Bounds on the Average Velocity of a Rigid Body in a Stokes Fluid

Bounds are established for the average velocity per unit force for the motion of a rigid body in a viscous fluid modeled with the Stokes equations. The translational velocity in the direction of an imposed force is considered in a fluid that is at rest at infinity, and the average is taken over all orientations of the force, with the orientations being uniform on the unit sphere. This average velocity for an arbitrary rigid body is shown to be bounded above and below by two simple, characteristic inverse distances associated with the surface of the body. Whereas the lower bound is implied by classic comparison results, the upper bound is of a different character and does not rely on any notion of an enclosing surface as in classic results, and may be useful in making comparisons between two bodies when neither can be enclosed by the other. In contrast to previous studies based on partial differential equations for the primary field variables, the bounds are established using a boundary integral formulation and an associated constrained extremum principle for the potential density. The results derived here additionally imply a purely geometric inequality for surfaces that may be of independent interest.