The motion of a balanced circular foil in the field of fixed point sources

A finite-dimensional model is developed, which describes the motion of a balanced circular foil with proper circulation in the field of a fixed vortex source. The motion of the foil has been studied in two special cases: that of a fixed vortex and that of a fixed source. It is shown that in the absence of proper circulation, the fixed vortex and the fixed source have the same impact on the motion of the foil. However, adding nonzero proper circulation leads to qualitative differences in the foil's dynamics. For a fixed vortex, there exist three types of motions: the fall on a vortex in finite time, periodic and quasiperiodic motion around the vortex. The investigation of this case reduces to analysis of a Hamiltonian system with one degree of freedom. Typical phase portraits and graphs of the effective potential of the system are plotted vs the distance between the geometric center of the foil and the vortex. For a fixed source, two types of motions are possible: the fall on the source in finite time and unbounded escape from the source. For small intensities of the source, the asymptotics of escape to infinity is constructed.

The dynamics of a system governing the controlled motion of an unbalanced circular foil in the presence of point vortices is considered. The foil motion is controlled by periodically changing the position of the center of mass, the gyrostatic momentum, and the moment of inertia of the system. A derivation of the equations of motion based on Sedov's approach is proposed, the equations of motion are presented in the Hamiltonian form. A periodic perturbation of the known integrable case is considered.

A plane unsteady-state linear problem of the immersion of an elastic plate of finite length in an ideal incompressible weightless fluid is considered. The deflection of the plate and the velocity of its points are known at the initial moment of time. The fluid occupies the lower halfplane, and its boundary outside the plate is free. The plate which is the bottom of a structure immersed in the fluid with a constant velocity is modeled by an Euler beam. At the initial stage of immersion, when the displacement of the structure is much smaller than the length of the plate, the plate deflection and the distribution of bending stresses in it are determined. The model used allows one to estimate the maximum stresses occurring in the elastic plate during its impact on water and to predict the moment and site of its occurrence. Calculations are performed under the conditions of the experiment carried out in MARINTEX (Norway). Qualitative agreement between the numerical and experimental results is shown.

This paper addresses the problem of the motion of an unbalanced circular foil and point vortices in an ideal incompressible fluid. Using Bernoulli's theorem for unsteady potential flow, the force due to the pressure from the fluid on the foil is obtained for an arbitrary vortex motion. A detailed analysis is made of the case of free vortex motion in which a Hamiltonian reduction by symmetries is performed. For the resulting system, relative equilibria corresponding to the motion of an unbalanced foil and a vortex in a circle or in a straight line are found and their stability is investigated. New examples of stationary configurations of a vortex and a foil are given. Using a Poincaré map, it is also shown that in the general case of an unbalanced circular foil the reduced system exhibits chaotic trajectories.

The object of this study is two thin elastic isotropic rectangular plates in an infinitely long rectangular parallelepiped with an ideal fluid. The first plate is the upper base of the rectangular parallelepiped, and the second one horizontally separates ideal fluids that have different densities. The subject of the study is the normal joint plane vibrations of elastic rectangular plates and an incompressible fluid and the conditions that enable the stability of these vibrations. In the linear statement, the frequency spectrum of normal plane vibrations of two elastic isotropic plates in an infinitely long rectangular parallelepiped with an ideal incompressible fluid has been investigated. The frequency equation of joint vibrations of the plates and the ideal fluid was reduced to the form of an eighth-order determinant for arbitrary cases of fixing the contours of the plates. The case of clamped contours of the plates and the case of rebirth of the plates into membranes is analyzed. Based on analytical studies of infinite series in the transcendental frequency equation, exact stability conditions for the combined oscillations of plates and liquid were established. It has been shown that instability of oscillations of plates and liquid occurs when a heavier liquid is above a less heavy liquid. The derived stability conditions for symmetric and asymmetric oscillations of plates and liquid do not depend on the elastic parameters of the upper plate, the mass characteristics of the plates and the depths of filling liquids. The analytically obtained exact stability conditions for the combined oscillations of the plate and liquid generalize the previously obtained approximate stability conditions for this problem. The numerical calculations of the frequency equation confirmed the analytical studies of the stability conditions. The results could be used in the calculation and design of mechanical objects related to the storage and transportation of liquid cargo

Riemannian geometry of the motion of an ideal incompressible magnetohydrodynamical fluid

In this paper we obtain equations of motion for a vortex pair and a circular foil with parametric excitation due to the periodic motion of a material point. Undoubtedly, such problems are, on the one hand, model problems and cannot be used for an exact quantitative description of real trajectories of the system. On the other hand, in many cases such 2D models provide a sufficiently accurate qualitative picture of the dynamics and, due to their simplicity, an estimate of the influence of different parameters. We describe relative equilibria that generalize Föppl solutions and collinear configurations when the material point does not move. We show that a stochastic layer forms in the neighborhood of relative equilibria in the case of periodic motion of the foil's center of mass.

In this chapter, we study the hydrodynamics of a viscous incompressible fluid. The Lagrangian hydrodynamical systems (LHSs) of a viscous incompressible fluid were introduced in [39] as generalizations of those of an ideal incompressible fluid. Namely, these LHSs were defined as systems on the group of volume-preserving diffeomorphisms with an additional right-invariant force field which depends on the velocity of the fluid. In the tangent space at id, the force field is v△ where △ is the Laplace-de Rham operator and v is the viscosity coefficient. Note, however, that the operator △ does not preserve the space of H8-smooth vector fields. In fact, △sends H8-smooth vector fields to fields which belong to a broader Sobolev class. As a consequence, the method relies heavily on the theory of partial differential equations, leading to the loss of many natural geometric properties of the LHSs of an ideal incompressible fluid (Chap. 8) in the passage to a viscous incompressible fluid.

In this paper we derive a necessary criterion for dissipation by mixing in ideal fluids. The experimental test of that criterion is simple and will help to discriminate real dissipation on microscopic scales from mixing relaxation in almost ideal fluids or 2-D magnetized electron plasmas. We will show how globally stable states do depend on the constraints resulting from incompressibility. And we will develop a model for the relaxation process on macroscopic scales bypassing the exact dynamics by using simple assumptions on filamentation of the density within the exact dynamics.

We review the role of dual pairs in mechanics and use them to derive particle-like solutions to regularized incompressible fluid systems. In our case we have a dual pair resulting from the action of diffeomorphisms on point particles (essentially by moving the points). We then augment our dual pair by considering the action of diffeomorphisms on Taylor series, also known as jets. The augmented weak dual pairs induce a hierarchy of particle-like solutions and conservation laws with particles carrying a copy of a jet group. We call these augmented particles jetlets. The jet groups serve as finite-dimensional models of the diffeomorphism group itself, and so the jetlet particles serve as a finite-dimensional model of the self-similarity exhibited by ideal incompressible fluids. The conservation law associated to jetlet solutions is shown to be a shadow of Kelvin's circulation theorem. Finally, we study the dynamics of infinite time particle mergers. We prove that two merging particles at the zeroth level in the hierarchy yield dynamics which asymptotically approach that of a single particle in the first level in the hierarchy. This merging behavior is then verified numerically as well as the exchange of angular momentum which must occur during a near collision of two particles. The resulting particle-like solutions suggest a new class of meshless methods which work in dimensions $n \geq 2$ and which exhibit a shadow of Kelvin's circulation theorem. More broadly, this provides one of the first finite-dimensional models of self-similarity in ideal fluids.

The problem of the interaction of two pulsating spheres in an ideal incompressible fluid was first investigated in detail by Bjerknes [1]. However, his and subsequent studies on this subject [2–5] were restricted to the interaction forces between the spheres, whereas the law of their motion was not considered because of the much greater complexity of the corresponding problem. The aim of the present paper is to find an approximate analytic solution to the problem of the motion of two pulsating spheres in an ideal incompressible fluid filling the entire space exterior to the spheres under the assumption that the flow of the fluid is irrotational.

Steady-state and periodic motions in the attraction field of a rotating triaxial ellipsoid

The problem on linear stability of steady–state three–dimensional (3D) flows of an inviscid incompressible fluid, completely filling a volume with a solid boundary, is studied in the absence mass forces. It is proved by the direct Lyapunov method that these flows are absolutely unstable with respect to small 3D perturbations. The a priori exponential estimate from below, which testifies to growth of perturbations under consideration in time, is constructed.

Emergence of singularity of vorticity at a single point, not related to any symmetry of the initial distribution, has been demonstrated numerically for the first time. Behaviour of the maximum of vorticity near the point of collapse closely follows the dependence (t 0−t)−1, where t 0 is the time of collapse. This agrees with the interpretation of collapse in an ideal incompressible fluid as of the process of vortex lines breaking.

A new approach to the solution of the self-similar problem of the entry of a wedge into an ideal fluid at an arbitrary angle to the free surface is proposed. The method is based on the construction of the expressions for the complex velocity and the derivative of the complex potential in a parametric variable domain. An integral and an integro-differential equation are obtained for determining the absolute magnitude and the angle of the velocity vector at the free boundary. The calculated results for the free surface shape, the angles of contact between the free surface and the wedge, and the coefficients of the hydrodynamic forces are presented.