Long-time collective motion of rigid bodies immersed in a viscous fluid

Abstract

In linear approximation a number of rigid bodies, immersed in a viscous incompressible fluid, to which at time zero arbitrary small translational and rotational impulses are imparted, move collectively at long times with a common translational velocity which slowly decays to zero. The proof of this statement is based on a study of the low frequency behavior of the friction and mobility matrices of the set of bodies. It is shown that the linear term in a Taylor expansion of the friction matrix in powers of the square root of frequency may be expressed in terms of the zero frequency friction matrix. The corresponding term for the mobility matrix is even simpler. It is independent of the shape or size of the rigid bodies. Its particular form gives rise to the theorem stated above. Related consequences for the theory of Brownian motion are discussed. For example, the long-time tail of the velocity autocorrelation function of a selected particle in a polydisperse suspension of spheres has universal character. In particular it is independent of the concentration of the suspension.