Motion stability of a deformable body in an ideal fluid with applications to the N spheres problem

The Liapunov stability problem of the translation or spiraling motion of an arbitrary deformable body (the deformation of which is governed by the corresponding Hamiltonian) is treated here using the modified Energy-Casimir approach. The apropriate stability criteria are derived. It is shown that some unstable translational motions can be stabilized by a deformational or rotational motion. This formulism is further applied to the stability problem related to the motion of N (generally unequal) rigid spheres embedded in a potential flow field. The assembly of N-spheres is treated as an entire N-connected single deformable body. The Liapunov stability of the motion of two spheres in the direction orthogonal to their lines of centers and that of three spheres in the direction orthogonal to their plane of centers, is demonstrated and proven as a special case. Some existing conditions of clustering for a bubble cloud are also rederived and extended.