatable bladder. The cross section of the dam or bladder is analyzed as an inextensible elastica. The governing equations and boundary conditions are formulated for each case, and shooting methods are utilized to obtain numerical solutions for the equilibrium shapes. The effects of the internal air pressure and the external water height are investigated.