Euler Problem Research Articles

Thermophoresis of Graphene Plates

Thermophoresis of graphene plates in an air medium is discussed within the framework of a molecular-kinetic approach. Its rate is found to be independent of the plate area and the aspect ratio of a rectangular graphene. It does depend on the plate orientation in space, which is controlled by the principle of least resistance. The dependence is expressed via a geometrical parameter σ, whose limiting values within the variation interval are found to be 0.46 and 0.65. A solution of the Euler problem on the Brownian rotation of a plate around its center of mass as a result of collisions of molecules in the temperature gradient field allowed us to obtain for the graphene plates a statistical average of σ = 0.5. This value turned out to be the same as the one for spherical nanoparticles, for which rotations are irrelevant.

Exponential Mapping in Euler’s Elastic Problem

The Euler problem on optimal configurations of elastic rod in the plane with fixed endpoints and tangents at the endpoints is considered. The global structure of the exponential mapping that parameterises extremal trajectories is described. It is proved that open domains cut out by the Maxwell strata in the preimage and image of the exponential mapping are mapped diffeomorphically. As a consequence, computation of globally optimal elasticae with given boundary conditions is reduced to solving systems of algebraic equations having unique solutions in the open domains. For certain special boundary conditions, optimal elasticae are presented.

A Newtonian problem as an insightful tool for the behavior of gravitational-wave sources

We present the intriguing mathematical and physical similarities arising from the study of the Kerr metric and the Euler's problem of two fixed gravitational centers. We show how one could extend these similarities to physical problems that emerge from the above integrable problems, when both are slightly perturbed.

Buckling load and critical length of nanowires on an elastic substrate

This paper considers the stability of nanowires on an elastic substrate. The problem is converted to a generalized Euler problem containing rotational spring restraint. When distributed loading and tip forces are simultaneously applied, the buckling problem of a heavy nanocolumn with rotational spring junction is reduced to an integral equation. An approximate buckling load equation is derived explicitly. The critical length of nanocantilevers is given in closed form. Results indicate that spring stiffness increases the critical length of nanowires. The effect of self-weight on the critical length is pronounced for small tip forces, and becomes weaker for larger tip forces.

Euler elasticae in the plane and the Whitney-Graustein theorem

In this paper, we apply classical energy principles to Euler elasticae, i.e., closed C^2 curves in the plane supplied with the Euler functional U (the integral of the square of the curvature along the curve). We study the critical points of U, find the shapes of the curves corresponding to these critical points and show which of the critical points are stable equilibrium points of the energy given by U, and which are unstable. It turns out that the set of stable equilibrium points coincides with the set of minima of U, so that the corresponding shapes of the curves obtained may be regarded as normal forms of Euler elasticae. In this way, we find the solution of the Euler problem (set in 1744) for plane closed elasticae. As a by-product, we obtain a "mechanical" proof of the Whitney--Graustein theorem on the classification of regular curves in the plane up to regular homotopy (in the particular case of C^2 curves). Besides mathematical theorems, our work includes a computer graphics software which shows, as an animation, how any plane curve evolves to its normal form under a discretized version of gradient descent along the (discretized) Euler functional.

On Euler's problem

We consider the classical problem on the tallest column which was posed by Euler in 1757. Bernoulli-Euler theory serves today as the basis for the design of high buildings. This problem is reduced to the problem of finding the potential for the Sturm-Liouville equation corresponding to the maximum of the first eigenvalue. The problem has been studied by many mathematicians but we give the first rigorous proof of the existence and uniqueness of the optimal column and we give new formulae which let us find it. Our method is based on a new approach consisting in the study of critical points of a related nonlinear functional.Bibliography: 6 titles.

Approximation of a Solution to the Euler Equation by Solutions of the Navier–Stokes Equation

We show that a smooth solution u0 of the Euler boundary value problem on a time interval (0, T0) can be approximated by a family of solutions of the Navier–Stokes problem in a topology of weak or strong solutions on the same time interval (0, T0). The solutions of the Navier–Stokes problem satisfy Navier’s boundary condition, which must be “naturally inhomogeneous” if we deal with the strong solutions. We provide information on the rate of convergence of the solutions of the Navier–Stokes problem to the solution of the Euler problem for ν → 0. We also discuss possibilities when Navier’s boundary condition becomes homogeneous.

Reliability Modeling in Some Elastic Stability Problems Via the Generalized Stochastic Finite Element Method / Modelowanie Niezawodnosci W Pewnych Zagadnieniach Statecznosci Sprezystej Z Wykorzystaniem Uogólnionej Stochastycznej Metody Elementów Skonczonych

Abstract The main idea of this work is to demonstrate an application of the generalized perturbation-based Stochastic Finite Element Method for a determination of the reliability indicators concerning elastic stability for a certain spectrum of the civil engineering structures. The reliability indicator is provided after the Eurocode according to the First Order Reliability Method, and computed using the higher order Taylor expansions with random coefficients. Computational implementation provided by the hybrid usage of the FEM system ROBOT and the computer algebra system MAPLE enables for reliability analysis of the critical forces in the most popular civil engineering structures like simple Euler beam, 2 and 3D single and multi-span steel frames, as well as polyethylene underground cylindrical shell. A contrast of the perturbation-based numerical approach with the Monte-Carlo simulation technique for the entire variability of the input random dispersion included into the Euler problem demonstrates the probabilistic efficiency of the perturbation method proposed.

An iterative method for the shape reconstruction of the inverse Euler problem

AbstractThis article is concerned with the shape reconstruction for the inviscid fluid governed by the Euler equations. By formulating the domain derivative of the Euler equations and applying a regularized Gauss‐Newton iterative algorithm, the numerical examples are given for recovering the shape. The results show that our theory is useful for practical purpose and the proposed algorithm is feasible. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 587–596, 2012

Euler's problem and its applications in celestial mechanics and space dynamics

The numerous generalizations of the classical problem of two fixed centres are analysed, starting from the formulation of the problem and its solution by Euler in 1760 to the present day. The role played by numerous researchers in analysing this problem is noted. The publications cited indicate conclusively that the main results of generalizations of the problem and analytical and qualitative investigations had already been obtained in the nineteenth century and at the beginning of the twentieth century. Present-day researchers can only lay claim to a few occasionally productive and at the same time effective applications of individual generalizations (the Gredeaks problem, for example).

Time-Periodic Linearized Solutions of the Compressible Euler Equations and a Problem of Small Divisors

It has been unknown since the time of Euler whether or not time-periodic sound wave propagation is physically possible in the compressible Euler equations, due mainly to the ubiquitous formation of shock waves. The existence of such waves would confirm the possibility of dissipation free long distance signaling. Following our work in [B. Temple and R. Young, A paradigm for time-periodic sound wave propagation in the compressible Euler equations, Methods Appl. Anal., 16 (2009), pp. 341–363], we derive exact linearized solutions that exhibit the simplest possible periodic wave structure that can balance compression and rarefaction along characteristics in the nonlinear Euler problem. These linearized waves exhibit interesting phase and group velocities analogous to linear dispersive waves. Moreover, when the spatial period is incommensurate with the time period, the sound speed is incommensurate with the period, and a new periodic wave pattern is observed in which the sound waves move in a quasiperiodic trajectory though a periodic configuration of states. This establishes a new way in which nonlinear solutions that exist arbitrarily close to these linearized solutions can balance compression and rarefaction along characteristics in a quasiperiodic sense. We then rigorously establish the spectral properties of the linearized operators associated with these linearized solutions. In particular we show that the linearized operators are invertible on the complement of a one-dimensional kernel containing the periodic solutions only in the case when the wave speeds are incommensurate with the periods, but these invertible operators have small divisors, analogous to KAM theory. Almost everywhere algebraic decay rates for the small divisors are proven. In particular this provides a solid framework for the problem of perturbing these linearized solutions to exact nonlinear periodic solutions of the fully compressible Euler equations.

Carter-like constants of motion in the Newtonian and relativistic two-center problems

In Newtonian gravity, a stationary axisymmetric system admits a third, Carter-like constant of motion if its mass multipole moments are related to each other in exactly the same manner as for the Kerr black-hole spacetime. The Newtonian source with this property consists of two point masses at rest a fixed distance apart. The integrability of motion about this source was first studied in the 1760s by Euler. We show that the general relativistic analog of the Euler problem, the Bach–Weyl solution, does not admit a Carter-like constant of motion, first, by showing that it does not possess a non-trivial Killing tensor, and secondly, by showing that the existence of a Carter-like constant for the two-center problem fails at the first post-Newtonian order.

Generalized stochastic finite element method in elastic stability problems

The main issue of this paper is the stability analysis of elastic systems with random parameters using the Generalized Stochastic Finite Element Method. The Taylor expansion with random coefficients of nth order is used to express all random functions and to determine up to fourth order probabilistic moments of the critical force or critical pressure. The response function method assists to determine higher order partial derivatives of the structural response instead of the Direct Differentiation Method employed widely before. This approach is examined on the classical Euler problem, 2D and 3D steel frames as well as in addition to the cylindrical shell with some geometrical parameters defined as the Gaussian variables. The comparison of the GSFEM versus the Monte-Carlo simulation on the Euler problem proves the probabilistic convergence of this new technique.

Local in time strong solvability of the non-steady Navier–Stokes equations with Navier's boundary condition and the question of the inviscid limit

In this Note, we prove the existence of strong solutions to the Navier–Stokes equations for incompressible viscous fluids in a general regular bounded domain of R 3 on a “short” time interval ( 0 , T 0 ) , independent of the viscosity and of the friction between the fluid and the boundary. The solutions to the Navier–Stokes problem satisfy the inhomogeneous Navier's boundary condition and they reveal a remarkable structure of approximation of the solution to the Euler problem, which enables us to solve completely the question of the inviscid limit of the family of obtained solutions on the time interval ( 0 , T 0 ) .

An operator method for studying the Euler problem on types of the loss of stability for a pivoted rod under buckling load

In this paper we propose a new method for defining the Euler critical loads. We construct a scheme that leads to asymptotic formulas defining the bending of a rod both for constant and variable rigidities. The obtained results are based on operator methods of the bifurcation theory.

Frequency-Domain Linearized Euler Model for Turbomachinery Noise Radiation Through Engine Exhaust

A numerical model for the exhaust noise radiation problem is presented. In the model, it is assumed that an incoming wave is propagating through the exhaust nozzle, or the fan duct, and radiating outside. The near-field propagation is based on the solution of the linearized Euler equations in the frequency domain: for each wave number, a linearized Euler problem is solved using a finite element method on unstructured grids for arbitrarily shaped axisymmetric geometries. The frequency-domain approach enables the suppression of the Kelvin-Helmholtz instability waves. Moreover, each single calculation, limited to a single frequency, is well suited to the exhaust noise radiation problem in which the incoming wave can be treated as a superposition of elementary duct modes. To reduce the memory requirements, a continuous Galerkin formulation with linear triangular and quadrangular elements is employed and the global matrix inversion is performed with a direct solver based on a parallel memory distributed multifrontal algorithm for sparse matrices. The acoustic near field is then radiated in the far field using the formulation of Ffowcs Williams and Hawkings. Numerical calculations for a validation test case, the Munt problem, and two turbomachinery configurations are compared with analytical solutions and experimental data.

On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition

Under assumptions on smoothness of the initial velocity and theexternal body force, we prove that there exists T0 > 0, V* > 0and a unique family of strong solutions uv of the Euler or Navier-Stokes initial-boundary value problem on the time interval (0, T0), depending continuously on the viscosity coefficient $\nu$for $0\leq\nu*.The solutions of the Navier-Stokes problem satisfy a Navier-typeboundary condition. We give the information on the rate of convergence of the solutions of the Navier-Stokes problem to the solution of the Euler problem for $\nu\to 0+$.

Solution to Euler’s elastic problem

Euler's problem on stationary configurations of elastic rod with fixed endpoints and tangents at the endpoints is considered. The corresponding optimal control problem is reduced to several systems of algebraic equations in Jacobi's functions. An algorithm and software for solving the optimal control problem are constructed.

Законы сохранения в задаче Эйлера о двух притягивающих центрах и интегрирование уравнений динамики

The article establishes that Euler's problem of two attractive centres contains another conservation law alongside with the energy conservation one, which is also quadratic in velocities. These conservation laws and elliptic coordinates considerably simplify the procedure of trajectory equations generation.

Boundary and internal conditions for adjoint fluid-flow problems

The ever-increasing robustness and reliability of flow-simulation methods have consolidated CFD as a major tool in virtually all branches of fluid mechanics. Traditionally, those methods have played a crucial role in the analysis of flow physics. In more recent years, though, the subject has broadened considerably, with the development of optimization and inverse design applications. Since then, the search for efficient ways to evaluate flow-sensitivity gradients has received the attention of numerous researchers. In this scenario, the adjoint method has emerged as, quite possibly, the most powerful tool for the job, which heightens the need for a clear understanding of its conceptual basis. Yet, some of its underlying aspects are still subject to debate in the literature, despite all the research that has been carried out on the method. Such is the case with the adjoint boundary and internal conditions, in particular. The present work aims to shed more light on that topic, with emphasis on the need for an internal shock condition. By following the path of previous authors, the quasi-1D Euler problem is used as a vehicle to explore those concepts. The results clearly indicate that the behavior of the adjoint solution through a shock wave ultimately depends upon the nature of the objective functional.