Dynamics of a rigid body in a Stokes fluid

Abstract

We demonstrate that the dynamics of a rigid body falling in an infinite viscous fluid can, in the Stokes limit, be reduced to the study of a three-dimensional system of ordinary differential equations ${\dot\Grav} \,{=}\, \Grav\times M_2\Grav$ where $M_2\,{\in}\,{{\mathbb R}^{3\times 3}}$ is a generally non-symmetric matrix containing certain hydrodynamic mobility coefficients. We further show that all steady states and their stability properties can be classified in terms of the Schur form of $M_2$. Steady states correspond to screw motions (or limits thereof) in which the centre of mass traces a helical path, while the body spins uniformly about the vertical. All rigid bodies have at least one such stable screw motion. Bodies for which $M_2$ has exactly one real eigenvalue have a unique globally attracting asymptotically stable screw motion, while other bodies can have multiple, stable and unstable steady motions. One application of our theory is to the case of rigid filaments, which in turn is a first step in modelling the sedimentation rate of flexible polymers such as DNA. For rigid filaments the matrix $M_2$ can be approximated using the Rotne–Prager theory, and we present various examples corresponding to certain ideal shapes of knots which illustrate the various possible multiplicities of steady states. Our simulations of rigid ideal knots in a Stokes fluid predict an approximate linear relation between sedimentation speed and average crossing number, as has been observed experimentally for the much more complicated system of real DNA knots in gel electrophoresis.