The mathematical theory of viscous multipolar fluids, based on the general ideas of Green and Rivlin [8], was proposed by Necas and Silhavý [17] (see also Necas et al. [15], [16] for relevant existence theory) in order to develop a general framework for studying viscous fluids and to present a suitable alternative to the boundary layer theory (see Bellout et al. [1]). The theory is compatible with the basic principles of thermodynamics as well as with the principle of material frame indifference. The present paper is concerned with the mathematical description of the motion of one or several rigid bodies immersed in a viscous multipolar fluid. The principal and very natural idea behind the analysis presented below is the fact that the dissipation of mechanical energy, being much stronger than for classical newtonian fluids, yields better estimates on the gradient of the velocity field, in particular, the streamlines are well defined, which seems crucial for this class of problems partially formulated in terms of the Lagrangian coordinate system.