Motion of a rigid body with cavities filled with viscous fluid at small Reynolds numbers

Abstract

PROBLEMS concerning the motion of a rigid body with cavities filled with viscous fluid are of theoretical and practical interest. Such problems involve the difficulties both of hydrodynamic problems and of problems of the dynamics of a rigid body. Zhukovskii [1] was the first to investigate such a topic. Several results on the stability of the motion of a body with cavities containing viscous fluid were obtained in [2]. An asymptotic method was proposed in [3] for studying the small oscillations of a viscous fluid in the cavity of a rigid body for large Reynolds numbers, and was applied in [4] to the solution of several specific problems. There is a considerable literature on the motion of a viscous incompressible fluid at small Reynolds numbers. The usual method of solution is linearization of the Navier — Stokes equations with subsequent solution of the linear non-stationary boundary value problems [5]. Several problems of the dynamics of a rigid body with cavities filled with viscous fluid at small Reynolds numbers were discussed in [6], where use was made of the familiar solutions of the non-stationary linear problems of hydrodynamics for an infinite cylinder and sphere. We consider in the present paper the general problem of the motion of a viscous incompressible fluid in the cavity of a rigid body, and of the motion of the rigid body itself. The cavity can be of any shape, and our only assumption is that the Reynolds number be small. The hydrodynamic problem is shown to reduce to the solution of three stationary linear boundary value problems, which have to be solved once for each given cavity shape. Certain integrals of the solutions obtained then have to be evaluated, after which a system of ordinary differential equations can be written for the motion of the body with the cavity. This system will be written in a general form. The manner of solution is thus similar to that for the problem of the motion of a rigid body with a cavity containing an ideal fluid (with irrotational motion). We know from [1] that, in the case of an ideal fluid, we again have to solve three boundary value problems for each type of cavity, then evaluate integrals of the solutions (associated moments of inertia). The solution of these stationary boundary value problems is given for certain types of cavity (sphere, triaxial ellipsoid and finite cylinder) filled with viscous fluid. The ordinary differential equations for the motion of the body plus cavity are discussed, and are shown to be solvable by asymptotic methods. Some examples are analysed. The damping of the plane rotations and nonlinear oscillations of the body plus fluid in a gravitational field is obtained. The three-dimensional free motion of the body plus fluid is examined. It is well known that the only stable motion in this case is uniform rotation about the axis of maximum moment of inertia [1, 2]. The entire transitional process leading to this stable motion is here examined, and the duration of the transitional process obtained.