Generalized Kirchhoff Equations Research Articles

The Family of Global Attractors for a Generalized Kirchhoff Equations

This paper deals with the existence and uniqueness of solutions of generalized Kirchhoff equations and the family of global attractors for the equation and its dimension estimation. First, the stress term of Kirchhoff equation is properly assumed. When certain conditions are met between the order m and the degree p of Banach space , the existence and uniqueness of the solution of equation are obtained by a prior estimation and Galerkin’s method; Then, the bounded absorption set is obtained by prior estimation, and it is proved that the solution semigroup generated by the equation has a family of global attractors in phase space by using Rellich-Kondrachov compact embedding theorem. Further, the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Fréchet differentiable on ; Finally, the upper bound of Hausdorff dimension and Fractal dimension of is estimated, and the Hausdorff dimension and Fractal dimension are finite.

A Family of the Exponential Attractors and the Inertial Manifolds for a Class of Generalized Kirchhoff Equations

In this paper, we studied a family of the exponential attractors and the inertial manifolds for a class of generalized Kirchhoff-type equations with strong dissipation term. After making appropriate assumptions for Kirchhoff stress term and nonlinear term, the existence of exponential attractor is obtained by proving the discrete squeezing property of the equation, then according to Hadamard’s graph transformation method, the spectral interval condition is proved to be true, therefore, the existence of a family of the inertial manifolds for the equation is obtained.

Global Attractors and Their Dimension Estimates for a Class of Generalized Kirchhoff Equations

In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term g (u) and Kirchhoff stress term M (s) in the equation, and the existence and uniqueness of the solution are proved by using uniform prior estimates of time and Galerkin’s finite element method. Then, abounded absorption set B0k is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup S (t) generated by the equation has a family of the global attractor Ak in the phase space . Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on Ek. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor Ak was obtained.

一类广义非线性Kirchhoff型方程的整体吸引子

一类广义非线性Kirchhoff型方程的整体吸引子

一类广义非线性Kirchhoff型方程的惯性流形

一类广义非线性Kirchhoff型方程的惯性流形

The quasilinear parabolic kirchhoff equation

Abstract In this paper the existence of solution of a quasilinear generalized Kirchhoff equation with initial – boundary conditions of Dirichlet type will be studied using the Leray – Schauder principle.

On a generalized Kirchhoff equation with sublinear nonlinearities

In this paper, we consider a generalized Kirchhoff equation in a bounded domain under the effect of a sublinear nonlinearity. Under suitable assumptions on the data of the problem, we show that, with a simple change of variable, the equation can be reduced to a classical semilinear equation and then studied with standard tools. Copyright © 2016 John Wiley & Sons, Ltd.

Existence of least energy nodal solution with two nodal domains for a generalized Kirchhoff problem in an Orlicz–Sobolev space

We show the existence of a nodal solution with two nodal domains for a generalized Kirchhoff equation of the type urn:x-wiley:0025584X:media:mana201500286:mana201500286-math-0001where Ω is a bounded domain in is a general C1 class function, f is a superlinear C1 class function with subcritical growth, Φ is defined for by setting is the operator . The proof is based on a minimization argument and a quantitative deformation lemma.

Experimental investigation of freely falling thin disks. Part 2. Transition of three-dimensional motion from zigzag to spiral

AbstractThe free-fall motion of a thin disk with small dimensionless moments of inertia (${I}^{\ast } \lt 1{0}^{- 3} $) was investigated experimentally. The transition from two-dimensional zigzag motion to three-dimensional spiral motion occurs due to the growth of three-dimensional disturbances. Oscillations in the direction normal to the zigzag plane increase with the development of this instability. At the same time, the oscillation of the nutation angle decreases to zero and the angle remains constant. The effects of initial conditions (release angle) were investigated. Two kinds of transition modes, zigzag–spiral transition and zigzag–spiral–zigzag intermittence transition, were observed to be separated by a critical Reynolds number. In addition, the solution of the generalized Kirchhoff equations shows that the small ${I}^{\ast } $ is responsible for the growth of disturbances in the third dimension (perpendicular to the planar motion).

Dynamical Model for the Buoyancy-Driven Zigzag Motion of Oblate Bodies

We describe a dynamical model that predicts the zigzag motion of disks and oblate spheroids moving freely in a viscous liquid over a continuous range of aspect ratios and Reynolds numbers. This model combines the generalized Kirchhoff equations to describe the linear and angular momentum balances for the fluid-body system with a dynamical model for the wake-induced force and torque that incorporates the main characteristics of the wake dynamics deduced from previous experimental observations. The resulting model is shown to be able to reproduce quantitatively the oscillatory paths measured experimentally.

Dynamics of axisymmetric bodies rising along a zigzag path

The forces and torques governing the planar zigzag motion of thick, slightly buoyant disks rising freely in a liquid at rest are determined by applying the generalized Kirchhoff equations to experimental measurements of the body motion performed for a single body-to-fluid density ratio ρs/ρf ≈ 1. The evolution of the amplitude and phase of the various contributions is discussed as a function of the two control parameters, i.e. the body aspect ratio (the diameter-to-thickness ratio χ = d/h ranges from 2 to 10) and the Reynolds number (100 < Re < 330), Re being based on the rise velocity and diameter of the body. The body oscillatory behaviour is found to be governed by the force balance along the transverse direction and the torque balance. In the transverse direction, the wake-induced force is mainly balanced by two forces that depend on the body inclination, i.e. the inertia force generated by the body rotation and the transverse component of the buoyancy force. The torque balance is dominated by the wake-induced torque and the restoring added-mass torque generated by the transverse velocity component. The results show a major influence of the aspect ratio on the relative magnitude and phase of the various contributions to the hydrodynamic loads. The vortical transverse force scales as fo = (ρf − ρs)ghπd2 whereas the vortical torque involves two contributions, one scaling as fod and the other as f1d with f1 = χfo. Using this normalization, the amplitudes and phases of the vortical loads are made independent of the aspect ratio, the amplitudes evolving as (Re/Rec1 − 1)1/2, where Rec1 is the threshold of the first instability of the wake behind the corresponding body held fixed in a uniform stream.

The generalized Kirchhoff equations and their application to the interaction between a rigid body and an arbitrary time-dependent viscous flow

Recent numerical and analytical studies have demonstrated that added-mass effects acting on bluff bodies moving in viscous, time-dependent flows are independent of the Reynolds number, acceleration strength and steady/unsteady nature of the flow field. We discuss the origin of this crucial result and show how it can be used to derive the equations governing the motion of a non-deformable body moving freely in an arbitrary time-dependent, viscous flow. Then we show how these equations can be employed in conjunction with the Navier–Stokes equations to solve numerically the coupled problem in which the presence of the body modifies the surrounding flow which itself determines the trajectory of the body. Numerical tests of this coupling are presented. We finally apply the coupled set of equations to analyze the path instability of ellipsoidal bubbles rising at high Reynolds number. We show that numerical results recover the main experimental trends, an agreement suggesting that path instability is primarily driven by the instability of the wake which is itself crucially dependent on the curvature of the bubble surface.

Problème de Cauchy global pour des équations de Kirchhoff

We consider the global Cauchy problem for generalized Kirchhoff equations. Under conditions upon the nonlinear part or upon the size of data, we show existence of a solution which is twice differentiable with respect to time and uniformly analytic with respect to spatial variables. When the nonlinear part does not depend of spatial variables, we give asymptotic estimates for lifespan of the solution and we prove the stability of the problem.

Probleme de cauchy pour des equations de kirchhoff generalisees

We consider the global Cauchy problem for generalized Kirchhoff equations with small non-linear terms or small data. We solve this problem in the space of functions which are twice differentiable with respect to time coordinate and uniformly analytic with respect to other coordinates. We determine, in two different situations, estimates of lifespan of solutions for some problems with perturbations and we give stability result of the solution for small perturbations.

Generalized Kirchhoff equations for a deformable body moving in a weakly non-uniform flow field

The classical Kirchhoff’s method provides an efficient way of calculating the hydrodynamical loads (forces and moments) acting on a rigid body moving with six-degrees of freedom in an otherwise quiescent ideal fluid in terms of the body’s added-mass tensor. In this paper we provide a versatile extension of such a formulation to account for both the presence of an imposed ambient non-uniform flow field and the effect of surface deformation of a non-rigid body. The flow inhomogeneity is assumed to be weak when compared against the size of the body. The corresponding expressions for the force and moment are given in a moving body-fixed coordinate system and are obtained using the Lagally theorem. The newly derived system of nonlinear differential equations of motion is shown to possess a first integral. This can be interpreted as an energy-type conservation law and is a consequence of an anti-symmetry property of the coefficient matrix reported here for the first time. A few applications of the proposed formulation are presented including comparison with some existing limiting cases.