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Consider a rigid body, $\mathscr B$, constrained to move by translationalmotion in an unbounded viscous liquid. The driving mechanism is a givendistribution of time-periodic velocity field, $\bfv_*$, at the interfacebody-liquid, of magnitude $\delta$ (in appropriate function class). The mainobjective is to find conditions on $\bfv_*$ ensuring that $\mathscr B$ performsa non-zero net motion, namely, $\mathscr B$ can cover any given distance in afinite time. The approach to the problem depends on whether the averaged valueof $\bfv_*$ over a period of time is (case (b)) or is not (case (a))identically zero. In case (a) we solve the problem in a relativelystraightforward way, by showing that, for small $\delta$, it reduces to thestudy of a suitable amd well-investigated time-dependent Stokes (linear)problem. In case (b), however, the question is much more complicated, becausewe show that it {\em cannot} be brought to the study of a linear problem.Therefore, in case (b), self-propulsion is a genuinely nonlinear issue that wesolve directly on the nonlinear system by a contradiction argument approach. Inthis way, we are able to give, also in case (b), sufficient conditions forself-propulsion (for small $\delta$). Finally, we demonstrate, by means ofcounterexamples, that such conditions are, in general, also necessary.

On the Self-Propulsion of a Rigid Body in a Viscous Liquid by Time-Periodic Boundary Data