Consider a rigid body, \({\mathscr {B}}\), constrained to move by translational motion in an unbounded viscous liquid. The driving mechanism is a given distribution of time-periodic velocity field, \({\varvec{v}}_*\), at the interface body-liquid, of magnitude \(\delta \) (in appropriate function class). The main objective is to find conditions on \({\varvec{v}}_*\) ensuring that \({\mathscr {B}}\) performs a non-zero net motion, namely, \({\mathscr {B}}\) can cover any given distance in a finite time. The approach to the problem depends on whether the averaged value of \({\varvec{v}}_*\) over a period of time is (case (b)) or is not (case (a)) identically zero. In case (a) we solve the problem in a relatively straightforward way, by showing that, for small \(\delta \), it reduces to the study of a suitable and well-investigated time-independent Stokes (linear) problem. In case (b), however, the question is much more complicated, because we show that it cannot be brought to the study of a linear problem. Therefore, in case (b), self-propulsion is a genuinely nonlinear issue that we solve directly on the nonlinear system by a contradiction argument. In this way, we are able to give, also in case (b), sufficient conditions for self-propulsion (for small \(\delta \)). Finally, we demonstrate, by means of counterexamples, that such conditions are, in general, also necessary.

考虑一个刚体\（{{mathscr {B}} \），该刚体受平移运动约束在无界粘性液体中移动。驱动机制是给定的时间周期速度场\（{\ varvec {v}} _ * \）在体液界面处的大小为\（\ delta \）（在适当的函数类别中）。主要目的是在\（{\ varvec {v}} _ * \）上查找条件，以确保\（{\ mathscr {B}} \）执行非零净运动，即\（{\ mathscr { B}} \）可以在有限时间内覆盖任何给定距离。解决问题的方法取决于\（{\ varvec {v}} _ * \）的平均值在一段时间内（情况（b））或不是（情况（a））等于零。在情况（a）中，我们通过显示出，对于较小的\（\ delta \），可以用一种相对简单的方式解决该问题，从而减少了对合适的且经过充分研究的与时间无关的Stokes（linear）问题的研究。但是，在情况（b）中，这个问题要复杂得多，因为我们表明不能将其带入线性问题的研究。因此，在情况（b）中，自推进是一个真正的非线性问题，我们可以通过矛盾论证直接在非线性系统上解决。这样，在情况（b）中，我们也能够提供足够的自我推进条件（对于较小的\（\ delta \））。最后，我们通过反例证明，一般来说，这种条件也是必要的。