Instability of a vertical rotation of a heavy body

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 9, pp. 1214–1221, September, 1989.     

Predominantly unidirectional rotation of a solid body and a viscous liquid

Under consideration are two problems of the time-periodic motion of a hydro-mechanical system consisting of a viscous liquid and solid bodies bordering it. A new hydro-mechanical effect is discovered.     

Interaction between internal and external friction in rotation of vane with viscous filling

Rotation of the four-blade axisymmetric vane around the fixed point is considered in the constant flow of medium. The vane contains the cavity that is filled by viscous filling. Aerodynamic force acts upon the blades of the vane. The quasi-steady approach is used for modeling an aerodynamic load. A case for blades with low fineness is considered. Nonlinear approximation of the aerodynamic fineness function is used. Internal friction is modeled as the linear function between the angular speed of the vane and the vortex vector of the filling. This is a mechanical system with variable dissipation. Corresponding dynamic system is simulated for different values of the density of external medium and the coefficient of the internal friction. Two types of the vane configuration (oblate and oblong) are considered. Interaction between the external aerodynamic load and the internal friction is observed. Areas are found, where the steady axial rotation is not globally attracting.     

On the fast rotation of a heavy rigid body about a fixed point

Fast rotation of a symmetric heavy rigid body about a fixed point (the kinetic energy is large in comparison with the potential) is considered in cases when the resonance equations of Euler's motion /1, 2/ are approximately satisfied at the initial instant (the body is assumed to effect turns, ε is small, during time . It is shown that during that time ( ) a finite deviation from inertial motion takes place. Such mechanical effect is similar to the precession of a fast top, except that it is more “early” (in the considered time scale the top precession is slow). Approximate equations that define the motion in the principal order and are integrable in quadratures. The formal process of derivation of higher approximations is indicated, and a geometric interpretation of motions is given.     

The motion of a body with a cavity partly filled with a viscous liquid

Many papers are concerned with the dynamics of a rigid body with a cavity filled with liquid (see the bibliography in [1]). The present paper deals with the motion of a rigid body having a cavity partly filled with a viscous incompressible liquid, and having a free surface. The shape of the cavity is arbitrary. The problem is considered in a linear formulation. The oscillations of the body with respect to its center of inertia and the motion of the liquid in the cavity are assumed small. The viscosity of the liquid is considered low. The solution of the problem of the oscillations of a body with a cavity partly filled with an ideal liquid is used as an initial approximation [1 to 6]. The viscosity is taken into consideration by the boundary layer method used before in similar problems [1 and 7 to 10). General equations are derived for the dynamics of a body filled with a liquid, for an arbitrary form of cavity. The coefficients of those integro-differential equations depend only on the solution of the problem of the oscillations of a body with a cavity of the given form filled with an ideal liquid. Since the corresponding problem has been solved for cavities of many forms [1 to 6, 11 and 12] in the case of an ideal liquid, the determination of the characteristic coefficients is reduced to the evaluation of quadratures. Several particular cases of motion are considered.     

The conditions for loss of stability of the rotation of a dynamically symmetrical rigid body suspended on a string with parametric perturbations

A dynamically symmetrical rigid body suspended on a string is considered. The suspension point performs periodic oscillations. The loss of stability of the system when it performs high-frequency rotations around the axis of dynamic symmetry is investigated. The sufficient conditions for loss of stability are obtained.     

The self-propulsion of a body with moving internal masses in a viscous fluid

An investigation of the characteristics of motion of a rigid body with variable internal mass distribution in a viscous fluid is carried out on the basis of a joint numerical solution of the Navier — Stokes equations and equations of motion for a rigid body. A nonstationary three-dimensional solution to the problem is found. The motion of a sphere and a drop-shaped body in a viscous fluid in a gravitational field, which is caused by the motion of internal material points, is explored. The possibility of self-propulsion of a body in an arbitrary given direction is shown.     

The motion of a compressible viscous fluid around rotating body

We consider the motion of compressible viscous fluids around a rotating rigid obstacle when the velocity at infinity is non zero and parallel to the axis of rotation. We prove the existence of weak solution.     

Slow rotation of a sphere with source at its centre in a viscous fluid

In this note, the problem of a sphere carrying a fluid source at its centre and rotating with slow uniform angular velocity about a diameter is studied. The analysis reveals that only the azimuthal component of velocity exists and is seen that the effect of the source is to decrease it. Also, the couple on the sphere is found to decrease on account of the source.     

Asymptotic stability of viscous contact wave for the one-dimensional compressible viscous gas with radiation

In this paper, we study the large-time behavior of solutions of the Cauchy problem to a one-dimensional Navier-Stokes-Poisson coupled system, modeling the dynamics of a viscous gas in the presence of radiation. When the far field states are suitably given, and the corresponding Riemann problem for the Euler system admits only a contact discontinuity wave solution with the far field states as Riemann initial data. Then, we can define a “viscous contact wave” for such a Navier-Stokes-Poisson coupled system. Based on elementary energy methods and ellipticity of the equation of the radiation flux, we can prove the “viscous contact wave” is stable provided the strength of the contact discontinuity wave and the perturbation of the initial data are suitably small.     

Permanent rotations of a body with a viscous filler suspended on a rod

Problems of the existence, stability, and branching of the permanent rotations of a heavy, dynamically symmetrical rigid body suspended on a rod and which has an axisymmetric ellipsoidal cavity filled with a fluid are discussed. The phenomenological model of the friction of the fluid against the cavity wall proposed by Samsonov is used. All the trivial permanent rotations of the system and the non-trivial rotations that branch off from the trivial ones are found. Their stability and branching are investigated using a modified Routh's theory. The results obtained are presented in the form of an atlas of bifurcation diagrams.     

Motion of a rigid body in a viscous fluid

We introduce a concept of weak solution for a boundary value problem modelling the motion of a rigid body immersed in a viscous fluid. The time variation of the fluid's domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem. Our main theorem asserts the existence of at least one weak solution for this problem. The result is global in time provided that the rigid body does not touch the boundary.     

A posteriori error estimates for viscous flow problems with rotation

A new functional type a posteriori error estimates for the Stokes problem with rotating term are presented. The estimates give guaranteed upper bounds for the energy norm of the error and provide reliable error indication. Computational properties of the estimates are demonstrated by a number of numerical examples. Bibliography: 37 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 23–34.