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A rotary motion is studied, which is used as a paradigm in both mathematics and dynamics. In this motion, the body moves around its axis. Important examples of this motion are the earth, wheels for cars or bicycles, cooling fans, airplanes, radars, and motion of the discs. In this article, we consider one of these motions such as the slow rotational motion of a disc of high natural frequency around a fixed point different from its center of mass. Suppose a constant gyrostatic couple acts on the body about the axis of symmetry. Let us consider the axis of symmetry coincide with one of the main axes of inertia. The controlled nonlinear independent system of equations of motion is obtained in the form of six nonlinear differential equations in the presence of three independent first integrals. This system is reduced to an alternative independent quasi-linear system consisting of two differential equations with only one integral. At first, when the body rotates slowly with a small angular speed about the axis of the symmetry, we achieve a large parameter for the motion which is proportional to the inverse of the angular velocity component. Then, we use the large parameter technique to obtain approximate and high-frequency analytic solutions for this motion. This technique enables us to introduce new conditions of the motion, save a lot of energy required to move the disc, and obtain new approximated solutions in a new domain of the disc parameters. We present a digital example of the problem to clarify the inferences of the motion depending on the parameters of the body. On the other hand, we use the Runge–Kutta algorithmic method to obtain the numerical solutions corresponding to the approximate solutions of the considered independent system. We investigate the inaccuracies and errors of analytic and numerical solutions by comparing them. This problem offers many applications in gyroscope dynamics, industrial engineering, astrophysics, navigation, satellites, and space engineering. [ABSTRACT FROM AUTHOR]

Stationary irrotational barotropic gas flows are investigated on the basis of the analysis of three-dimensional Euler equations. Critical flows in the article are those in which the Mach number is everywhere less than or equal to one, and at least at one point the Mach number reaches one. In 1954, D.Gilbarg and M.Shiffman showed that if an internal (not lying on the streamlined surface) sonic point exists in a critical flow, then it lies on a flat sonic surface, which at all its points is perpendicular to the gas velocity vector and cannot end inside the flow (theorem about the sonic point). Using this theorem, D.Gilbarg and M.Shiffman obtained a conclusion that is important for the problems of maximizing the critical Mach number. It consists in the fact that in a critical flow for a wide class of bodies in flow, sonic points can be located only on its surface. This conclusion is essentially used in constructing the shapes of streamlined bodies with the maximum value of the critical Mach number (for given isoperimetric conditions). In this paper, the question of the curvature of streamlines at the internal sonic points of critical flows is considered. It is shown that this curvature is zero. The result is a new necessary condition for the existence of an interior sonic point (and sonic surface). It consists in the fact that at the point of intersection with the sonic surface, the normal curvature of the streamlined surface in the direction normal to the sonic surface should be equal to zero. Examples of streamlined bodies are given for which the theorem by D. Gilbarg and M. Shiffman (on the sonic point) does not answer the question of the location of the sonic points, at the same time a new necessary condition makes it possible to prove that the existence of internal sonic points in a critical flow around these bodies is impossible.

We study axially symmetric Euler's system of equations for the real gas in two space dimensions. This problem poses various difficulties for arbitrary equation of state and general problem is still open. Deviation from ideal gas law generally involves change of phase and more number of chemical species. This affects number of degrees of freedom which is governed by phase rule. We analyze self-similar solutions of Euler's system of equations for the real gas (alternatively called non-ideal gas) which is in one phase but may contain vapor or moisture. We find that structure of solutions in this case is governed by fundamental derivative of underlying gas. Emergence of fundamental derivative in calculations carried out by the authors suggest that structure of solutions to Euler's equations in gas dy namics is fundamentally dependent on the thermodynamics of underlying gas. We have considered case of zero swirl in this article. [ABSTRACT FROM AUTHOR]

The purpose of this study is essentially pedagogical and aims to provide an additional argument in clarification of a question often raised by first-year undergraduate mechanical engineering students concerning the reason for using two frames of reference—one fixed in space and one fixed in the rigid-body—to describe its motion. The reasoning employed to illustrate the inappropriateness of using a single reference frame entails showing that the equations of motion, thus obtained, are far more complex than the equations resulting from application of the traditional Euler Method. This point is illustrated through the well-known frictionless symmetrical spinning top problem. [ABSTRACT FROM AUTHOR]

We study, in a unified manner the motion of a uniformly advancing piston in the planar ( m = 0 ), cylindrically symmetric ( m = 1 ) and spherically symmetric ( m = 2 ) isentropic flow governed by Euler's equations. In the three dimensional space the piston is replaced by an expanding cylinder or a sphere which produces a motion in the medium outside. Under the hypothesis of self symmetry pair of ordinary differential equations is derived from Euler's equations of fluid flow. Based on the analysis of critical solutions of this system of ordinary differential equations, unified treatment of motion of uniformly advancing piston is given with different geometries viz. planar, cylindrically symmetric and spherically symmetric flow configuration. In case of nonplanar flows, solutions in the neighbourhood of nontransitional critical points approach these critical points in finite time. Transitional solutions exist in nonplanar cases which doesn't correspond to the motion of a piston. In the planar case, it is found that solutions take infinite time to reach critical line. [ABSTRACT FROM AUTHOR]

The results of numerical simulation of the interaction of supersonic flow with a permeable screen in form of an infinite plane cascade (lattice) of circular cylinders are given. The interaction regime in which the shocks ahead of the cylinders are localized on the scale of the cascade step is considered. The multi-block computational technique in which the viscous boundary layers are resolved by means of local grids using the Navier–Stokes equations, while the effects of inteferrence between the shock-wave structures in supersonic wake are described within the framework of Euler's equations. The action of shock waves induced by the neighboring elements of lattice to the near-wake region behind the intermediate elements can ambiguously affect the aerodynamic lattice performance as well as generate time-dependent phenomena in the wake. The flow regimes are classified depending on continuous increase and decrease in the free-stream supersonic air flow in the Mach number range from 2.4 to 4.2 with reference to the lattice of the 80% permeability. The sources of the hysteresis behavior of the lattice aerodynamic drag with respect to the Mach number and the mechanisms of the onset of self-oscillating wake flow regimes are discussed. [ABSTRACT FROM AUTHOR]

The work is devoted to the study of gyroscopic phenomena in the interaction of a rotating fullerene molecule and a xenon atom incident on it. The methods of classical molecular physics are used: intermolecular potentials, Newton's equations for describing the motion of particles, and the Runge–Kutta numerical method of high order of accuracy. A mathematical model is constructed and implemented for the rotation frequencies of fullerene up to 1014 Hz and the speed of the incident xenon atom of the order of 103 m s−1. For such parameters of the problem, the de Broglie wavelength of the incident atom and the fullerene molecule become smaller than the diameter of the carbon atomic nucleus. This made it possible to apply the Newtonian approach without involving quantum mechanics. The aim of this work is the consistent application of the apparatus of classical mechanics to reveal the effect of the precession of rotating fullerene inside fullerite. [ABSTRACT FROM AUTHOR]

In recent years, there has been an increased use of large displacement vessels for cargo transport around the world. Due to space constraints in the matured ports, the clearance available between the approach channels and berths could not be increased or modified. Hence, the vessels passing in close vicinity to the berthed vessels has become a routine. It is essential to study the passing vessel and berthed vessel interaction for ensuring safety so that the mooring systems could be designed for the increased forces. In this paper, a study envisaging such interaction using Computational Fluid Dynamics (CFD) simulations has been attempted to determine the passing vessel effect on a moored tanker. The CFD solution used is based on a three-dimensional, unsteady, inviscid flow. Initially, the CFD simulation has been carried out for existing experimental data from published paper for validation and has been extended to a range of parameters such as passing vessel displacement, velocity, separation distance and water depth. The effect of each parameter on passing vessel forces is explained in detail. It is observed from the results that the longitudinal and transverse forces on the moored vessel become negligible for separation distance more than three times the width of the vessel. Also, the study provides a proper insight into the effect of full hull forms of various displacements on the passing vessel forces. [ABSTRACT FROM AUTHOR]

In this article, we propose a mathematical model using the port-Hamiltonian formalism for a satellite’s three-axis attitude system comprising fluid rings. Fluid rings are an alternative to reaction wheels used for the same purpose, since, for the same mass, they can exert a greater torque than a reaction wheel as the fluid can circulate the periphery of the satellite. The port-Hamiltonian representation lays the foundation for a posterior controller that is feasible, stable, and robust based on the interconnection of the system to energy shaping and/or damping injection components, and by adding energy routing controllers. The torques exerted by the fluid rings are modeled using linear regression analysis on the experimental data got from a prototype of a fluid ring. Since the dynamics of turbulent flows is complex, the torques obtained by the prototype lead to a simpler first approach, leaving its uncertainties to a controller. Thus, the attitude system model could be tested in a future prototype before considering a spatial environment.

The problem about the interaction between a planar shock wave and cylinders of different mass is considered; the cylinders may move under the action of the pressure forces. This problem qualitatively corresponds to the problem about particle relaxation behind a shock wave. The mathematical model is based on the two-dimensional system of Euler equations. The computational algorithm is based on the Cartesian grid method for calculating the flows with the shock waves in the domains with varying geometry. The algorithm and its program realization are tested on the problem about the cylinder rising behind the shock wave. The curves of variation in the cylinder velocity are drawn, and the explanation on the qualitative form of the curves for a different cylinder mass is given. For a particular mass, the dynamics of the relaxation process are analyzed from the viewpoint of the transient shock-wave patterns realized in the interaction between the shock wave and the cylinder. [ABSTRACT FROM AUTHOR]

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The motion of a rigid body is described in Classical Mechanics with the venerable Euler's equations which are based on the assumption that the relative distances among the constituent particles are fixed in time. Real bodies, however, cannot satisfy this property, as a consequence of thermal fluctuations. We generalize Euler's equations for a free body in order to describe dissipative and thermal fluctuation effects in a thermodynamically consistent way. The origin of these effects is internal, i.e. not due to an external thermal bath. The stochastic differential equations governing the orientation and central moments of the body are derived from first principles through the theory of coarse-graining. Within this theory, Euler's equations emerge as the reversible part of the dynamics. For the irreversible part, we identify two distinct dissipative mechanisms; one associated with diffusion of the orientation, whose origin lies in the difference between the spin velocity and the angular velocity, and one associated with the damping of dilations, i.e. inelasticity. We show that a deformable body with zero angular momentum will explore uniformly, through thermal fluctuations, all possible orientations. When the body spins, the equations describe the evolution towards the alignment of the body's major principal axis with the angular momentum vector. In this alignment process, the body increases its temperature. We demonstrate that the origin of the alignment process is not inelasticity but rather orientational diffusion. The theory also predicts the equilibrium shape of a spinning body. • Euler's equations are generalized to include dissipation and thermal fluctuations. • Thermal fluctuations induce change in orientation of small non-spinning bodies. • Nutation damping leading to re-orientation of the body to its major axis is captured by the theory. • Nutation damping results from orientational diffusion, not body viscoelasticity. [ABSTRACT FROM AUTHOR]

This work introduces a hybridizable discontinuous Galerkin formulation for the simulation of ideal plasmas governed by the Euler–Maxwell equations. The approach is based on a monolithic source-based coupling of the fluid and electromagnetic subproblems, along with a fully implicit time integration scheme, and a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. The numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation and validate it against problems of increasing complexity, ranging from single-physics cases to weakly and fully coupled electromagnetic plasma simulations. • Novel Hybridizable Discontinuous Galerkin formulation for Euler-Maxwell plasma model. • Monolithic source-based coupling and fully implicit time integration scheme. • Projection-based divergence correction method to enforce Gauss' laws. • Simulation of high-frequency plasma phenomena. • Accuracy, efficiency and robustness demonstrated on problems of increasing complexity. [ABSTRACT FROM AUTHOR]

This work focuses on the motion of a dynamical model that consists of a symmetric rigid body (gyro) that rotates about a fixed point similar to Lagrange’s gyroscope. This body is acted upon by external forces represented by a Newtonian force field, gyro torques about the principal axes of inertia of the gyro and perturbing moments acting on the same axes. Assuming that, the gyro initially has a high angular velocity about the dynamic axis of symmetry. The averaging technique is used to obtain a more appropriate averaging system for the governing system of equations of motion in terms of a small parameter. Therefore, the analytical solutions of this system for two applications, depending on different forms of perturbing moments, are presented. These solutions are represented graphically to clarify the effectiveness of the different parameters of the body on the motion. [ABSTRACT FROM AUTHOR]

The recently proposed low degree-of-freedom model of Moffatt and Kimura [1,2] for describing the approach to finite-time singularity of the incompressible Euler fluid equations is investigated. The model assumes an initial finite-energy configuration of two vortex rings placed symmetrically on two tilted planes. The Hamiltonian structure of the inviscid limit of the model is obtained. The associated noncanonical Poisson bracket [3] and two invariants, one that serves as the Hamiltonian and the other a Casimir invariant, are discovered. It is shown that the system is integrable with a solution that lies on the intersection for the two invariants, just as for the free rigid body of mechanics whose solution lies on the intersection of the kinetic energy and angular momentum surfaces. Also, a direct quadrature is given and used to demonstrate the Leray form for finite-time singularity in the model. To the extent the Moffatt and Kimura model accurately represents Euler's ideal fluid equations of motion, we have shown the existence of finite-time singularity. • A low degree-of freedom model for the interaction of two tilted vortex rings is described. • Noncanonical Hamiltonian structure is described, yielding two previously unknown invariants, the Hamiltonian and Casimir. • The Hamiltonian system is integrable providing analytical and numerical solutions for all initial conditions. • Leray-like solutions that diverge in finite time are given, suggesting finite-time singularity for Euler's fluid equations. [ABSTRACT FROM AUTHOR]

The aim of this work is to investigate the analytical solutions for the equations of motion of a rigid body about a fixed point through the process of decoupling Euler's dynamic equations. This body is acted upon by a gyrostatic torque ℓ = (ℓ1, ℓ2, ℓ3) about the axes of rotation, and in the presence of a moment about the same axes, it depends on an external loading in which its components have been expressed as a harmonic function of time. The achieved analytical solutions for the equations of motion are obtained under conditions consistent with the physical nature of the body, and the uniqueness of the solution is proved. Some new theoretical applications are presented when the body is symmetric about one of its principal axes and when the body is in complete symmetry. The graphical representations for the motion of the body are represented to show the effectiveness of the physical parameters of the body. Moreover, the phase plane plots are given to ensure that the considered motion is free of chaos. [ABSTRACT FROM AUTHOR]

The purpose of the present work is to use a finite volume method for solving Euler equations in the presence of shocks and discontinuities, with a generalized equation of state. This last choice allows to treat both compressible and incompressible fluids. The first results of the work are presented. They consist in simulating two-dimensional single-specie flows in the presence of shocks. The results obtained are compared with the analytical results considered as benchmarks in the domain.

In this work, the rotational motion of a body fixed at one point is investigated when the effectiveness of a Newtonian field is taken into consideration. It is assumed that the center of mass of the body lies on one of the principal planes of inertia. The attained solutions are obtained when the body carrying rotating masses represented by a flywheel or may be multiply-connected cavities containing circulated liquid. Two special cases are evaluated. The Runge-Kutta algorithms from fourth order is applied to achieve the numerical solution of the equations of motion and represented graphically to show the variation of the body’s parameters on the motion. The importance of this work is due to its wide applications in many fields such as in physics, engineering and industrial applications. [ABSTRACT FROM AUTHOR]

Euler's equations for a two-dimensional fluid can be written in the Hamiltonian form, where the Poisson bracket is the Lie-Poisson bracket associated with the Lie algebra of divergence-free vector fields. For the two-dimensional hydrodynamics of ideal fluids, we propose a derivation of the Poisson brackets using a reduction from the bracket associated with the full algebra of vector fields. Taking the results of some recent studies of the deformations of Lie-Poisson brackets of vector fields into account, we investigate the dispersive deformations of the Poisson brackets of Euler's equation: we show that they are trivial up to the second order. [ABSTRACT FROM AUTHOR]

The generation of acoustic-gravity waves and their propagation in the Earth's atmosphere is analyzed numerically on the basis of a computer model of the stratified atmosphere with dissipation. Atmospheric and ionospheric wavelike disturbances from different surface sources such as earthquakes, explosions, seiches, temperature heating, and tsunamis are studied. [ABSTRACT FROM AUTHOR]

Abstract: It is shown that the three nonlinear dynamic Euler ordinary differential equations (ODEs), concerning the motion of a rigid body free to rotate about a fixed point, are reduced, by means of a subsidiary function which is to be determined, to three Abel equations of the second kind of the normal form. Based on a recently developed mathematical construction concerning exact analytic solutions of the Abel nonlinear ODEs of the second kind, we perform a new mathematical solution for the classical dynamic Euler nonlinear ODEs. [Copyright &y& Elsevier]

Abstract: We present artificial boundary conditions for the numerical simulation of compressible flows using high-order accurate discretizations with the discontinuous Galerkin (DG) finite element method. The construction of the proposed boundary conditions is based on characteristic analysis and applied for boundaries with arbitrary shape and orientation. Numerical experiments demonstrate that the proposed boundary treatment enables to convect out of the computational domain complex flow features with little distortion. In addition, it is shown that small-amplitude acoustic disturbances could be convected out of the computational domain, with no significant deterioration of the overall accuracy of the method. Furthermore, it was found that application of the proposed boundary treatment for viscous flow over a cylinder yields superior performance compared to simple extrapolation methods. [Copyright &y& Elsevier]

We present a novel, cell-local shock detector for use with discontinuous Galerkin (DG) methods. The output of this detector is a reliably scaled, element-wise smoothness estimate which is suited as a control input to a shock capture mechanism. Using an artificial viscosity in the latter role, we obtain a DG scheme for the numerical solution of nonlinear systems of conservation laws. Building on work by Persson and Peraire, we thoroughly justify the detector’s design and analyze its performance on a number of benchmark problems. We further explain the scaling and smoothing steps necessary to turn the output of the detector into a local, artificial viscosity. We close by providing an extensive array of numerical tests of the detector in use. [ABSTRACT FROM PUBLISHER]

The application of the optimized dispersion relation preserving scheme (DRP-scheme) in combination with the explicit optimized two-layer Runge-Kutta scheme is presented to solve a system of one-dimensional and quasi-one-dimensional Euler equations using as an example the solution of four test problems, namely, discontinuity disintegration in a tube (Sod's problem); transfer of the lowamplitude Gaussian pulse; acoustic wave propagation through the transonic nozzle; acoustic wave-shock interaction. Also given are the comparison of the calculation results using different schemes: DRP, CABARET, CE-SE and the standard Lax-Wendroff schemes as well as the solutions obtained with the use of software packages. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for the nonlinear Laplace operator with data on a part of boundary. To this end, we use purely nonlinear methods, such as successive iterations of a Zaremba-type mixed boundary value problem for the p-Laplace operator. [ABSTRACT FROM AUTHOR]

We discuss the Cauchy problem for the so-called Chaplygin system which often appears in gas, aero- and hydrodynamics. This system can be thought of as a nonlinear analogue of the Cauchy-Riemann system in the plane. We pose Cauchy data on a part of the boundary and apply variational approach to construct a solution to this ill-posed problem. The problem actually gives insight to fundamental questions related to instable problems for nonlinear equations. [ABSTRACT FROM AUTHOR]

This paper is aimed at presenting a method to generate online jumping patterns, which can be applied to one-legged jumping robots and optionally to humanoid robots. Our proposed method is based on ensuring the overall dynamic balance through the complete jumping cycle. To be able to reach this goal, we discretized the zero moment point equation in polar coordinates so that we are able to include angular momentum information in a natural way. Thus, undesired torso angle fluctuation is expected to be more restrainable compared to other methods in which angular momentum information is ignored or zero referenced. Moreover, we unified support and flight phases in terms of motion generation. Having obtained successful simulation results and vertical jumping experiments in our previous work, we conducted forward jumping experiments. As the result, we obtained successful and repetitive jumping cycles, which satisfactorily verify the proposed method. [ABSTRACT FROM AUTHOR]

We discuss the relaxation of a class of nonlinear elliptic Cauchy problems with data on a piece S of the boundary surface by means of a variational approach known in the optimal control literature as “equation error method”. By the Cauchy problem is meant any boundary value problem for an unknown function y in a domain χ with the property that the data on S, if combined with the differential equations in χ, allow one to determine all derivatives of y on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy–Kovalevskaya theorem. We also admit overdetermined elliptic systems, in which case the set of those Cauchy data on S for which the Cauchy problem is solvable is very “thin”. For this reason we discuss a variational setting of the Cauchy problem which always possesses a generalised solution. [ABSTRACT FROM AUTHOR]

A wide class of linear ordinary differential equations reducible to algebraic equations is studied. The method for solving all these equations is given. The new class depends essentially on two arbitrary functions and contains the constant coefficient equations and Euler's equations as particular cases. [ABSTRACT FROM AUTHOR]

We consider a simplified acoustic model to describe nonlinear phenomena occurring in loudspeakers. The first simplification is that we restrict to the one-dimensional isentropic Euler equations in a slab, where on the right end a membrane is moving periodically with frequency ω and maximal displacement ε ≪ 1. Moreover we apply a perturbation method to the nonlinear model based on the small parameter ε, which yields linear hyperbolic first order systems coupled by nonlinear source terms of lower order. The asymptotic model is investigated numerically for two different frequencies ω. [ABSTRACT FROM AUTHOR]

In this article, we propose a mathematical model using the port-Hamiltonian formalism for a satellite's three-axis attitude system comprising fluid rings. Fluid rings are an alternative to reaction wheels used for the same purpose, since, for the same mass, they can exert a greater torque than a reaction wheel as the fluid can circulate the periphery of the satellite. The port-Hamiltonian representation lays the foundation for a posterior controller that is feasible, stable, and robust based on the interconnection of the system to energy shaping and/or damping injection components, and by adding energy routing controllers. The torques exerted by the fluid rings are modeled using linear regression analysis on the experimental data got from a prototype of a fluid ring. Since the dynamics of turbulent flows is complex, the torques obtained by the prototype lead to a simpler first approach, leaving its uncertainties to a controller. Thus, the attitude system model could be tested in a future prototype before considering a spatial environment. [ABSTRACT FROM AUTHOR]