Theorems on the Stokesian Hydrodynamics of a Rigid Filament in the Limit of Vanishing Radius

The transport dynamics of a rigid filament in a slow viscous flow modeled by the Stokes equations in a three-dimensional domain are considered. The relation between the transport velocities, external loads, and far-field flow around the filament is studied in the singular limit as the radius of the filament tends to zero, and the filament collapses to a curve, which may be an open arc or closed loop. Beginning from an initially implicit relation defined through the solution of a boundary-value problem, a potential theoretic representation is used to show that the relation between the transport velocities, external loads, and far-field flow remains well defined and remarkably becomes explicit in the limit of vanishing radius. Theorems establishing the form of this relation in different cases are stated, proved, and illustrated with examples. Special considerations at the ends of the filament when the limiting curve is open are discussed throughout.