Global attractors for the three-dimensional Navier-Stokes equations

In this paper we show that the weak solutions of the Navier-Stokes equations on any bounded, smooth three-dimensional domain have a global attractor for any positive value of the viscosity. The proof of this result, which bypasses the two issues of the possible nonuniqueness of the weak solutions and the possible lack of global regularity of the strong solutions, is based on a new point of view for the construction of the semiflow generated by these equations. We also show that, under added assumptions, this global attractor consists entirely of strong solutions.     

On the wave-front approximation in three-dimensional gas dynamics

Summary Since the differential equations governing steady supersonic flow of an inviscid gas are hyperbolic, the fluid acceleration must satisfy some compatibility relations along characteristic surfaces. These relations are here obtained and integrated for a characteristic surface bounding a region of uniform flow, and it is then shown that the same relations are satisfied to a first approximation in a region adjacent to the region of uniform flow. The singularities predicted in this manner are discussed, and an approximate method of solution, complementary to the Linearized Theory, is briefly explained.     

On two models in the three-dimensional theory of stability of composites

A comparative analysis is made of the infinite-fiber and finite-fiber models in the three-dimensional theory of stability of composites. The results analyzed have been obtained using the three-dimensional linearized theory of stability of deformable bodies. A historical sketch is given of the theory of stability for and approaches used in the mechanics of laminated and fibrous composite materials Translated from Prikladnaya Mekhanika, Vol. 44, No. 8, pp. 3–31, August 2008.     

A method for flow visualization and measurement of velocity vectors in three-dimensional flow patterns in water models by using colour photography

Summary A method is described which permits the measurement of velocity vectors in a three-dimensional flow field. This method, which is especially useful for the study of water models, uses colour as an indication of the third dimension, thus enabling complete determination of direction and magnitude of the velocity by means of a single colour-photograph.     

On path-independent integrals in three-dimensional nonlinear fracture dynamics

This paper deals with the path-independent integrals in non-linear three-dimensional fracture dynamics.Both the nonli-near elastic case and the elastic-plastic case are considered,and some path-independent integrals have been worked out.Forexplaining the physical meaning of these integrals,a speci-men with plane notch is considered,and the relation betweenthe integral and dynamical crack extension force is establish-ed.Thus.such integrals may serve as a fracture criterionin nonlinear fracture dynamics.     

Control of three-dimensional phase dynamics in a cylinder wake

Recently there has been a surge of new interest in three-dimensional wake patterns. In the present work, we have devised a method to control the spanwise end conditions and wake patterns using “end suction”, which is both continuously-variable and admits transient control. Classical steady-state patterns, such as parallel or oblique shedding or the “chevron” patterns are simply induced. The wake, at a given Reynolds number, is receptive to a continuous range of oblique shedding angles (θ), rather than to discrete angles, and there is excellent agreement with the “cos θ” formula for oblique-shedding frequencies. We show that the laminar shedding regime exists up to Reynolds numbers (Re) of 205, and that the immense disparity among reported critical Re for wake transition (Re = 140–190) can be explained in terms of spanwise end contamination. Our transient experiments have resulted in the discovery of new phenomena such as “phase shocks” and “phase expansions”, which can be explained in terms of a simple model assuming constant normal wavelength of the wake pattern. Peter Monkewitz (Lausanne) also predicts such transient phenomena from a Guinzburg-Landau model for the wake.     

Nonlinear dynamics of three-dimensional belt drives using the finite-element method

In this paper, new nonlinear dynamic formulations for belt drives based on the three-dimensional absolute nodal coordinate formulation are developed. Two large deformation three-dimensional finite elements are used to develop two different belt-drive models that have different numbers of degrees of freedom and different modes of deformation. Both three-dimensional finite elements are based on a nonlinear elasticity theory that accounts for geometric nonlinearities due to large deformation and rotations. The first element is a thin-plate element that is based on the Kirchhoff plate assumptions and captures both membrane and bending stiffness effects. The other three-dimensional element used in this investigation is a cable element obtained from a more general three-dimensional beam element by eliminating degrees of freedom which are not significant in some cable and belt applications. Both finite elements used in this investigation allow for systematic inclusion or exclusion of the bending stiffness, thereby enabling systematic examination of the effect of bending on the nonlinear dynamics of belt drives. The finite-element formulations developed in this paper are implemented in a general purpose three-dimensional flexible multibody algorithm that allows for developing more detailed models of mechanical systems that include belt drives subject to general loading conditions, nonlinear algebraic constraints, and arbitrary large displacements. The use of the formulations developed in this investigation is demonstrated using two-roller belt-drive system. The results obtained using the two finite-element formulations are compared and the convergence of the two finite-element solutions is examined.     

Global qualitative analysis of tippe top dynamics

The tippe top is a dynamically and geometrically symmetric body supported by a horizontal plane. If one twists the tippe top rapidly about the symmetry axis so that the symmetry axis is vertical and its center of mass takes the lowest position, then it turns upside down by 180° and stats to rotate about the same symmetry axis with the center of mass occupying the highest position. A local analysis of tippe top dynamics (in a neighborhood of its rotations about the vertical symmetry axis) is given in [1, 2]. The simplest model of the tippe top is a dynamically symmetric inhomogeneous ball whose center of mass lies on the dynamic symmetry axis but does not coincide with its geometric center. Such a model allows global qualitative analysis of the top dynamics.     

Global dynamics of a flexible asymmetrical rotor

Nonlinear Dynamics - Global dynamics of a flexible asymmetrical rotor resting on vibrating supports is investigated. Hamilton’s principle is used to derive the partial differential governing...