Euler Equations Of Dynamics Research Articles

GENERALIZATION OF KIRCHHOFF KINETIC ANALOGY TO THIN ELASTIC SHELLS

The Kirchhoff kinetic analogy is generalized from thin elastic rods to thin elastic shells. The generalization makes thin shell deformations physically correspond and mathematically equivalent to rigid body motions. Hence theories and methods of rigid body dynamics can be applied to investigate deformations of thin elastic shell, and also provide a novel discretization for continuous thin elastic shells. An orthogonal spatial axis system is established along the coordinate lines of the middle surface under the straight normal assumption. The moving of the axis system along the coordinate lines in unit velocity forms its angular velocity, which is the curvature-twist vector with two independent variables. The curvature-twist vector along two coordinate lines expresses the deformation and the configuration of a thin elastic shell. It is demonstrated that curvature-twist vectors are compatible, and curvature-twist vectors and tangential vectors of middle surface are compatible. Nonholonomic constraints and differential equations of middle surfaces are established in the Euler angles and the Lam$\acute{e}$ coefficient form. The strain, the stress and the internal forces are formulated in the curvature-twist vectors and the Lam$\acute{e}$ coefficients. The equilibrium partial differential equations are presented with distributed internal forces intensity of thin elastic shells. The forms of the equations are similar to the Euler equations of rigid body dynamics and Kirchhoff equations of thin elastic rods. The fact means that the Kirchhoff kinetic analogy of thin elastic rods is generalized to thin elastic shells. The analogy relations between thin elastic shells and dynamics of rigid body or thin elastic rods are concluded. Finally, an example is given to show the application of this method. The proposed analogy leads to novel views and approaches to model and to analyze deformation of thin elastic shells. It is possible to generalize further the analogy for dynamics of thin elastic shells.

Multiphase SPH Modelling of Supercooled Large Droplets Freezing on Aircraft Surfaces

Solidification is an important issue that needs to be addressed when modelling supercooled large droplets (SLD) impacting aircraft surfaces. This work develops a 3D triple-phase smoothed particle hydrodynamics (SPH) solver that can simultaneously simulate SLD impingement and solidification on cold surfaces by solving the Euler equations for fluid dynamics, the energy equation for heat transfer, and a latent heat model for phase changes. The solver is validated against a Stefan problem in which a liquid is cooled and solidified by a cold boundary, and against a still droplet freezing on a cold plate. It is then applied to droplets impacting on cold surfaces at aeronautical speeds, analysing the freezing time and ice particle distributions as the droplet velocity and impact angle are varied. This is an important step towards a macroscopic numerical model of SLD post-impact behaviour for in-flight icing simulations.

Reply on the comments on “on the dynamics of non-rigid asteroid rotation”

We have presented here our reply on the comments on our recent article “ON THE DYNAMICS OF NON-RIGID ASTEROID ROTATION”. We would restate again: our partial solution stems from fundamental law of angular momentum conservation for the non-rigid asteroid rotation (where asteroids were considered preferably as deformed ‘rubble pile’ asteroids, not as elastic bodies), meanwhile authors of the aforementioned comments have considered other partial solving procedure which can be associated with commensurable elongations of the asteroid during the deformations. This assumption contradicts the physical nature of most known asteroids (‘rubble pile’ asteroids are known to have a non-elastic internal rheology), therefore as such the solutions of Euler equations for dynamics of non-rigid asteroid rotation, obtained by authors of the comments, does not have real meaning in astrophysical sense. Despite the fact that it is useless in astrophysical area, the suggested strategy or solving procedure for the constructing a new class of special kind of solutions for Euler equations (dynamics of non-rigid elastic body's rotation) can be of partial interest to a journals for theoretical mechanics but not first-class astrophysical journals or journal in the field of celestial mechanics. So, we wish the best for the authors to possible publishing their full solution (not a simple a note, which seems to be “following-in-wake” in regard to our article) where they could discuss additionally the stability of solutions if it is deemed required.

Solving the problem of moving a solid body around a fixed point in a resisting medium

In this paper, we consider the problem of motion of an axisymmetric solid around a fixed point in a resistive medium, described by nonlinear dynamic Euler equations. An analytical solution to the problem when the moments of external resistance forces are proportional to the corresponding projections of the angular velocity of the body is obtained by partial discretization of nonlinear differential equations based on the theory of generalized functions. Curves of angular velocity change are constructed and graphs of comparison of these curves with previously known exact solutions are given.

Gyroscopic effect in intermolecular collision with the C60 fullerene molecule

The work is devoted to the study of the gyroscopic phenomena of a rotating C60 fullerene molecule on interaction with a moving particle. To determine the precession characteristics of the C60 fullerene molecule, the pairwise interaction model of nonpolar Lennard-Jones molecules was used. The rotational motion of fullerene was described by the dynamic Euler equations. The influence of the angular velocity and direction of rotation of the fullerene molecule on the deviation from the initial position is investigated.

Solving procedure for the Kelvin–Kirchhoff equations in case of buoyant (or the falling) ellipsoid of rotation

We have presented in this communication a new solving procedure for Kelvin–Kirchhoff equations, considering the dynamics of rising the quasi-rigid ellipsoid of rotation in an ideal incompressible fluid (up to the surface), assuming additionally the dynamical symmetry of rotation for the rising body, I1=I2. Fundamental law of angular momentum conservation has been used for the aforementioned solving procedure. The system of Euler equations for dynamics of non-rigid ellipsoid rotation has been explored in regard to the existence of an analytic way of presentation for the approximated solution (where we suppose that components of fluid’s torque vector {Ti} are approximately proportional to the appropriate components of angular velocity {Ωi}). The results of calculations for the components of angular velocity {Ωi} should then be used for solving momentum equation of Kelvin–Kirchhoff system. Thus, the full system of equations of Kelvin–Kirchhoff problem has been explored with respect to the existence of an analytic way of presentation of the general solution. The last but not least, we have pointed out the 1-st integral of Kelvin–Kirchhoff system under the aforesaid additional assumption for fluid’s torque vector {Ti} when {Ii} = const (but without the additional restriction of dynamical symmetry onto the form of the rising body, I1=I2).

Md-Simulation of Fullerene Rotations in Molecular Crystal Fullerite

The present paper describes rotations of C60 fullerene molecules in the solid phase of a fullerite. The conducted studies show that these relatively large molecules rotate according to the same laws as macroscopic bodies, i.e., according to the laws of classical mechanics. The performed calculations confirm that fullerene rotations do not cause friction. We suggest a method for a strong increase in the internal energy of the material that does not lead to its destruction. It is theoretically shown that in standard fullerite, in the absence of electric and magnetic fields, fullerene rotations occur with an average angular frequency of 0.34·× 1012 rad·s−1, which is consistent with the experimental data obtained using nuclear magnetic resonance. By means of calculations, we found that alternating magnetic fields of a certain configuration wind fullerenes encapsulated by iron. In this case, two temperatures arise in the fullerite crystal: a high rotational temperature and a vibrational temperature close to normal. For the purpose of determining this velocity, as well as the nature of rotations, the present paper suggests a way of integrating the dynamic Euler equations for the projections of a molecule’s angular velocity vector onto the coordinate axes associated with the fullerene. The stages of computer simulation of fullerene movements, which was carried out without using previously developed packages of molecular-dynamic modelling, are consistently described.

The Effect of Random Wind Forcing in the Nonlinear Schrödinger Equation

The influence of a strong and gusty wind field on ocean waves is investigated. How the random wind affects solitary waves is analyzed in order to obtain insights about wave generation by randomly time varying wind forcing. Using the Euler equations of fluid dynamics and the method of multiple scales, a random nonlinear Schrödinger equation and a random modified nonlinear Schrödinger equation are obtained for randomly wind forced nonlinear deep water waves. Miles theory is used for modeling the pressure variation at the wave surface resulting from the wind velocity field. The nonlinear Schrödinger equation and the modified nonlinear Schrödinger equation are computed using a relaxation pseudo spectral scheme. The results show that the influence of gusty wind on solitary waves leads to a randomly increasing ocean wave envelope. However, in a laboratory setup with much smaller wave amplitudes and higher wave frequencies, the influence of water viscosity is much higher. This leads to fluctuating solutions, which are sensitive to wind forcing.

Dispersion of stratified dust layers by a moving shock wave

Simulations of a shock passing over stratified dust layers containing two different types of particles were conducted to explore the effect of particle size and dust layer thickness on particle dispersion with application to coal mine explosion conditions. The simulations solve the Euler equations of fluid dynamics and granular flow, and accounts for different particle types using a binning approach. Test cases with a 2.7-mm layer consisting of particles with a smaller diameter (ds,2=10, 20, and 40 µm) placed on top of a 10-mm layer consisting particles with a larger diameter (ds,1=80 µm) were performed to study the effect of particle size on dust dispersion. The results indicate that larger particles from the lower layer can be lifted higher than the smaller particles from the upper layer if the two types of particles have a large size difference. In addition, dust dispersion of a shock passing over a rock-dust layer on top of a coal-dust layer were also investigated. Here the coal dust has a diameter of 30 µm and a density of 1330 kg/m3, and the rock dust has a diameter of 15 µm and a density of 2680 kg/m3. The rock-layer thickness of 1, 2, and 3 mm were considered, and the coal-dust layer underneath had a thickness of 4 mm. The results suggest that the rock particles work more effectively on suppressing the coal dust from lifting with increasing rock-layer thickness. Placing another layer of coal dust on top of the rock-coal layer shows that the rock dust underneath fails to suppress the coal particles above from being lifted, and this suggests the need to simultaneously or regularly apply rock dust during the mining operation.

Emergence of Self-Excited Oscillations in Flows of Inviscid Fluids in a Channel

The mechanism of self-excited oscillations arising in an ideal incompressible fluid flowing through a rectangular channel is studied numerically. The problem is formulated in the form of the Euler equations for ideal fluid dynamics in terms of vorticity and stream function with Yudovich’s boundary conditions. The vorticity intensity at the inlet of the channel is used as a bifurcation parameter. Grid approximations are employed to search for steady-state regimes and to analyze their stability, while the nonstationary problem is solved by applying the vortex-in-cell method. It is shown that a steady flow through the channel is established when the vorticity intensity at the inlet is low. As the vorticity intensity at the inlet grows, the steady-state regime becomes unstable in an oscillatory manner and self-excited oscillations emerge in its neighborhood. The evolution of the self-excited oscillations with an increasing bifurcation parameter is studied. In the case of high supercriticality, a chaotic flow regime is observed in the channel.

On the dynamics OF NON-RIGID asteroid rotation

We have presented in this communication a new solving procedure for the dynamics of non-rigid asteroid rotation, considering the final spin state of rotation for a small celestial body (asteroid). The last condition means the ultimate absence of the applied external torques (including short-term effect from torques during collisions, long-term YORP effect, etc.).Fundamental law of angular momentum conservation has been used for the aforementioned solving procedure. The system of Euler equations for dynamics of non-rigid asteroid rotation has been explored with regard to the existence of an analytic way of presentation of the approximated solution.Despite of various perturbations (such as collisions, YORP effect) which destabilize the rotation of asteroid via deviating from the current spin state, the inelastic (mainly, tidal) dissipation reduces kinetic energy of asteroid. So, evolution of the spinning asteroid should be resulting by the rotation about maximal-inertia axis with the proper spin state corresponding to minimal energy with a fixed angular momentum.Basing on the aforesaid assumption (component K1 is supposed to be fluctuating near the given appropriate constant of the fixed angular momentum), we have obtained that 2-nd component K2 is the solution of appropriate Riccati ordinary differential equation of 1-st order, whereas component K3 should be determined via expression for K2.

Sub-Symmetries and Conservation Laws

The concept of sub-symmetry of a differential system was introduced in [ 40 ], where it was shown that a sub-symmetry is a considerably more powerful tool than a regular symmetry with regard to deformation of conservation laws. In this paper, we study the nature of a correspondence between sub-symmetries and conservation laws of a differential system. We show that for a large class of non-Lagrangian systems, there is a natural association between sub-symmetries and local conservation laws based on the Noether operator identity, and we prove an analogue of the first Noether theorem for sub-symmetries. We also demonstrate that a similar association can be established for infinite sub-symmetries of the system. We discuss the role of sub-symmetries in generation of infinite series of conservation laws for the constrained Maxwell system and the incompressible Euler equations of fluid dynamics. Despite the fact that infinite symmetries (with arbitrary functions of dependent variables) are not known for the Euler equations, we find infinite sub-symmetries for the two- and three-dimensional Euler equations with certain constraints. We show that these sub-symmetries generate known series of infinite conservation laws, and obtain new classes of infinite conservation laws.

Algorithm of the Time-Optimal Reorientation of an Axially Symmetric Spacecraft in the Class of Conical Motions

The problem of the time-optimal turn of a spacecraft as a rigid body with one axis of symmetry and bounded control function in absolute value is considered in the quaternion statement. For simplifying problem (concerning dynamic Euler equations), we change the variables reducing the original optimal turn problem of axially symmetric spacecraft to the problem of optimal turn of the rigid body with spherical mass distribution including one new scalar equation. Using the Pontryagin maximum principle, a new analytical solution of this problem in the class of conical motions is obtained. Algorithm of the optimal turn of a spacecraft is given. An explicit expression for the constant in magnitude optimal angular velocity vector of a spacecraft is found. The motion trajectory of a spacecraft is a regular precession. The conditions for the initial and terminal values of a spacecraft angular velocity vector are formulated. These conditions make it possible to solve the problem analytically in the class of conical motions. The initial and the terminal vectors of spacecraft angular velocity must be on the conical surface generated by arbitrary given constant conditions of the problem. The numerical example is presented. The example contain optimal reorientation of the Space Shuttle in the class of conical motions.

Feedback Liniarization Method for Problem of Control of a Part of Variables in Uncontrolled Disturbances

The paper studies a problem of guaranteed transfer within a finite amount of time of a nonlinear dynamical system subjected to uncontrolled disturbances to a state where a given part of the variables equals zero. The bounded controls are offered to be generated by means of a feedback in form of nonlinear functions of phase variables of a given nonlinear controlled system of differential equations. The method of exact feedback linearization of the nonlinear system is used. As a result, the solution of the original nonlinear problem is narrowed down to solve the linear game-theoretic antagonistic control problem. Sufficient conditions are obtained with ensure that the problem has a guaranteed solution for the given domain of initial conditions. 
 As an example, problem of the space turn of an asymmetric rigid bode (spacecraft) is considered within the framework of the method. Three reaction wheels are employed to produce necessary torque in the axes of the spacecraft. External uncontrolled disturbances, that have no statistical description, are taken into consideration in the process of reorientation. In this case the initial nonlinear controlled systems consists of dynamic Euler equations and Rodriges – Hamilton kinematic equations based on the quaternion parameterization of attitude kinematics. Two problems of the space turn of the spacecraft are considered. 1) The rest - to - rest reorientation problem. 2) The space turn from a stationary state to a given angular position; it is not assumed that the turn takes the spacecraft to a stationary state. The proposed approach allows common positions to give some already well-known solutions of these problems. A new solution of the reorientation problem is also given. For this new solution an estimation of the admissible domain of uncontrolled disturbances is found. Results of a numerical calculations are considered.

Geometric effects in the design of catalytic converters in car exhaust pipes

IntroductionWe solve the gas dynamics (Euler) equations, augmented by adding a fourth equation governing the fraction of unburnt gas, in a number of cylindrically symmetric configurations of the pipe system.Case descriptionThe purpose is to test several duct profiles to see which one favors a higher reduction of the residual noxious gases, at the end of a car’s exhaust pipe.Discussion and evaluationIt is found that this purely geometric factor does play a role in the environment’s purification accomplished by the catalytic converter. This is possibly due to the longer time spent by the noxious gases resident inside the device when this has certain profiles, though at the price of a little higher temperature attained.ConclusionsIt seems that geometric factors play a role in reducing cars’ noxious gases by means of catalytic converters. A more precise analysis should be formulated as a mathematical inverse problem.

High-order residual distribution scheme for the time-dependent Euler equations of fluid dynamics

In the present work, a high order finite element type residual distribution scheme is designed in the framework of multidimensional compressible Euler equations of gas dynamics. The strengths of the proposed approximation rely on the generic spatial discretization of the model equations using a continuous finite element type approximation technique, while avoiding the solution of a large linear system with a sparse mass matrix which would come along with any standard ODE solver in a classical finite element approach to advance the solution in time. In this work, we propose a new Residual Distribution (RD) scheme, which provides an arbitrary explicit high order approximation of the smooth solutions of the Euler equations both in space and time. The design of the scheme via the coupling of the RD formulation (Ricchiuto and Abgrall, 2010; Abgrall, 2006) with a Deferred Correction (DeC) type method (Liu et al., 2008; Minion, 2003), allows to have the matrix associated to the update in time, which needs to be inverted, to be diagonal. The use of Bernstein polynomials as shape functions, guarantees that this diagonal matrix is invertible and ensures strict positivity of the resulting diagonal matrix coefficients. This work is the extension of Abgrall et al. (2016) and Abgrall (2017) to multidimensional systems. We have assessed our method on several challenging benchmark problems for one- and two-dimensional Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.

Analytical Solution of the Minimum Time Slew Maneuver Problem for an Axially Symmetric Spacecraft in the Class of Conical Motions

The conventional problem of the time-optimal slew of a spacecraft considered as a solid body with a single symmetry axis subject to arbitrary boundary conditions for the attitude and angular velocity is considered in the quaternion statement. By making certain changes of variables, the original dynamic Euler equations are simplified, and the problem turns into the optimal slew problem for a solid body with a spherical distribution of mass containing one additional scalar differential equation. For this problem, a new analytical solution in the class of conical motions is found; in this solution, the initial and terminal attitudes of the space vehicle belong to the same cone realized under a bounded control. A modification of the optimal slew problem in the class of generalized conical motions is made that makes it possible to obtain its analytical solution under arbitrary boundary conditions for the attitude and angular velocity of the spacecraft. A numerical example of a spacecraft’s conical motion and examples demonstrating the proximity of the solutions of the conventional and modified optimal slew problems of an axially symmetric spacecraft are discussed.

Adaptive refinement strategies for the simulation of gas flow in networks using a model hierarchy

A model hierarchy that is based on the one-dimensional isothermal Euler equations of fluid dynamics is used for the simulation and optimisation of natural gas flow through a pipeline network. Adaptive refinement strategies have the aim of bringing the simulation error below a prescribed tolerance while keeping the computational costs low. While spatial and temporal stepsize adaptivity is well studied in the literature, model adaptivity is a new field of research. The problem of finding an optimal refinement strategy that combines these three types of adaptivity is a generalisation of the unbounded knapsack problem. A refinement strategy that is currently used in gas flow simulation software is compared to two novel greedy-like strategies. Both a theoretical experiment and a realistic gas flow simulation show that the novel strategies significantly outperform the current refinement strategy with respect to the computational cost incurred.

From the Kinematics of Precession Motion to Generalized Rabi Cycles

We use both vector-parameter and quaternion techniques to provide a thorough description of several classes of rotations, starting with coaxial angular velocity Ω of varying magnitude. Then, we fix the magnitude and let Ω precess at constant rate about the z-axis, which yields a particular solution to the free Euler dynamical equations in the case of axially symmetric inertial ellipsoid. The latter appears also in the description of spin precessions in NMR and quantum computing. As we show below, this problem has analytic solutions for a much larger class of motions determined by a simple condition relating the polar angle and z-projection of Ω (expressed in cylindrical coordinates), which are both time-dependent in the generic case. Relevant physical examples are also provided.

Alternative Description of Rigid Body Kinematics and Quantum Mechanical Angular Momenta

In the present paper we investigate an alternative two-axes decomposition method for rotations that has been proposed in our earlier research. It is shown to provide a convenient parametrization for many important physical systems. As an example, the kinematics of a rotating rigid body is considered and a specific class of solutions to the Euler dynamical equations are obtained in the case of symmetric inertial ellipsoid. They turn out to be related to the Rabi oscillator in spin systems well known in quantum computation. The corresponding quantum mechanical angular momentum and Laplace operator are derived as well with the aid of infinitesimal variations. Curiously, the coefficients in this new representation happen to depend only on one of the angles, which simplifies the corresponding system of ODE's emerging from separation of variables. Some applications of the hyperbolic and complex analogues of this construction in quantum mechanics and relativity are considered in a different paper cited below.